arxiv:1105.0823v4 [math.lo] 21 feb 2013 · arxiv:1105.0823v4 [math.lo] 21 feb 2013 borel conjecture...

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BOREL CONJECTURE AND DUAL BOREL CONJECTURE

MARTIN GOLDSTERN, JAKOB KELLNER, SAHARON SHELAH, AND WOLFGANG WOHOFSKY

Dedicated to the memory of Richard Laver

Abstract. We show that it is consistent that the Borel Conjecture and the dual Borel Conjecture hold simulta-neously.

Introduction

History. A set X of reals1 is called “strong measure zero” (smz), if for all functionsf : ω → ω there areintervalsIn of measure≤ 1/ f (n) coveringX. Obviously, a smz set is a null set (i.e., has Lebesgue measurezero), and it is easy to see that the family of smz sets forms aσ-ideal and that perfect sets (and thereforeuncountable Borel or analytic sets) are not smz.

At the beginning of the 20th century, Borel [Bor19, p. 123] conjectured:

Every smz set is countable.

This statement is known as the “Borel Conjecture” (BC). In the 1970s it was proved that BC isindependent,i.e., neither provable nor refutable.

Let us very briefly comment on the notion of independence: A sentenceϕ is called independent of a setT of axioms, if neitherϕ nor¬ϕ follows fromT. (As a trivial example, (∀x)(∀y)x · y = y · x is independentfrom the group axioms.) The set theoretic (first order) axiomsystem ZFC (Zermelo Fraenkel with the axiomof choice) is considered to be the standard axiomatization of all of mathematics: A mathematical proof isgenerally accepted as valid iff it can be formalized in ZFC. Therefore we just say “ϕ is independent” ifϕis independent of ZFC. Several mathematical statements areindependent, the earliest and most prominentexample is Hilbert’s first problem, the Continuum Hypothesis (CH).

BC is independent as well: Sierpinski [Sie28] showed that CH implies¬BC (and, since Godel showedthe consistency of CH, this gives us the consistency of¬BC). Using the method of forcing, Laver [Lav76]showed that BC is consistent.

Galvin, Mycielski and Solovay [GMS73] proved the followingconjecture of Prikry:

X ⊆ 2ω is smz if and only if every comeager (denseGδ) set contains a translate ofX.

Prikry also defined the following dual notion:

X ⊆ 2ω is called “strongly meager” (sm) if every set of Lebesgue measure 1 containsa translate ofX.

The dual Borel Conjecture (dBC) states:

Every sm set is countable.

Prikry noted that CH implies¬dBC and conjectured dBC to be consistent (and therefore independent),which was later proved by Carlson [Car93].

Date: 2011-12-27.2000Mathematics Subject Classification.Primary 03E35; secondary 03E17, 28E15.We gratefully acknowledge the following partial support: US National Science Foundation Grant No. 0600940 (all authors);

US-Israel Binational Science Foundation grant 2006108 (third author); Austrian Science Fund (FWF): P21651-N13 and P23875-N13and EU FP7 Marie Curie grant PERG02-GA-2207-224747 (secondand fourth author); FWF grant P21968 (first and fourth author);OAW Doc fellowship (fourth author). This is publication 969of the third author.We are grateful to the anonymous referee for pointing out several unclarities in the original version.

1In this paper, we use 2ω as the set of reals. (ω = {0,1, 2, . . .}.) By well-known results both the definition and the theorem alsowork for the unit interval [0, 1] or the torusR/Z. Occasionally we also write “x is a real” for “x ∈ ωω”.

1

http://arxiv.org/abs/1105.0823v4

2 MARTIN GOLDSTERN, JAKOB KELLNER, SAHARON SHELAH, AND WOLFGANG WOHOFSKY

Numerous additional results regarding BC and dBC have been proved: The consistency of variants ofBC or of dBC, the consistency of BC or dBC together with certain assumptions on cardinal characteristics,etc. See [BJ95, Ch. 8] for several of these results. In this paper, we prove the consistency (and thereforeindependence) of BC+dBC (i.e., consistently BC and dBC hold simultaneously).

The problem. The obvious first attempt to force BC+dBC is to somehow combine Laver’s and Carlson’sconstructions. However, there are strong obstacles:

Laver’s construction is a countable support iteration of Laver forcing. The crucial points are:

• Adding a Laver real makes every old uncountable setX non-smz.• And this setX remains non-smz after another forcingP, provided thatP has the “Laver property”.

So we can start with CH and use a countable support iteration of Laver forcing of lengthω2. In the finalmodel, every setX of reals of sizeℵ1 already appeared at some stageα < ω2 of the iteration; the next Laverreal makesX non-smz, and the rest of the iteration (as it is a countable support iteration of proper forcingswith the Laver property) has the Laver property, and thereforeX is still non-smz in the final model.

Carlson’s construction on the other hand addsω2 many Cohen reals in a finite support iteration (orequivalently: finite support product). The crucial points are:

• A Cohen real makes every old uncountable setX non-sm.• And this setX remains non-sm after another forcingP, provided thatP has precaliberℵ1.

So we can start with CH, and use more or less the same argument as above: Assume thatX appears atα < ω2. Then the next Cohen makesX non-sm. It is enough to show thatX remains non-sm at allsubsequent stagesβ < ω2. This is guaranteed by the fact that a finite support iteration of Cohen reals oflength< ω2 has precaliberℵ1.

So it is unclear how to combine the two proofs: A Cohen real makes all old sets smz, and it is easyto see that whenever we add Cohen reals cofinally often in an iteration of length, say,ω2, all sets of anyintermediate extension will be smz, thus violating BC. So wehave to avoid Cohen reals,2 which alsoimplies that we cannot use finite support limits in our iterations. So we have a problem even if we find areplacement for Cohen forcing in Carlson’s proof that makesall old uncountable setsX non-sm and thatdoes not add Cohen reals: Since we cannot use finite support, it seems hopeless to get precaliberℵ1, anessential requirement to keepX non-sm.

Note that it is theproofsof BC and dBC that are seemingly irreconcilable; this is not clear for themodels. Of course Carlson’s model, i.e., the Cohen model, cannot satisfy BC, but it is not clear whethermaybe already the Laver model could satisfy dBC. (It is even still open whether a single Laver forcingmakes every old uncountable set non-sm.) Actually, Bartoszynski and Shelah [BS03] proved that the Lavermodel does satisfy the following weaker variant of dBC (notethat the continuum has sizeℵ2 in the Lavermodel):

Every sm set has size less than the continuum.

In any case, it turns out that onecanreconcile Laver’s and Carlson’s proof, by “mixing” them “generi-cally”, resulting in the following theorem:

Theorem. If ZFC is consistent, then ZFC+BC+dBC is consistent.

Prerequisites. To understand anything of this paper, the reader

• should have some experience with finite and countable support iteration, proper forcing,ℵ2-cc,σ-closed, etc.,• should know what a quotient forcing is,• should have seen some preservation theorem for proper countable support iteration,• should have seen some tree forcings (such as Laver forcing).

To understand everything, additionally the following is required:

• The “case A” preservation theorem from [She98], more specifically we build on the proof of [Gol93](or [GK06]).• In particular, some familiarity with the property “preservation of randoms” is recommended. We

will use the fact that random and Laver forcing have this property.

2An iteration that forces dBC without adding Cohen reals was given in [BS10], using non-Cohen oracle-cc.

BOREL CONJECTURE AND DUAL BOREL CONJECTURE 3

• We make some claims about (a rather special case of) ord-transitive models in Section 3.A. Thereaders can either believe these claims, or check them themselves (by some rather straightforwardproofs), or look up the proofs (of more general settings) in [She04] or [Kelar].

From the theory of strong measure zero and strongly meager, we only need the following two results(which are essential for our proofs of BC and dBC, respectively):

• Pawlikowski’s result from [Paw96a] (which we quote as Theorem 0.2 below), and• Theorem 8 of Bartoszynski and Shelah’s [BS10] (which we quote as Lemma 2.1).

We do not need any other results of Bartoszynski and Shelah’s paper [BS10]; in particular we do not use thenotion of non-Cohen oracle-cc (introduced in [She06]); andthe reader does not have to know the originalproofs of Con(BC) and Con(dBC), by Laver and Carlson, respectively.

The third author claims that our construction is more or lessthe same as a non-Cohen oracle-cc con-struction, and that the extended version presented in [She10] is even closer to our preparatory forcing.

Notation and some basic facts on forcing, strongly meager (sm) and strong measure zero (smz) sets.We call a lemma “Fact” if we think that no proof is necessary — either because it is trivial, or because it iswell known (even without a reference), or because we give an explicit reference to the literature.

Stronger conditions in forcing notions are smaller, i.e.,q ≤ p means thatq is stronger thanp.Let P ⊆ Qbe forcing notions. (As usual, we abuse notation by not distinguishing between the underlying

set and the quasiorder on it.)

• For p1, p2 ∈ P we write p1 ⊥P p2 for “ p1 andp2 are incompatible”. Otherwise we writep1 ‖P p2.(We may just write⊥ or ‖ if P is understood.)• q ≤∗ p (or: q ≤∗P p) means thatq forces thatp is in the generic filter, or equivalently that every

q′ ≤ q is compatible withp. And q =∗ p meansq ≤∗ p ∧ p ≤∗ q.• P is separative, if≤ is the same as≤∗, or equivalently, if for allq ≤ p with q , p there is anr ≤ p

incompatible withq. Given anyP, we can define its “separative quotient”Q by first replacing (inP) ≤ by ≤∗ and then identifying elementsp, q wheneverp =∗ q. ThenQ is separative and forcingequivalent toP.• “P is a subforcing of Q” means that the relation≤P is the restriction of≤Q to P.• “P is an incompatibility-preserving subforcing of Q” means thatP is a subforcing ofQ and that

p1 ⊥P p2 iff p1 ⊥Q p2 for all p1, p2 ∈ P.

Let additionallyM be a countable transitive3 model (of a sufficiently large subset of ZFC) containingP.

• “P is anM-complete subforcing ofQ” (or: P⋖M Q) means thatP is a subforcing ofQ and: ifA ⊆ Pis in M a maximal antichain, then it is a maximal antichain ofQ as well. (Or equivalently:P isan incompatibility-preserving subforcing ofQ and every predense subset ofP in M is predensein Q.) Note that this means that everyQ-generic filterG overV induces aP-generic filter overM,namelyGM

≔ G ∩ P (i.e., every maximal antichain ofP in M meetsG ∩ P in exactly one point).In particular, we can interpret aP-nameτ in M as aQ-name. More exactly, there is aQ-nameτ′

such thatτ′[G] = τ[GM] for all Q-generic filtersG. We will usually just identifyτ andτ′.• Analogously, ifP ∈ M andi : P→ Q is a function, theni is called anM-complete embedding if it

preserves≤ (or at least≤∗) and⊥ and moreover: IfA ∈ M is predense inP, theni[A] is predensein Q.

There are several possible characterizations of sm (“strongly meager”) and smz (“strong measure zero”)sets; we will use the following as definitions:

A setX is not sm if there is a measure 1 set into whichX cannot be translated; i.e., if there is a null setZ such that (X + t) ∩ Z , ∅ for all realst, or, in other words,Z + X = 2ω. To summarize:

(0.1) X is notsm iff there is a Lebesgue null setZ such thatZ + X = 2ω.

We will call such aZ a “witness” for the fact thatX is not sm (or say thatZ witnesses thatX is not sm).The following theorem of Pawlikowski [Paw96a] is central for our proof4 that BC holds in our model:

3We will also use so-called ord-transitive models, as definedin Section 3.A.4We thank Tomek Bartoszynski for pointing out Pawlikowski’s result to us, and for suggesting that it might be useful for our

proof.


Theorem 0.2. X ⊆ 2ω is smz iff X + F is null for every closed null set F.Moreover, for every dense Gδ set H we canconstruct(in an absolute way) a closed null set F such that forevery X⊆ 2ω with X+ F null there is t∈ 2ω with t+ X ⊆ H.

In particular, we get:

(0.3) X is notsmz iff there is a closed null setF such thatX+F has positive outer Lebesguemeasure.

Again, we will say that the closed null setF “witnesses” thatX is not smz (or callF a witness for thisfact).

Annotated contents.

Section 1, p. 4: We introduce the family of ultralaver forcing notions and prove some properties.Section 2, p. 15: We introduce the family of Janus forcing notions and prove some properties.Section 3, p. 21: We define ord-transitive models and mentionsome basic properties. We define the

“almost finite” and “almost countable” support iteration over a model. We show that in manyrespects they behave like finite and countable support, respectively.

Section 4, p. 33: We introduce the preparatory forcing notionRwhich adds a generic forcing iterationP.Section 5, p. 41: Putting everything together, we show thatR∗Pω2 forces BC+dBC, i.e., that an uncount-

ableX is neither smz nor sm. We show this under the assumptionX ∈ V, and then introduce afactorization ofR ∗ P that this assumption does not result in loss of generality.

Section 6, p. 45: We briefly comment on alternative ways some notions could be defined.

An informal overview of the proof, including two illustrations, can be found athttp://arxiv.org/abs/1112.4424/.

1. Ultralaver forcing

In this section, we define the family ofultralaver forcingsLD, variants of Laver forcing which dependon a systemD of ultrafilters.

In the rest of the paper, we will use the following propertiesof LD. (And we will useonly theseproperties. So readers who are willing to take these properties for granted could skip to Section 2.)

(1) LD is σ-centered, hence ccc.(This is Lemma 1.2.)

(2) LD is separative.(This is Lemma 1.3.)

(3) Ultralaver kills smz: There is a canonicalLD-name¯˜ℓ for a fast growing real inωω called the

ultralaver real. From this real, we can define (in an absoluteway) a closed null setF such thatX+F is positive for all uncountableX in V (and thereforeF witnesses thatX is not smz, accordingto Theorem 0.2).(This is Corollary 1.21.)

(4) WheneverX is uncountable, thenLD forces thatX is not “thin”.(This is Corollary 1.24.)

(5) If (M, ∈) is a countable model of ZFC∗ and if LDM is an ultralaver forcing inM, then for anyultrafilter systemD extendingDM, LDM is anM-complete subforcing of the ultralaver forcingLD.(This is Lemma 1.5.)Moreover, the real

˜ℓ of item (3) is so “canonical” that we get: If (inM) ¯

˜ℓM is theLDM -name for

theLDM -generic real, and if (inV) ¯˜ℓ is theLD-name for theLD-generic real, and ifH isLD-generic

overV and thusHM≔ H ∩ LDM is the inducedLDM -generic filter overM, then ¯

˜ℓ[H] is equal to

¯˜ℓM[HM].Since the closed null setF is constructed from

˜ℓ in an absolute way, the same holds forF, i.e., the

Borel codesF[H] andF[HM] are the same.(6) Moreover, givenM andLDM as above, and a random realr over M, we can chooseD extending

DM such thatLD forces that randomness ofr is preserved (in a strong way that can be preserved ina countable support iteration).(This is Lemma 1.30.)

http://arxiv.org/abs/1112.4424/


1.A. Definition of ultralaver.

Notation. We use the following fairly standard notation:A treeis a nonempty setp ⊆ ω<ω which is closed under initial segments and has no maximal elements.5

The elements (“nodes”) of a tree are partially ordered by⊆.For each sequences ∈ ω<ω we write lh(s) for the length ofs.For any treep ⊆ ω<ω and anys ∈ p we write succp(s) for one of the following two sets:

{k ∈ ω : s⌢k ∈ p} or {t ∈ p : (∃k ∈ ω) t = s⌢k}

and we rely on the context to help the reader decide which set we mean.A branchof p is either of the following:

• A function f : ω→ ω with f↾n ∈ p for all n ∈ ω.• A maximal chain in the partial order (p,⊆). (As our trees do not have maximal elements, each

such chainC determines a branch⋃

C in the first sense, and conversely.)

We write [p] for the set of all branches ofp.For any treep ⊆ ω<ω and anys ∈ p we write p[s] for the set{t ∈ p : t ⊇ sor t ⊆ s}, and we write [s] for

either of the following sets:{t ∈ p : s⊆ t} or {x ∈ [p] : s⊆ x}.

The stem of a treep is the shortests ∈ p with | succp(s)| > 1. (The trees we consider will never bebranches, i.e., will always have finite stems.)

Definition 1.1. • For treesq, p we writeq ≤ p if q ⊆ p (“q is stronger thanp”), and we say that “qis a pure extension of p” (q ≤0 p) if q ≤ p and stem(q) = stem(p).• A filter systemD is a family (Ds)s∈ω<ω of filters onω. (All our filters will contain the Frechet filter

of cofinite sets.) We writeD+s for the collection ofDs-positive sets (i.e., sets whose complement isnot in Ds).• We defineLD to be the set of all treesp such that succp(t) ∈ D+t for all t ∈ p above the stem.• The generic filter is determined by the generic branchℓ = (ℓi)i∈ω ∈ ω

ω, called thegeneric real:{ℓ} =

⋂

p∈G[p] or equivalently,ℓ =⋃

p∈G stem(p).• An ultrafilter system is a filter system consisting of ultrafilters. (Since all our filters contain the

Frechet filter, we only consider nonprincipal ultrafilters.)• An ultralaver forcing is a forcingLD defined from an ultrafilter system. The generic real for an

ultralaver forcing is also called theultralaver real.

Recall that a forcing notion (P,≤) is σ-centeredif P =⋃

n Pn, where for alln, k ∈ ω and for allp1, . . . , pk ∈ Pn there isq ≤ p1, . . . , pk.

Lemma 1.2. All ultralaver forcingsLD areσ-centered (hence ccc).

Proof. Every finite set of conditions sharing the same stem has a common lower bound. �

Lemma 1.3. LD is separative.6

Proof. If q ≤ p, andq , p, then there iss ∈ p \ q. Now p[s] ⊥ q. �

If eachDs is the Frechet filter, thenLD is Laver forcing (often just writtenL).

1.B. M-complete embeddings.Note that for all ultrafilter systemsD we have:

(1.4)Two conditions inLD are compatible if and only if their stems are comparable andmoreover, the longer stem is an element of the condition withthe shorter stem.

Lemma 1.5. Let M be countable.7 In M, let LDM be an ultralaver forcing. LetD be (in V) a filter systemextending8 DM. ThenLDM is an M-complete subforcing ofLD.

5Except for the proof of Lemma 1.5, where we also allow trees with maximal elements, and even empty trees.6See page 3 for the definition.7Here, we can assume thatM is a countable transitive model of a sufficiently large finite subset ZFC∗ of ZFC. Later, we will also

use ord-transitive models instead of transitive ones, which does not make any difference as far as properties ofLD are concerned, asour arguments take place in transitive parts of such models.

8I.e.,DMs ⊆ Ds for all s ∈ ω<ω.


Proof. For any tree9 T, any filter systemE = (Es)s∈ω<ω , and anys0 ∈ T we define a sequence (TαE,s0

)α∈ω1 of“derivatives” (where we may abbreviateTα

E,s0to Tα) as follows:

• T0≔ T [s0].

• GivenTα, we letTα+1≔ Tα \

⋃

{[s] : s ∈ Tα, s0 ⊆ s, succTα (s) < E+s }, where [s] ≔ {t : s⊆ t}.• For limit ordinalsδ > 0 we letTδ ≔

⋂

α<δ Tα.

Then we have

(a) EachTα is closed under initial segments. Also:α < β impliesTα ⊇ Tβ.(b) There is anα0 < ω1 such thatTα0 = Tα0+1 = Tβ for all β > α0. We writeT∞ or T∞

E,s0for Tα0.

(c) If s0 ∈ T∞E,s0

, thenT∞E,s0∈ LE with stems0.

Conversely, if stem(T) = s0, andT ∈ LE, thenT∞ = T.(d) If T contains a treeq ∈ LE with stem(q) = s0, thenT∞ containsq∞ = q, so in particulars0 ∈ T∞.(e) Thus:T contains a condition inLE with stems0 iff s0 ∈ T∞

E,s0.

(f) The computation ofT∞ is absolute between any two models containingT andE. (In particular,any transitive ZFC∗-model containingT andE will also containα0.)

(g) Moreover: LetT ∈ M, E ∈ M, and letE′ be a filter system extendingE such that for alls0 andall A ∈ P(ω) ∩ M we have:A ∈ (Es0)

+ iff A ∈ (E′s0)+. (In particular, this will be true for anyE′

extendingE, provided that eachEs0 is anM-ultrafilter.)Then for eachα ∈ M we haveTα

E,s0= Tα

E′ ,s0(and henceTα

E′ ,s0∈ M). (Proved by induction onα.)

Now let A = (pi : i ∈ I ) ∈ M be a maximal antichain inLDM , and assume (inV) that q ∈ LD. Lets0 ≔ stem(q).

We will show thatq is compatible with somepi (in LD). This is clear if there is somei with s0 ∈ pi andstem(pi) ⊆ s0, by (1.4). (In this case,pi ∩ q is a condition inLD with stems0.)

So for the rest of the proof we assume that this is not the case,i.e.:

(1.6) There is noi with s0 ∈ pi and stem(pi) ⊆ s0.

Let J ≔ {i ∈ I : s0 ⊆ stem(pi)}. We claim that there isj ∈ J with stem(p j) ∈ q (which as above impliesthatq andp j are compatible).

Assume towards a contradiction that this is not the case. Then q is contained in the following treeT:

T ≔ (ω<ω)[s0] \⋃

j∈J

[stem(p j)].(1.7)

Note thatT ∈ M. In V we have:

(1.8) The treeT contains a conditionq with stems0.

So by (e) (applied inV), followed by (g), and again by (e) (now inM) we get:

(1.9) The treeT also contains a conditionp ∈ M with stems0.

Now p has to be compatible with somepi . The sequencess0 = stem(p) and stem(pi) have to be comparable,so by (1.4) there are two possibilities:

(1) stem(pi) ⊆ stem(p) = s0 ∈ pi . We have excluded this case in our assumption (1.6).(2) s0 = stem(p) ⊆ stem(pi) ∈ p. Soi ∈ J. By construction ofT (see (1.7)), we conclude stem(pi) < T,

contradicting stem(pi) ∈ p ⊆ T (see 1.9). �

1.C. Ultralaver kills strong measure zero. The following lemma appears already in [Bla88, Theorem 9].We will give a proof below in Lemma 1.35.

Lemma 1.10. If A is a finite set,˜α anLD-name, p∈ LD, and p

˜α ∈ A, then there isβ ∈ A and a pure

extension q≤0 p such that q ˜α = β.

Definition 1.11. Let ℓ be an increasing sequence of natural numbers. We say thatX ⊆ 2ω is smz withrespect toℓ, if there exists a sequence (Ik)k∈ω of basic intervals of 2ω of measure≤ 2−ℓk (i.e., eachIk is ofthe form [sk] for somesk ∈ 2ℓk) such thatX ⊆

⋂

m∈ω⋃

k≥m Ik.

Remark 1.12. It is well known and easy to see that the properties

9Here we also allow empty trees, and trees with maximal nodes.


• For all ℓ there exists exists a sequence (Ik)k∈ω of basic intervals of 2ω of measure≤ 2−ℓk such thatX ⊆⋃

k∈ω Ik.• For all ℓ there exists exists a sequence (Ik)k∈ω of basic intervals of 2ω of measure≤ 2−ℓk such that

X ⊆⋂

m∈ω⋃

k≥m Ik.

are equivalent. Hence, a setX is smz iff X is smz with respect to allℓ ∈ ωω.

The following lemma is a variant of the corresponding lemma (and proof) for Laver forcing (see forexample [Jec03, Lemma 28.20]): Ultralaver makes old uncountable sets non-smz.

Lemma 1.13. Let D be a system of ultrafilters, and let¯˜ℓ be theLD-name for the ultralaver real. Then each

uncountable set X∈ V is forced to be non-smz (witnessed by the ultralaver real¯˜ℓ).

More precisely, the following holds:

(1.14) LD∀X ∈ V ∩ [2ω]ℵ1 ∀(xk)k∈ω ⊆ 2ω X *

⋂

m∈ω

⋃

k≥m

[xk↾˜ℓk].

We first give two technical lemmas:

Lemma 1.15. Let p∈ LD with stem s∈ ω<ω, and let˜x be aLD-name for a real in2ω. Then there exists a

pure extension q≤0 p and a realτ ∈ 2ω such that for every n∈ ω,

(1.16) {i ∈ succq(s) : q[s⌢i] ˜x↾n = τ↾n} ∈ Ds.

Proof. For eachi ∈ succp(s), let qi ≤0 p[s⌢ i] be such thatqi decides˜x↾i, i.e., there is ati of lengthi such

thatqi ˜x↾i = ti (this is possible by Lemma 1.10).

Now we define the realτ ∈ 2ω as theDs-limit of the ti ’s. In more detail: For eachn ∈ ω there is a(unique)τn ∈ 2n such that{i : ti↾n = τn} ∈ Ds; sinceDs is a filter, there is a realτ ∈ 2ω with τ↾n = τn foreachn. Finally, letq≔

⋃

i qi . �

Lemma 1.17. Let p ∈ LD with stem s, and let(˜xk)k∈ω be a sequence ofLD-names for reals in2ω. Then

there exists a pure extension q≤0 p and a family of reals(τη)η∈q, η⊇s ⊆ 2ω such that for eachη ∈ q above s,and every n∈ ω,

(1.18) {i ∈ succq(η) : q[η⌢i] ˜x|η|↾n = τη↾n} ∈ Dη.

Proof. We apply Lemma 1.15 to each nodeη in p aboves (and to˜x|η|) separately: We first get ap1 ≤0 p

and aτs ∈ 2ω; for every immediate successorη ∈ succp1(s), we getqη ≤0 p[η]1 and aτη ∈ 2ω, and let

p2 ≔⋃

η qη; in this way, we get a (fusion) sequence (p, p1, p2, . . .), and letq≔⋂

k pk. �

Proof of Lemma 1.13.We want to prove (1.14). Assume towards a contradiction thatX is an uncountableset inV, and that (

˜xk)k∈ω is a sequence of names for reals in 2ω andp ∈ LD such that

(1.19) p X ⊆⋂

m∈ω

⋃

k≥m

[˜xk↾

˜ℓk].

Let s ∈ ω<ω be the stem ofp.By Lemma 1.17, we can fix a pure extensionq ≤0 p and a family (τη)η∈q, η⊇s ⊆ 2ω such that for each

η ∈ q above the stems and everyn ∈ ω, condition (1.18) holds.SinceX is (in V and) uncountable, we can find a realx∗ ∈ X which is different from each real in the

countable family (τη)η∈q, η⊇s; more specifically, we can pick a family of natural numbers (nη)η∈q, η⊇s suchthatx∗↾nη , τη↾nη for anyη.

We can now findr ≤0 q such that:

• For allη ∈ r abovesand alli ∈ succr (η) we havei > nη.• For allη ∈ r abovesand alli ∈ succr (η) we haver [η⌢ i]

˜x|η|↾nη = τη↾nη , x∗↾nη.

So for allη ∈ r aboveswe have, writingk for |η|, thatr [η⌢ i] forcesx∗ < [˜xk↾nη] ⊇ [

˜xk↾ℓk]. We conclude

thatr forcesx∗ <⋃

k≥|s|[˜xk↾ℓk], contradicting (1.19). �


Corollary 1.20. Let (tk)k∈ω be a dense subset of2ω.Let D be a system of ultrafilters, and let

˜ℓ be theLD-name for the ultralaver real. Then the set

˜H ≔

⋂

m∈ω

⋃

k≥m

[tk↾˜ℓk]

is forced to be a comeager set with the property that˜H does not contain any translate of any old uncount-

able set.

Pawlikowski’s theorem 0.2 gives us:

Corollary 1.21. There is a canonical name F for a closed null set such that X+ F is positive for alluncountable X in V.

In particular, no uncountable ground model set is smz in the ultralaver extension.

1.D. Thin sets and strong measure zero.For the notion of “(very) thin” set, we use an increasing func-tion B∗(k) (the function we use will be described in Corollary 2.2). Wewill assume thatℓ∗ = (ℓ∗k)k∈ω isan increasing sequence of natural numbers withℓ∗k+1 ≫ B∗(k). (We will later use a subsequence of theultralaver realℓ asℓ∗, see Lemma 1.23).

Definition 1.22. For X ⊆ 2ω andk ∈ ω we writeX↾[ℓ∗k, ℓ∗k+1) for the set{x↾[ℓ∗k, ℓ

∗k+1) : x ∈ X}. We say that

• X ⊆ 2ω is “very thin with respect toℓ∗ and B∗”, if there are infinitely manyk with |X↾[ℓ∗k, ℓ∗k+1)| ≤

B∗(k).• X ⊆ 2ω is “thin with respect toℓ∗ and B∗”, if X is the union of countably many very thin sets.

Note that the family of thin sets is aσ-ideal, while the family of very thin sets is not even an ideal. Also,every very thin set is covered by a closed very thin (in particular nowhere dense) set. In particular, everythin set is meager and the ideal of thin sets is a proper ideal.

Lemma 1.23. Let B∗ be an increasing function. Letℓ be an increasing sequence of natural numbers. Wedefine a subsequenceℓ∗ of ℓ in the following way:ℓ∗k = ℓnk where nk+1 − nk = B∗(k) · 2ℓ

∗k .

Then we get: If X is thin with respect toℓ∗ and B∗, then X is smz with respect toℓ.

Proof. Assume thatX =⋃

i∈ω Yi , eachYi very thin with respect toℓ∗ andB∗. Let (X j) j∈ω be an enumerationof {Yi : i ∈ ω} where eachYi appears infinitely often. SoX ⊆

⋂

m∈ω⋃

j≥m X j .By induction onj ∈ ω, we find for all j > 0 somek j > k j−1 such that

|X j↾[ℓ∗k j, ℓ∗k j+1)| ≤ B∗(k j) hence |X j↾[0, ℓ

∗k j+1)| ≤ B∗(k j) · 2

ℓ∗kj = nk j+1 − nk j .

So we can enumerateX j↾[0, ℓ∗k j+1) as (si)nkj≤i<nkj+1. HenceX j is a subset of⋃

nkj≤i<nkj+1[si ]; and eachsi has

lengthℓ∗k j+1 ≥ ℓi , sinceℓ∗k j+1 = ℓnkj+1 andi < nk j+1. This implies

X ⊆⋂

m∈ω

⋃

j≥m

X j ⊆⋂

m∈ω

⋃

i≥m

[si ].

HenceX is smz with respect toℓ. �

Lemma 1.13 and Lemma 1.23 yield:

Corollary 1.24. Let B∗ be an increasing function. LetD be a system of ultrafilters, and˜ℓ the name for the

ultralaver real. Let˜ℓ∗ be constructed from B∗ and

˜ℓ as in Lemma 1.23.

ThenLD forces that for every uncountable X⊆ 2ω:

• X is not smz with respect to˜ℓ.

• X is not thin with respect to˜ℓ∗ and B∗.

1.E. Ultralaver and preservation of Lebesgue positivity. It is well known that both Laver forcing andrandom forcing preserve Lebesgue positivity; in fact they satisfy a stronger property that is preservedunder countable support iterations. (So in particular, a countable support iteration of Laver and randomalso preserves positivity.)

Ultralaver forcingLD will in general not preserve positivity. Indeed, if all ultrafiltersDs are equal to thesame ultrafilterD∗, then the rangeL ≔ {ℓ0, ℓ1, . . .} ⊆ ω of the ultralaver realℓ will diagonalizeD∗, so everyground model realx ∈ 2ω (viewed as a subset ofω) will either almost containL or be almost disjoint toL,


which implies that the set 2ω ∩ V of old reals is covered by a null set in the extension. However, later inthis paper it will become clear that if we choose the ultrafiltersDs in a sufficiently generic way, then manyold positive sets will stay positive. More specifically, in this section we will show (Lemma 1.30): IfDM isan ultrafilter system in a countable modelM andr a random real overM, then we can find an extensionDsuch thatLD forces thatr remains random overM[HM] (whereHM denotes theLD-name for the restrictionof theLD-generic filterH to LDM ∩M). Additionally, some “side conditions” are met, which are necessaryto preserve the property in forcing iterations.

In Section 3.D we will see how to use this property to preserverandoms in limits.The setup we use for preservation of randomness is basicallythe notation of “Case A” preservation

introduced in [She98, Ch.XVIII], see also [Gol93, GK06] or the textbook [BJ95, 6.1.B]:

Definition 1.25. We writeclopen for the collection of clopen sets on 2ω. We say that the functionZ : ω→clopen is a code for a null set, if the measure ofZ(n) is at most 2−n for eachn ∈ ω.

For such a codeZ, the set nullset(Z) coded byZ is

nullset(Z) ≔⋂

n

⋃

k≥n

Z(k).

The set nullset(Z) obviously is a null set, and it is well known that every null set is contained in such aset nullset(Z).

Definition 1.26. For a realr and any codeZ, we defineZ ⊏n r by:

(∀k ≥ n) r < Z(k).

We writeZ ⊏ r if Z ⊏n r holds for somen; i.e., if r < nullset(Z).

For later reference, we record the following trivial fact:

(1.27) p ˜Z ⊏ r iff there is a name

˜n for an element ofω such thatp

˜Z ⊏

˜n r.

Let P be a forcing notion, and˜Z a P-name of a code for a null set. An interpretation of

˜Z below p is

some codeZ∗ such that there is a sequencep = p0 ≥ p1 ≥ p2 ≥ . . . such thatpm forces˜Z↾m = Z∗↾m.

Usually we demand (which allows a simpler proof of the preservation theorem at limit stages) that thesequence (p0, p1, . . . ) is inconsistent, i.e.,p forces that there is anmsuch thatpm < G. Note that wheneverP adds a newω-sequence of ordinals, we can find such an interpretation forany

˜Z.

If˜Z = (

˜Z1, . . . ,

˜Zm) is a tuple of names of codes for null sets, then an interpretation of ¯

˜Z belowp is some

tuple (Z∗1, . . . ,Z∗m) such that there is a single sequencep = p0 ≥ p1 ≥ p2 ≥ . . . interpreting each

˜Zi asZ∗i .

We now turn to preservation of Lebesgue positivity:

Definition 1.28. (1) A forcing notionP preserves Borel outer measure, if P forces Leb∗(AV) = Leb(AV[GP])for every codeA for a Borel set. (Leb∗ denotes the outer Lebesgue measure, and for a Borel codeA and a set-theoretic universeV, AV denotes the Borel set coded byA in V.)

(2) P strongly preserves randoms, if the following holds: LetN ≺ H(χ∗) be countable for a sufficientlylarge regular cardinalχ∗, let P, p, ¯

˜Z = (

˜Z1, . . . ,

˜Zm) ∈ N, let p ∈ P and letr be random overN.

Assume that inN, Z∗ is an interpretation of˜Z, and assumeZ∗i ⊏ki r for eachi. Then there is an

N-genericq ≤ p forcing thatr is still random overN[G] and moreover,˜Zi ⊏ki r for eachi. (In

particular,P has to be proper.)(3) Assume thatP is absolutely definable.P strongly preserves randoms over countable modelsif (2)

holds for all countable (transitive10) modelsN of ZFC∗.

It is easy to see that these properties are increasing in strength. (Of course (3)⇒(2) works only if ZFC∗

is satisfied inH(χ∗).)In [KS05] it is shown that (1) implies (3), provided thatP is nep (“non-elementary proper”, i.e., nicely

definable and proper with respect to countable models). In particular, every Suslin ccc forcing notion suchas random forcing, and also many tree forcing notions including Laver forcing, are nep. HoweverLD is notnicely definable in this sense, as its definition uses ultrafilters as parameters.

10Later we will introduce ord-transitive models, and it is easy to see that it does not make any difference whether we demandtransitive or not; this can be seen using a transitive collapse.


Lemma 1.29. Both Laver forcing and random forcing strongly preserve randoms over countable models.

Proof. For random forcing, this is easy and well known (see, e.g., [BJ95, 6.3.12]).For Laver forcing: By the above, it is enough to show (1). Thiswas done by Woodin (unpublished) and

Judah-Shelah [JS90]. A nicer proof (including a variant of (2)) is given by Pawlikowski [Paw96b]. �

Ultralaver will generally not preserve Lebesgue positivity, let alone randomness. However, we getthe following “local” variant of strong preservation of randoms (which will be used in the preservationtheorem 3.33). The rest of this section will be devoted to theproof of the following lemma.

Lemma 1.30. Assume that M is a countable model,DM an ultrafilter system in M and r a random realover M. Then there is (in V) an ultrafilter systemD extending11 DM, such that the following holds:If

• p ∈ LDM ,• in M,

˜Z = (

˜Z1, . . . ,

˜Zm) is a sequence ofLDM -names for codes for null sets,12 and Z∗1, . . . ,Z

∗m are

interpretations under p, witnessed by a sequence(pn)n∈ω with strictly increasing13 stems,• Z∗i ⊏ki r for i = 1, . . . ,m,

then there is a q≤ p in LD forcing that

• r is random over M[GM],•

˜Zi ⊏ki r for i = 1, . . . ,m.

For the proof of this lemma, we will use the following concepts:

Definition 1.31. Let p ⊆ ω<ω be a tree. A “front name below p” is a function14 h : F → clopen, whereF ⊆ p is a front (a set that meets every branch ofp in a unique point). (For notational simplicity we alsoallow h to be defined on elements< p; this way, every front name belowp is also a front name belowqwheneverq ≤ p.)

If h is a front name andD is any filter system withp ∈ LD, we define the correspondingLD-name (inthe sense of forcing)

˜zh by

˜zh≔ {(y, p[s]) : s ∈ F, y ∈ h(s)}.(1.32)

(This does not depend on theD we use, since we set ˇy≔ {(x, ω<ω) : x ∈ y}.)Up to forced equality, the name

˜zh is characterized by the fact thatp[s] forces (in anyLD) that

˜zh = h(s),

for everys in the domain ofh.

Note that the same objecth can be viewed as a front name belowp with respect to different forcingsLD1

, LD2, as long asp ∈ LD1

∩ LD2.

Definition 1.33. Let p ⊆ ω<ω be a tree. A “continuous name below p” is either of the following:

• An ω-sequence of front names belowp.• A ⊆-increasing functiong : p → clopen<ω such that limn→∞ lh(g(c↾n)) = ∞ for every branch

c ∈ [p].

For eachn, the set of minimal elements in{s ∈ p : lh(g(s)) > n} is a front, so each continuous name in thesecond sense naturally defines a name in the first sense, and conversely. Being a continuous name belowpdoes not involve the notion of nor does it depend on the filter systemD.

If g is a continuous name andD is any filter system, we can again define the correspondingLD-name

˜Zg (in the sense of forcing); we leave a formal definition of

˜Zg to the reader and content ourselves with this

characterization:

(∀s ∈ p) : p[s] LDg(s) ⊆

˜Zg.(1.34)

Note that a continuous name belowp naturally corresponds to a continuous functionF : [p] → clopenω,and

˜Zg is forced (byp) to be the value ofF at the generic real

˜ℓ.

11This implies, by Lemma 1.5, that theLD-generic filterG induces anLDM -generic filter overM, which we callGM .12Recall that nullset(

˜Z) =

⋂

n⋃

k≥n ˜Z(k) is a null set in the extension.

13It is enough to assume that the lengths of the stems diverge toinfinity; any thin enough subsequence will then have strictlyincreasing stems and will still interpret each

˜Zi asZ∗i .

14Instead ofclopen we may also consider other ranges of front names, such as the class of all ordinals, or the setω.


Lemma 1.35. LD has the following “pure decision properties”:

(1) Whenever˜y is a name for an element ofclopen, p ∈ LD, then there is a pure extension p1 ≤0 p

such that˜y =

˜zh (is forced) for a front name h below p1.

(2) Whenever˜Y is a name for a sequence of elements ofclopen, p ∈ LD, then there is a pure extension

q ≤0 p such that˜Y =

˜Zg (is forced) for some continuous name g below q.

(3) (This is Lemma 1.10.) If A is a finite set,˜α a name, p∈ LD, and p forces

˜α ∈ A, then there isβ ∈ A

and a pure extension q≤0 p such that q ˜α = β.

Proof. Let p ∈ LD, s0 ≔ stem(p),˜y a name for an element ofclopen.

We call t ∈ p a “good node inp” if˜y is a front name belowp[t] (more formally: forced to be equal to

˜zh

for a front nameh). We can findp1 ≤0 p such that for allt ∈ p1 aboves0: If there isq ≤0 p[t]1 such thatt is

good inq, thent is already good inp1.We claim thats0 is now good (inp1). Note that for any bad nodes the set{ t ∈ succp1(s) : t bad} is

in D+s . Hence, ifs0 is bad, we can inductively constructp2 ≤0 p1 such that all nodes ofp2 are bad nodesin p1. Now letq ≤ p2 decide

˜y, s≔ stem(q). Thenq ≤0 p[s]

1 , sos is good inp1, contradiction. This finishesthe proof of (1).

To prove (2), we first constructp1 as in (1) with respect to˜y0. This gives a frontF1 ⊆ p1 deciding

˜y0.

Above each node inF1 we now repeat the construction from (1) with respect to˜y1, yieldingp2, etc. Finally,

q≔⋂

n pn.To prove (3): Similar to (1), we can findp1 ≤0 p such that for eacht ∈ p1: If there is a pure extension

of p[t]1 deciding

˜α, thenp[t]

1 decides˜α; in this case we again callt good. Since there are only finitely many

possibilities for the value of˜α, any bad nodet hasD+t many bad successors. So if the stem ofp1 is bad, we

can again reach a contradiction as in (1). �

Corollary 1.36. Let D be a filter system, and let G⊆ LD be generic. Then every Y∈ clopenω in V[G] isthe evaluation of a continuous name

˜Zg by G.

Proof. In V, fix a p ∈ LD and a name˜Y for an element ofclopenω. We can findq ≤0 p and a continuous

nameg belowq such thatq ˜Y =

˜Zg. �

We will need the following modification of the concept of “continuous names”.

Definition 1.37. Let p ⊆ ω<ω be a tree,b ∈ [p] a branch. An “almost continuous name below p (withrespect to b)” is a ⊆-increasing functiong : p → clopen<ω such that limn→∞ lh(g(c↾n)) = ∞ for everybranchc ∈ [p], except possibly forc = b.

Note that “except possibly forc = b” is the only difference between this definition and the definition ofa continuous name.

Since for anyD it is forced15 that the generic real (forLD) is not equal to the exceptional branchb, weagain get a name

˜Zg of a function inclopenω satisfying:

(∀s ∈ p) : p[s] LDg(s) ⊆

˜Zg.

An almost continuous name naturally corresponds to a continuous functionF from [p] \ {b} into clopenω.Note that being an almost continuous name is a very simple combinatorial property ofg which does

not depend onD, nor does it involve the notion . Thus, the same functiong can be viewed as an almostcontinuous name for two different forcing notionsLD1

, LD2simultaneously.

Lemma 1.38. Let D be a system of filters (not necessarily ultrafilters).Assume thatp = (pn)n∈ω witnesses that Y∗ is an interpretation of

˜Y, and that the lengths of the stems of

the pn are strictly increasing.16 Then there exists a sequenceq = (qn)n∈ω such that

(1) q0 ≥ q1 ≥ · · · .(2) qn ≤ pn for all n.(3) q also interprets

˜Y as Y∗. (This follows from the previous two statements.)

15 This follows from our assumption that all our filters containthe Frechet filter.16It is easy to see that for everyLD-name

˜Y we can find such ¯p andY∗: First find p which interprets both

˜Y and ¯

˜ℓ, and then thin

out to get a strictly increasing sequence of stems.


(4)˜Y is almost continuous below q0, i.e., there is an almost continuous name g such that q0 forces

˜Y =

˜Zg.

(5)˜Y is almost continuous below qn, for all n. (This follows from the previous statement.)

Proof. Let b be the branch described by the stems of the conditionspn:

b≔ {s : (∃n) s⊆ stem(pn)}.

We now construct a conditionq0. For everys ∈ b satisfying stem(pn) ⊆ s ( stem(pn+1) we setsuccq0(s) = succpn(s), and for allt ∈ succq0(s) except for the one inb we letq[t]

0 ≤0 p[t]n be such that

˜Y is

continuous belowq[t]0 . We can do this by Lemma 1.35(2).

Now we setqn ≔ pn ∩ q0 = q[stem(pn)]

0 ≤ pn.

This takes care of (1) and (2). Now we show (4): Any branchc of q0 not equal tob must contain a nodes⌢k < b with s ∈ b, soc is a branch inq[s⌢k]

0 , below which˜Y was continuous. �

The following lemmas and corollaries are the motivation forconsidering continuous and almost contin-uous names.

Lemma 1.39. Let D be a system of filters (not necessarily ultrafilters). Let p∈ LD, let b be a branch, andlet g : p→ clopen<ω be an almost continuous name below p with respect to b; write

˜Zg for the associated

LD-name.Let r ∈ 2ω be a real, n0 ∈ ω. Then the following are equivalent:

(1) p LDr <⋃

n≥n0 ˜Zg(n), i.e.,

˜Zg ⊏n0 r.

(2) For all n ≥ n0 and for all s∈ p for which g(s) has length> n we have r< g(s)(n).

Note that (2) does not mention the notion and does not depend onD.

Proof. ¬(2)⇒ ¬(1): Assume that there iss ∈ p for which g(s) = (C0, . . . ,Cn, . . . ,Ck) andr ∈ Cn. Thenp[s] forces that the generic sequence

˜Zg = (

˜Z(0),

˜Z(1), . . .) starts withC0, . . . ,Cn, sop[s] forcesr ∈

˜Zg(n).

¬(1)⇒ ¬(2): Assume thatp does not forcer <⋃

n≥n0 ˜Zg(n). So there is a conditionq ≤ p and some

n ≥ n0 such thatq r ∈˜Zg(n). By increasing the stem ofq, if necessary, we may assume thats≔ stem(q)

is not onb (the “exceptional” branch), and thatg(s) has already length> n. Let Cn ≔ g(s)(n) be then-thentry ofg(s). So p[s] already forces

˜Zg(n) = Cn; now q[s] ≤ p[s], andq[s] forces the following statements:

r ∈˜Zg(n),

˜Zg(n) = Cn. Hencer ∈ Cn, so (2) fails. �

Corollary 1.40. Let D1 and D2 be systems of filters, and assume that p is inLD1∩ LD2

. Let g : p →clopen<ω be an almost continuous name of a sequence of clopen sets, andlet

˜Zg

1 and˜Zg

2 be the associatedLD1

-name andLD2-name, respectively.

Then for any real r and n∈ ω we have

p LD1 ˜Zg

1 ⊏n r ⇔ p LD2 ˜Zg

2 ⊏n r.

(We will use this corollary for the special case thatLD1is an ultralaver forcing, andLD2

is Laver forcing.)

Lemma 1.41. Let D1 andD2 be systems of filters, and assume that p is inLD1∩LD2

. Let g : p→ clopen<ω

be a continuous name of a sequence of clopen sets, let F⊆ p be a front and let h: F → ω be a front name.Again we will write

˜Zg

1, ˜Zg

2 for the associated names of codes for null sets, and we will writeñ1 and

ñ2 for

the associatedLD1- andLD2

-names, respectively, of natural numbers.Then for any real r we have:

p LD1 ˜Zg

1 ⊏ñ1 r ⇔ p LD2 ˜

Zg2 ⊏˜

n2 r.

Proof. Assumep LD1 ˜Zg

1 ⊏ñ1 r. So for eachs ∈ F we have:p[s] LD1 ˜

Zg1 ⊏h(s) r. By Corollary 1.40, we

also havep[s] LD2 ˜Zg

2 ⊏h(s) r. So alsop[s] LD2 ˜Zg

2 ⊏ñ2 r for eachs ∈ F. Hencep LD2 ˜

Zg2 ⊏˜

n2 r. �

Corollary 1.42. Assume q∈ L forces in Laver forcing that˜Zgk ⊏ r for k = 1, 2, . . ., where each gk is a

continuous name of a code for a null set. Then there is a Laver condition q′ ≤0 q such that for all filtersystemsD we have:

If q′ ∈ LD, then q′ forces (in ultralaver forcingLD) that˜Zgk ⊏ r for all k.


Proof. By (1.27) we can find a sequence (ñk)∞k=1 of L-names such thatq

˜Zgk ⊏

ñk r for eachk. By

Lemma 1.35(2) we can findq′ ≤0 q such that this sequence is continuous belowq′. Since eachñk is now a

front name belowq′, we can apply the previous lemma. �

Lemma 1.43. Let M be a countable model, r∈ 2ω, DM ∈ M an ultrafilter system,D a filter systemextendingDM, q ∈ LD. For any V-generic filter G⊆ LD we write GM for the (M-generic, by Lemma 1.5)filter onLDM .

The following are equivalent:

(1) q LDr is random over M[GM].

(2) For all names˜Z ∈ M of codes for null sets: q LD ˜

Z ⊏ r.(3) For all continuous names g∈ M: q LD ˜

Zg ⊏ r.

Proof. (1)⇔(2) holds because every null set is contained in a set of the form nullset(Z), for some codeZ.(2)⇔(3): Every code for a null set inM[GM] is equal to

˜Zg[GM], for someg ∈ M, by Corollary 1.36. �

The following lemma may be folklore. Nevertheless, we proveit for the convenience of the reader.

Lemma 1.44. Let r be random over a countable model M and A∈ M. Then there is a countable modelM′ ⊇ M such that A is countable in M′, but r is still random over M′.

Proof. We will need the following forcing notions, all defined inM:

MC

//

B1

��

MC

˜B2

��

MB1

˜P=C∗

˜B2/B1

// MC∗˜B2

• Let C be the forcing that collapses the cardinality ofA toω with finite conditions.• Let B1 be random forcing (treesT ⊆ 2<ω of positive measure).• Let

˜B2 be theC-name of random forcing.

• Let i : B1→ C ∗˜B2 be the natural complete embeddingT 7→ (1C,T).

• Let˜P be aB1-name for the forcingC∗

˜B2/i[GB1], the quotient ofC∗

˜B2 by the complete subforcing

i[B1].

The random realr is B1-generic overM. In M[r] we letP≔˜P[r]. Now let H ⊆ P be generic overM[r].

Thenr ∗ H ⊆ B1 ∗˜P ≃ C ∗

˜B2 induces anM-generic filterJ ⊆ C and anM[J]-generic filterK ⊆

˜B2[J]; it

is easy to check thatK interprets the˜B2-name of the canonical random real as the given random realr.

Hencer is random over the countable modelM′ ≔ M[J], andA is countable inM′.

MJ

//

r

��

M[J]

K

��

M[r]H

// M[r][H]

�

Proof of Lemma 1.30.We will first describe a construction that deals with a singletriple (p, ¯˜Z, Z∗) (where

p is a sequence of conditions with strictly increasing stems which interprets˜Z asZ∗); this construction will

yield a conditionq′ = q′(p, ¯˜Z, Z∗). We will then show how to deal with all possible triples.

So letp be a condition, and let ¯p = (pk)k∈ω be a sequence interpreting˜Z asZ∗, where the lengths of the

stems ofpn are strictly increasing andp0 = p. It is easy to see that it is enough to deal with a single nullset, i.e.,m= 1, and withk1 = 0. We write

˜Z andZ∗ instead of

˜Z1 andZ∗1.

Using Lemma 1.38 we may (strengthening the conditions in ourinterpretation) assume (inM) that thesequence (

˜Z(k))k∈ω is almost continuous, witnessed byg : p→ clopen<ω. By Lemma 1.44, we can find a

modelM′ ⊇ M such that (2ω)M is countable inM′, but r is still random overM′.We now work inM′. Note thatg still defines an almost continuous name, which we again call

˜Z.


Each filter inDMs is now countably generated; letAs be a pseudo-intersection ofDM

s which additionallysatisfiesAs ⊆ succp(s) for all s ∈ p above the stem. LetD′s be the Frechet filter onAs. Let p′ ∈ LD′ be thetree with the same stem asp which satisfies succp′ (s) = As for all s ∈ p′ above the stem.

By Lemma 1.5, we know thatLDM is anM-complete subforcing ofLD′ (in M′ as well as inV). We writeGM for the induced filter onLDM .

We now work inV. Note that below the conditionp′, the forcingLD′ is just Laver forcingL, and thatp′ ≤L p. Using Lemma 1.29 we can find a conditionq ≤ p′ (in Laver forcingL) such that:

q is M′-generic.(1.45)

q L r is random overM′[GL] (hence also overM[GM]).(1.46)

Moreover,q L˜Z ⊏0 r.(1.47)

Enumerate all continuousLDM -names of codes for null sets fromM as˜Zg1 ,

˜Zg2, . . . Applying Corol-

lary 1.42 yields a conditionq′ ≤ q such that for all filter systemsE satisfyingq′ ∈ LE, we haveq′ LE

˜Zgi ⊏ r for all i. Corollary 1.40 and Lemma 1.43 now imply:

(1.48) For every filter systemE satisfyingq′ ∈ LE, q′ forces inLE that r is random overM[GM] and that

˜Z ⊏0 r.

By thinning outq′ we may assume that

(1.49) For eachν ∈ ωω ∩ M there isk such thatν↾k < q′.

We have now described a construction ofq′ = q′(p,˜Z,Z∗).

Let (pn,˜Zn,Z∗n) enumerate all triples ( ¯p,

˜Z,Z∗) ∈ M where p interprets

˜Z asZ∗ (and consists of con-

ditions with strictly increasing stems). For eachn write νn for⋃

k stem(pnk), the branch determined by the

stems of the sequence ¯pn. We now define by induction a sequenceqn of conditions:

• q0≔ q′(p0,

˜Z0,Z∗0).

• Givenqn−1 and (pn,˜Zn,Z∗n), we findk0 such thatνn↾k0 < q0 ∪ · · · ∪ qn−1 (using (1.49)). Letk1 be

such that stem(pnk1

) has length> k0. We replace ¯pn by p′ ≔ (pnk)k≥k1. (Obviously,p′ still interprets

˜Zn asZ∗n.) Now letqn

≔ q′(p′,˜Zn,Z∗n).

Note that the stem ofqn is at least as long as the stem ofpnk1

, and is therefore not inq0 ∪ · · · ∪ qn−1, sostem(qi) and stem(q j) are incompatible for alli , j. Therefore we can choose for eachs an ultrafilterDs

extendingDMs such that stem(qi) ⊆ s implies succqi (s) ∈ Ds.

Note that allqi are inLD. Therefore, we can use (1.48). Also,qi ≤ pi0. �

Below, in Lemma 3.33, we will prove a preservation theorem using the following “local” variant of“random preservation”:

Definition 1.50. Fix a countable modelM, a realr ∈ 2ω and a forcing notionQM ∈ M. Let QM be anM-complete subforcing ofQ. We say that “Q locally preserves randomness of r over M”, if there is in Ma sequence (DQM

n )n∈ω of open dense subsets ofQM such that the following holds:Assume that

• M thinks thatp ≔ (pn)n∈ω interprets (˜Z1, . . . ,

˜Zm) as (Z∗1, . . . ,Z

∗m) (so each

˜Zi is a QM-name of a

code for a null set and eachZ∗i is a code for a null set, both inM);

• moreover, eachpn is in DQM

n (we call such a sequence (pn)n∈ω, or the according interpretation,“quick”);• r is random overM;• Z∗i ⊏ki r for i = 1, . . . ,m.

Then there is aq ≤Q p0 forcing that

• r is random overM[GM];•

˜Zi ⊏ki r for i = 1, . . . ,m.

Note that this is trivially satisfied ifr is not random overM.For a variant of this definition, see Section 6.SettingDQM

n to be the set of conditions with stem of length at leastn, Lemma 1.30 gives us:


Corollary 1.51. If QM is an ultralaver forcing in M and r a real, then there is an ultralaver forcing Qover17 QM locally preserving randomness of r over M.

2. Janus forcing

In this section, we define a family of forcing notions that hastwo faces (hence the name “Janus forcing”):Elements of this family may be countable (and therefore equivalent to Cohen), and they may also beessentially random.

In the rest of the paper, we will use the following propertiesof Janus forcing notionsJ. (And we willuseonly these properties. So readers who are willing to take these properties for granted could skip toSection 3.)

Throughout the whole paper we fix a functionB∗ : ω → ω given by Corollary 2.2. The Janus forcingswill depend on a real parameterℓ∗ = (ℓ∗m)m∈ω ∈ ω

ω which grows fast with respect toB∗. (In our application,ℓ∗ will be given by a subsequence of an ultralaver real.)

The sequenceℓ∗ and the functionB∗ together define a notion of a “thin set” (see Definition 1.22).

(1) There is a canonicalJ-name for a (code for a) null set˜Z∇.

WheneverX ⊆ 2ω is not thin, andJ is countable, thenJ forces thatX is not strongly meager,witnessed18 by nullset(

˜Z∇) (the set we get when we evaluate the code

˜Z∇). Moreover, for anyJ-

name˜Q of aσ-centered forcing, alsoJ ∗

˜Q forces thatX is not strongly meager, again witnessed

by nullset(˜Z∇).

(This is Lemma 2.9; “thin” is defined in Definition 1.22.)(2) Let M be a countable transitive model andJM a Janus forcing inM. ThenJM is a Janus forcing in

V as well (and of course countable inV). (Also note that trivially the forcingJM is anM-completesubforcing of itself.)(This is Fact 2.8.)

(3) WheneverM is a countable transitive model andJM is a Janus forcing inM, then there is a Janusforcing J such that• JM is anM-complete subforcing ofJ.• J is (in V) equivalent to random forcing (actually we just need thatJ preserves Lebesgue

positivity in a strong and iterable way).(This is Lemma 2.16 and Lemma 2.20.)

(4) Moreover, the name˜Z∇ referred to in (1) is so “canonical” that it evaluates to the same code in the

J-generic extension overV as in theJM-generic extension overM.(This is Fact 2.7.)

2.A. Definition of Janus. A Janus forcingJ will consist of:19

• A countable “core” (or: backbone)∇which is defined in a combinatorial way from a parameterℓ∗.(In our application, we will use a Janus forcing immediatelyafter an ultralaver forcing, andℓ∗ willbe a subsequence of the ultralaver real.) This core is of course equivalent to Cohen forcing.• Some additional “stuffing” J \ ∇ (countable20 or uncountable). We allow great freedom for this,

we just require that the core∇ is a “sufficiently” complete subforcing (in a specific combinatorialsense, see Definition 2.5(3)).

We will use the following combinatorial theorem from [BS10]:

Lemma 2.1([BS10, Theorem 8]21). For everyε, δ > 0 there exists Nε,δ ∈ ω such that for all sufficiently

large finite sets I⊆ ω there is a familyAI with |AI | ≥ 2 consisting of sets A⊆ 2I with|A|

2|I |≤ ε such that if

17“Q overQM” just means thatQM is anM-complete subforcing ofQ.18in the sense of (0.1)19We thank Andreas Blass and Jindrich Zapletal for their comments that led to an improved presentation of Janus forcing.20Also the trivial caseJ = ∇ is allowed.21The theorem in [BS10] actually says “for a sufficiently largeI”, but the proof shows that this should be read as “forall sufficiently

large I”. Also, the quoted theorem only claims thatAI will be nonempty, but forε ≤ 12 and|I | > Nε,δ it is easy to see thatAI cannot

be a singleton{A}: The setX := 2I \ A has size≥ 2|I |−1 ≥ Nε,δ but satisfiesX + A , 2I , as the constant sequence0 is not inX + A.


X ⊆ 2I , |X| ≥ Nε,δ then

|{A ∈ AI : X + A = 2I }|

|AI |≥ 1− δ.

(Recall that X+ A≔ {x+ a : x ∈ X, a ∈ A}.)

Rephrasing and specializing toδ = 14 andε = 1

2i we get:

Corollary 2.2. For every i ∈ ω there exists B∗(i) such that for all finite sets I with|I | ≥ B∗(i) there is anonempty familyAI with |AI | ≥ 2 satisfying the following:

• AI consists of sets A⊆ 2I with|A|

2|I |≤

12i

.

• For every X⊆ 2I satisfying|X| ≥ B∗(i), the set{A ∈ AI : X + A = 2I } has at least34 |AI | elements.

Assumption 2.3. We fix a sufficiently fast increasing sequenceℓ∗ = (ℓ∗i )i∈ω of natural numbers; moreprecisely, the sequenceℓ∗ will be a subsequence of an ultralaver realℓ, defined as in Lemma 1.23 using thefunctionB∗ from Corollary 2.2. Note that in this caseℓ∗i+1 − ℓ

∗i ≥ B∗(i); so we can fix for eachi a family

Ai ⊆P(2Li ) on the intervalLi ≔ [ℓ∗i , ℓ∗i+1) according to Corollary 2.2.

Definition 2.4. First we define the “core”∇ = ∇ℓ∗ of our forcing:

∇ =⋃

i∈ω

∏

j<i

A j .

In other words,σ ∈ ∇ iff σ = (A0, . . . ,Ai−1) for somei ∈ ω, A0 ∈ A0, . . . , Ai−1 ∈ Ai−1. We will denote thenumberi by height(σ).

The forcing notion∇ is ordered by reverse inclusion (i.e., end extension):τ ≤ σ if τ ⊇ σ.

Definition 2.5. Let ℓ∗ = (ℓ∗i )i∈ω be as in the assumption above. We say thatJ is a Janus forcing based onℓ∗

if:

(1) (∇,⊇) is an incompatibility-preserving subforcing ofJ.(2) For eachi ∈ ω the set{σ ∈ ∇ : height(σ) = i} is predense inJ. So in particular,J adds a

branch through∇. The union of this branch is called˜C∇ = (

˜C∇0 , ˜

C∇1 , ˜C∇2 , . . .), where

˜C∇i ⊆ 2Li with

˜C∇i ∈ Ai .

(3) “Fatness”:22 For all p ∈ J and all real numbersε > 0 there are arbitrarily largei ∈ ω such that thereis a core conditionσ = (A0, . . . ,Ai−1) ∈ ∇ (of lengthi) with

|{A ∈ Ai : σ⌢A ‖J p }||Ai |

≥ 1− ε.

(Recall thatp ‖J q means thatp andq are compatible inJ.)(4) J is ccc.(5) J is separative.23

(6) (To simplify some technicalities:)J ⊆ H(ℵ1).

We now define˜Z∇, which will be a canonicalJ-name of (a code for) a null set. We will use the sequence

˜C∇ added byJ (see Definition 2.5(2)).

Definition 2.6. Each˜C∇i defines a clopen set

˜Z∇i = {x ∈ 2ω : x↾Li ∈

˜C∇i } of measure at most12i . The

sequence˜Z∇ = (

˜Z∇0 , ˜

Z∇1 , ˜Z∇2 , . . .) is (a name for) a code for the null set

nullset(˜Z∇) =

⋂

n<ω

⋃

i≥n˜Z∇i .

Since˜C∇ is defined “canonically” (see in particular Definition 2.5(1),(2)), and

˜Z∇ is constructed in an

absolute way from˜C∇, we get:

22This is the crucial combinatorial property of Janus forcing. Actually, (3) implies (2).23Separative is defined on page 3.


Fact 2.7. If J is a Janus forcing,M a countable model andJM a Janus forcing inM which is anM-completesubset ofJ, if H is J-generic overV andHM the inducedJM-generic filter overM, then

˜C∇ evaluates to the

same real inM[HM] as inV[H], and therefore˜Z∇ evaluates to the same code (but of course not to the same

set of reals).

For later reference, we record the following trivial fact:

Fact 2.8. Being a Janus forcing is absolute. In particular, ifV ⊆ W are set theoretical universes andJ is aJanus forcing inV, thenJ is a Janus forcing inW. In particular, ifM is a countable model inV andJ ∈ Ma Janus forcing inM, thenJ is also a Janus forcing inV.Let (Mn)n∈ω be an increasing sequence of countable models, and letJn ∈ Mn be Janus forcings. AssumethatJn is Mn-complete inJn+1. Then

⋃

n Jn is a Janus forcing, and anMn-complete extension ofJn for all n.

2.B. Janus and strongly meager.Carlson [Car93] showed that Cohen reals make every uncountableset X of the ground model not strongly meager in the extension (andthat not being strongly meager ispreserved in a subsequent forcing with precaliberℵ1). We show that acountableJanus forcingJ does thesame (for a subsequent forcing that is evenσ-centered, not just precaliberℵ1). This sounds trivial, sinceany (nontrivial) countable forcing is equivalent to Cohen forcing anyway. However, we show (and willlater use) that the canonical null set

˜Z∇ defined above witnesses thatX is not strongly meager (and not

just some null set that we get out of the isomorphism betweenJ and Cohen forcing). The point is thatwhile∇ is not a complete subforcing ofJ, the condition (3) of the Definition 2.5 guarantees that Carlson’sargument still works, if we assume thatX is non-thin (not just uncountable). This is enough for us, sinceby Corollary 1.24 ultralaver forcing makes any uncountableset non-thin.

Recall that we fixed the increasing sequenceℓ∗ = (ℓ∗i )i∈ω andB∗. In the following, whenever we say“(very) thin” we mean “(very) thin with respect toℓ∗ andB∗” (see Definition 1.22).

Lemma 2.9. If X is not thin,J is a countable Janus forcing based onℓ∗, and˜R is aJ-name for aσ-centered

forcing notion, thenJ ∗˜R forces that X is not strongly meager witnessed by the null set

˜Z∇.

Proof. Let˜c be aJ-name for a function

˜c :

˜R→ ω witnessing that

˜R isσ-centered.

Recall that “˜Z∇ witnesses thatX is not strongly meager” means thatX +

˜Z∇ = 2ω. Assume towards

a contradiction that (p, r) ∈ J ∗˜R forces thatX +

˜Z∇ , 2ω. Then we can fix a (J ∗

˜R)-name

˜ξ such that

(p, r) ˜ξ < X +

˜Z∇, i.e., (p, r) (∀x ∈ X)

˜ξ < x+

˜Z∇. By definition of

˜Z∇, we get

(p, r) (∀x ∈ X) (∃n ∈ ω) (∀i ≥ n)˜ξ↾Li < x↾Li +

˜C∇i .

For eachx ∈ X we can find (px, rx) ≤ (p, r) and natural numbersnx ∈ ω andmx ∈ ω such thatpx forcesthat

˜c(rx) = mx and

(px, rx) (∀i ≥ nx)˜ξ↾Li < x↾Li +

˜C∇i .

SoX =⋃

p∈J,m∈ω,n∈ω Xp,m,n, whereXp,m,n is the set of allx with px = p, mx = m, nx = n. (Note thatJ iscountable, so the union is countable.) AsX is not thin, there is somep∗,m∗, n∗ such thatX∗ ≔ Xp∗ ,m∗ ,n∗ isnot very thin. So we get for allx ∈ X∗:

(2.10) (p∗, rx) (∀i ≥ n∗)˜ξ↾Li < x↾Li +

˜C∇i .

SinceX∗ is not very thin, there is somei0 ∈ ω such that for alli ≥ i0

(2.11) the (finite) setX∗↾Li has more thanB∗(i) elements.

Due to the fact thatJ is a Janus forcing (see Definition 2.5 (3)), there are arbitrarily large i ∈ ω such thatthere is a core conditionσ = (A0, . . . ,Ai−1) ∈ ∇ with

(2.12)|{A ∈ Ai : σ⌢A ‖J p∗}|

|Ai |≥

23.

Fix such ani larger than bothi0 andn∗, and fix a conditionσ satisfying (2.12).We now consider the following two subsets ofAi :

(2.13) {A ∈ Ai : σ⌢A ‖J p∗} and {A ∈ Ai : X∗↾Li + A = 2Li }.

By (2.12), the relative measure (inAi) of the left one is at least23; due to (2.11) and the definition ofAi

according to Corollary 2.2, the relative measure of the right one is at least34; so the two sets in (2.13) arenot disjoint, and we can pick anA belonging to both.


Clearly,σ⌢A forces (inJ) that˜C∇i is equal toA. Fix q ∈ J witnessingσ⌢A ‖J p∗. Then

(2.14) q J X∗↾Li +˜C∇i = X∗↾Li + A = 2Li .

Sincep∗ forces that for eachx ∈ X∗ the color˜c(rx) = m∗, we can find anr∗ which is (forced byq ≤ p∗

to be) a lower bound of thefiniteset{rx : x ∈ X∗∗}, whereX∗∗ ⊆ X∗ is any finite set withX∗∗↾Li = X∗↾Li .By (2.10),

(q, r∗) ˜ξ↾Li < X∗∗↾Li +

˜C∇i = X∗↾Li +

˜C∇i ,

contradicting (2.14). �

Recall that by Corollary 1.24, every uncountable setX in V will not be thin in theLD-extension. Hencewe get:

Corollary 2.15. Let X be uncountable. IfLD is any ultralaver forcing adding an ultralaver realℓ, and ℓ∗

is defined fromℓ as in Lemma 1.23, and if˜J is a countable Janus forcing based onℓ∗,

˜Q is anyσ-centered

forcing, thenLD ∗ ˜J ∗

˜Q forces that X is not strongly meager.

2.C. Janus forcing and preservation of Lebesgue positivity.We show that every Janus forcing in acountable modelM can be extended to locally preserve a given random real overM. (We showed the samefor ultralaver forcing in Section 1.E.)

We start by proving that every countable Janus forcing can beembedded into a Janus forcing which isequivalent to random forcing, preserving the maximality ofcountably many maximal antichains. (In thefollowing lemma, the letterM is just a label to distinguishJM from J, and does not necessarily refer to amodel.)

Lemma 2.16. Let JM be a countable Janus forcing (based onℓ∗) and let {Dk : k ∈ ω} be a countablefamily of open dense subsets ofJM. Then there is a Janus forcingJ (based on the sameℓ∗) such that

• JM is an incompatibility-preserving subforcing ofJ.• Each Dk is still predense inJ.• J is forcing equivalent to random forcing.

Proof. Without loss of generality assumeD0 = JM. Recall that∇ = ∇J

Mwas defined in Definition 2.4.

Note that for eachj the set{σ ∈ ∇ : height(σ) = j} is predense inJM, so the set

E j ≔ {p ∈ JM : ∃σ ∈ ∇ : height(σ) = j, p ≤ σ}(2.17)

is dense open inJM ; hence without loss of generality eachE j appears in our list ofDk’s.Let {rn : n ∈ ω} be an enumeration ofJM.We now fixn for a while (up to (2.19)). We will construct a finitely splitting treeSn ⊆ ω<ω and a family

(σns, p

ns, τ∗ns )s∈Sn satisfying the following (suppressing the superscriptn):

(a) σs ∈ ∇, σ〈〉 = 〈〉, s⊆ t impliesσs ⊆ σt, ands⊥Sn t impliesσs ⊥∇ σt.(So in particular the set{σt : t ∈ succSn(s)} is a (finite) antichain aboveσs in ∇.)

(b) ps ∈ JM, p〈〉 = rn; if s⊆ t thenpt ≤JM ps (hencept ≤ rn); s⊥Sn t implies ps ⊥JM pt.

(c) ps ≤JM σs.(d) σs ⊆ τ

∗s ∈ ∇, and{σt : t ∈ succSn(s)} is the set of allτ ∈ succ∇(τ∗s) which are compatible withps.

(e) The set{σt : t ∈ succSn(s)} is a subset of succ∇(τ∗s) of relative size at least 1− 1lh(s)+10.

(f) Eachs ∈ Sn has at least 2 successors (inSn).(g) If k = lh(s), thenps ∈ Dk (and therefore also in allDl for l < k).

Setσ〈〉 = 〈〉 and p〈〉 = rn. Given s, σs and ps, we construct succSn(s) and (σt, pt)t∈succSn (s): We applyfatness 2.5(3) tops with ε = 1

lh(s)+10. So we get someτ∗s ∈ ∇ of height bigger than the height ofσs suchthat the setB of elements of succ∇(τ∗s) which are compatible withps has relative size at least 1− ε. Sinceps ≤JM σs we get thatτ∗s is compatible with (and therefore stronger than)σs. EnumerateB as{τ0, . . . , τl−1}.Set succSn(s) = {s⌢i : i < l} andσs⌢ i = τi . For t ∈ succSn(s), choosept ∈ J

M stronger than bothσt andps

(which is obviously possible sinceσt andps are compatible), and moreoverpt ∈ Dlh(t). This concludes theconstruction of the family (σn

s, pns, τ∗ns )s∈Sn.

So (Sn,⊆) is a finitely splitting nonempty tree of heightω with no maximal nodes and no isolatedbranches. [Sn] is the (compact) set of branches ofSn. The closed subsets of [Sn] are exactly the sets of


the form [T], whereT ⊆ Sn is a subtree ofSn with no maximal nodes. [Sn] carries a natural (“uniform”)probability measureµn, which is characterized by

µn((Sn)[t]) =1

|succSn(s)|· µn((Sn)[s])

for all s ∈ Sn and allt ∈ succSn(s). (We just writeµn(T) instead ofµn([T]) to increase readability.)We callT ⊆ Sn positive ifµn(T) > 0, and we callT pruned ifµn(T [s]) > 0 for all s ∈ T. (Clearly every

positive treeT contains a pruned treeT′ of the same measure, which can be obtained fromT by removingall nodesswith µn(T [s]) = 0.)

Let T ⊆ Sn be a positive pruned tree andε > 0. Then on all but finitely many levelsk there is ans ∈ Tsuch that

(2.18) succT (s) ⊆ succSn(s) has relative size≥ 1− ε.

(This follows from Lebesgue’s density theorem, or can easily be seen directly: SetCm =⋃

t∈T, lh(t)=m (Sn)[t] .ThenCm is a decreasing sequence of closed sets, each containing [T]. If the claim fails, thenµn(Cm+1)) ≤µn(Cm) · (1− ε) infinitely often; soµn(T) ≤ µn(

⋂

mCm) = 0.)It is well known that the set of positive, pruned subtrees ofSn, ordered by inclusion, is forcing equivalent

to random forcing (which can be defined as the set of positive,pruned subtrees of 2<ω).We have now constructedSn for all n. Define

J = JM ∪⋃

n

{

(n,T) : T ⊆ Sn is a positive pruned tree}

(2.19)

with the following partial order:

• The order onJ extends the order onJM.• (n′,T′) ≤ (n,T) if n = n′ andT′ ⊆ T.• For p ∈ JM: (n,T) ≤ p if there is ak such thatpn

t ≤ p for all t ∈ T of lengthk. (Note that this willthen be true for all biggerk as well.)• p ≤ (n,T) never holds (forp ∈ JM).

The lemma now easily follows from the following properties:

(1) The order onJ is transitive.(2) JM is an incompatibility-preserving subforcing ofJ.

In particular,J satisfies item (1) of Definition 2.5 of Janus forcing.(3) For allk: the set{(n,T [t]) : t ∈ T, lh(t) = k} is a (finite) predense antichain below (n,T).(4) (n,T [t]) is stronger thanpn

t for eacht ∈ T (witnessed, e.g., byk = lh(t)). Of course, (n,T [t]) isstronger than (n,T) as well.

(5) Sincepnt ∈ Dk for k = lh(t), this implies that eachDk is predense below each (n,Sn) and therefore

in J.Also, since each setE j appeared in our list of open dense subsets (see (2.17)), the set {σ ∈ ∇ :height(σ) = j} is still predense inJ, i.e., item (2) of the Definition 2.5 of Janus forcing is satisfied.

(6) The condition (n,Sn) is stronger thanrn, so {(n,Sn) : n ∈ ω} is predense inJ andJ \ JM is densein J.Below each (n,Sn), the forcingJ is isomorphic to random forcing.Therefore,J itself is forcing equivalent to random forcing. (In fact, the complete Boolean algebragenerated byJ is isomorphic to the standard random algebra, Borel sets modulo null sets.) Thisproves in particular thatJ is ccc, i.e., satisfies property 2.5(4).

(7) It is easy (but not even necessary) to check thatJ is separative, i.e., property 2.5(5). In any case,we could replace≤J by ≤∗

J, thus makingJ separative without changing≤JM , sinceJM was already

separative.(8) Property 2.5(6), i.e.,J ∈ H(ℵ1), is obvious.(9) The remaining item of the definition of Janus forcing, fatness 2.5(3), is satisfied.

I.e., given (n,T) ∈ J andε > 0 there is an arbitrarily highτ∗ ∈ ∇ such that the relative size of theset{τ ∈ succ∇(τ∗) : τ ‖ (n,T)} is at least 1− ε. (We will show≥ (1− ε)2 instead, to simplify thenotation.)


We show (9): Given (n,T) ∈ J andε > 0, we use (2.18) to get an arbitrarily highs ∈ T such that succT(s)is of relative size≥ 1− ε in succSn(s). We may choosesof length> 1

ε. We claim thatτ∗s is as required:

• Let B≔ {σt : t ∈ succSn(s)}. Note thatB = {τ ∈ succ∇(τ∗s) : τ ‖ ps}.B has relative size≥ 1− 1

lh(s) ≥ 1− ε in succ∇(τ∗s) (according to property (e) ofSn).• C ≔ {σt : t ∈ succT(s)} is a subset ofB of relative size≥ 1− ε according to our choice ofs.• SoC is of relative size (1− ε)2 in succ∇(τ∗s).• Eachσt ∈ C is compatible with (n,T), as (n,T [t]) ≤ pt ≤ σt (see (4)). �

So in particular ifJM is a Janus forcing in a countable modelM, then we can extend it to a Janus forcingJwhich is in fact random forcing. Since random forcing strongly preserves randoms over countable models(see Lemma 1.29), it is not surprising that we get local preservation of randoms for Janus forcing, i.e., theanaloga of Lemma 1.30 and Corollary 1.51. (Still, some additional argument is needed, since the fact thatJ

(which is now random forcing) “strongly preserves randoms”just means that a random realr over M ispreserved with respect to random forcing inM, not with respect toJM.)

Lemma 2.20. If JM is a Janus forcing in a countable model M and r a random real over M, then there isa Janus forcingJ such thatJM is an M-complete subforcing ofJ and the following holds:If

• p ∈ JM,• in M,

˜Z = (

˜Z1, . . . ,

˜Zm) is a sequence ofJM-names for codes for null sets, and Z∗1, . . . ,Z

∗m are

interpretations under p, witnessed by a sequence(pn)n∈ω,• Z∗i ⊏ki r for i = 1, . . . ,m,

then there is a q≤ p in J forcing that

• r is random over M[HM],•

˜Zi ⊏ki r for i = 1, . . . ,m.

Remark 2.21. In the version for ultralaver forcings, i.e., Lemma 1.30, wehad to assume that the stems ofthe witnessing sequence are strictly increasing. In the Janus version, we do not have any requirement ofthat kind.

Proof. LetD be the set of dense subsets ofJM in M. According to Lemma 1.44, we can first find somecountableM′ such thatr is still random overM′ and such that inM′ both JM andD are countable. Ac-cording to Fact 2.8,JM is a (countable) Janus forcing inM′, so we can apply Lemma 2.16 to the setDto construct a Janus forcingJM

′

which is equivalent to random forcing such that (from the point of V)JM ⋖M J

M′ . In V, let24 J be random forcing.JM′

is anM′-complete subforcing ofJ and thereforeJM ⋖M J.Moreover, as was noted in Lemma 1.29, we even know that randomforcing strongly preserves randomsoverM′ (see Definition 1.50). To show thatJ is indeed a Janus forcing, we have to check the fatness con-dition 2.5(3); this follows easily fromΠ1

1-absoluteness (recall that incompatibility of random conditions isBorel).

So assume that (inM) the sequence (pn)n∈ω of JM-conditions interprets˜Z asZ∗. In M′, JM-names can

be reinterpreted asJM′

-names, and theJM′

-name˜Z is interpreted asZ∗ by the same sequence (pn)n∈ω. Let

k1, . . . , km be such thatZ∗i ⊏ki r for i = 1, . . . ,m. So by strong preservation of randoms, we can inV findsomeq ≤ p0 forcing thatr is random overM′[HM′ ] (and therefore also over the subsetM[HM]), and that

˜Zi ⊏ki r (where

˜Zi can be evaluated inM′[HM′ ] or equivalently inM[HM]). �

So Janus forcing is locally preserving randoms (just as ultralaver forcing):

Corollary 2.22. If QM is a Janus forcing in M and r a real, then there is a Janus forcing Q over QM (whichis in fact equivalent to random forcing) locally preservingrandomness of r over M.

Proof. In this case, the notion of “quick” interpretations is trivial, i.e.,DQM

k = QM for all k, and the claimfollows from the previous lemma. �

24More precisely: Densely embedJM′

into (Borel/null)M′ , the complete Boolean algebra associated with random forcing in M′,and letJ := (Borel/null)V . Using the embedding,JM

′can now be viewed as anM′-complete subset ofJ.


3. Almost finite and almost countable support iterations

A main tool to construct the forcing for BC+dBC will be “partial countable support iterations”, moreparticularly “almost finite support” and “almost countablesupport” iterations. A partial countable supportiteration is a forcing iteration (Pα,Qα)α<ω2 such that for each limit ordinalδ the forcing notionPδ is a subsetof the countable support limit of (Pα,Qα)α<δ which satisfies some natural properties (see Definition 3.6).

Instead of transitive models, we will use ord-transitive models (which are transitive when ordinals areconsidered as urelements). Why do we do that? We want to “approximate” the generic iterationP of lengthω2 with countable models; this can be done more naturally with ord-transitive models (since obviouslycountable transitive models only see countable ordinals).We call such an ord-transitive model a “candi-date” (provided it satisfies some nice properties, see Definition 3.1). A basic point is that forcing extensionswork naturally with candidates.

In the next few paragraphs (and also in Section 4),x = (Mx, Px) will denote a pair such thatMx is acandidate andPx is (in Mx) a partial countable support iteration; similarly we write, e.g.,y = (My, Py) orxn = (Mxn, Pxn).

We will need the following results to prove BC+dBC. (However, as opposed to the case of the ultralaverand Janus section, the reader will probably have to read thissection to understand the construction in thenext section, and not just the following list of properties.)

Given x = (Mx, Px), we can construct by induction onα a partial countable support iterationP =(Pα,Qα)α<ω2 satisfying:

There is a canonicalMx-complete embedding fromPx to P.

In this construction, we can use at each stageβ any desiredQβ, as long asPβ forces thatQxβ

is (evaluatedas) anMx[Hx

β]-complete subforcing ofQβ (whereHx

β⊆ Px

βis theMx-generic filter induced by the generic

filter Hβ ⊆ Pβ).Moreover, we can demand either of the following two additional properties25 of the limit of this iterationP:

(1) If all Qβ are forced to beσ-centered, andQβ is trivial for all β < Mx, thenPω2 isσ-centered.(2) If r is random overMx, and allQβ locally preserve randomness ofr over Mx[Hx

β] (see Defini-

tion 1.50), then alsoPω2 locally preserves the randomness ofr.

Actually, we need the following variant: Assume that we already havePα0 for someα0 ∈ Mx, and thatPxα0

canonically embeds intoPα0, and that the respective assumption onQβ holds for allβ ≥ α0. Then we getthatPα0 forces that the quotientPω2/Pα0 satisfies the respective conclusion.

We also need:26

(3) If instead of a singlex we have a sequencexn such that eachPxn canonically (andMxn-completely)embeds intoPxn+1, then we can find a partial countable support iterationP into which allPxn embedcanonically (and we can again use any desiredQβ, assuming thatQxn

βis an Mxn[Hxn

β]-complete

subforcing ofQβ for all n ∈ ω).(4) (A fact that is easy to prove but awkward to formulate.) Ifa ∆-system argument produces two

x1, x2 as in Lemma 4.7(3), then we can find a partial countable support iterationP such thatPxi

canonically (andMxi -completely) embeds intoP for i = 1, 2.

3.A. Ord-transitive models. We will use “ord-transitive” models, as introduced in [She04] (see also thepresentation in [Kelar]). We briefly summarize the basic definitions and properties (restricted to the rathersimple case needed in this paper):

Definition 3.1. Fix a suitable finite subset ZFC∗ of ZFC (that is satisfied byH(χ∗) for sufficiently largeregularχ∗).

(1) A setM is called acandidate, if• M is countable,• (M, ∈) is a model of ZFC∗,• M is ord-absolute:M |= α ∈ Ord iff α ∈ Ord, for allα ∈ M,• M is ord-transitive: ifx ∈ M \Ord, thenx ⊆ M,

25Theσ-centered version is central for the proof of dBC; the randompreserving version for BC.26This will give σ-closure andℵ2-cc for the preparatory forcingR.


• ω + 1 ⊆ M.• “α is a limit ordinal” and “α = β + 1” are both absolute betweenM andV.

(2) A candidateM is callednice, if “ α has countable cofinality” and “the countable setA is cofinalin α” both are absolute betweenM andV. (So if α ∈ M has countable cofinality, thenα ∩ M iscofinal inα.) Moreover, we assumeω1 ∈ M (which impliesω1

M = ω1) andω2 ∈ M (but we donot requireω2

M = ω2).(3) Let PM be a forcing notion in a candidateM. (To simplify notation, we can assume without loss

of generality thatPM ∩ Ord = ∅ (or at least⊆ ω) and that thereforePM ⊆ M and alsoA ⊆ MwheneverM thinks thatA is a subset ofPM.) Recall that a subsetHM of PM is M-generic (or:PM-generic overM), if |A∩ HM | = 1 for all maximal antichainsA in M.

(4) LetHM bePM-generic overM and˜τ a PM-name inM. We define the evaluation

˜τ[HM]M to bex if

M thinks thatp PM

˜τ = x for somep ∈ HM andx ∈ M (or equivalently just forx ∈ M∩Ord), and

{˜σ[HM]M : (

˜σ, p) ∈

˜τ, p ∈ HM} otherwise. Abusing notation we write

˜τ[HM] instead of

˜τ[HM]M,

and we writeM[HM] for {˜τ[HM] :

˜τ is aPM-name inM}.

(5) For any setN (typically, an elementary submodel of someH(χ)), the ord-collapsek (or kN) is arecursively defined function with domainN: k(x) = x if x ∈ Ord, andk(x) = {k(y) : y ∈ x ∩ N}otherwise.

(6) We define ordclos(α) := ∅ for all ordinalsα. The ord-transitive closure of a non-ordinalx is definedinductively on the rank:

ordclos(x) = x∪⋃

{ordclos(y) : y ∈ x \Ord} = x∪⋃

{ordclos(y) : y ∈ x}.

So forx < Ord, the set ordclos(x) is the smallest ord-transitive set containingx as a subset. HCONis the collection of all setsx such that the ord-transitive closure ofx is countable.x is in HCON iffx is element of some candidate. In particular, all reals and all ordinals are HCON.

We write HCONα for the family of all setsx in HCON whose transitive closure only containsordinals< α.

The following facts can be found in [She04] or [Kelar] (they can be proven by rather straightforward, iftedious, inductions on the ranks of the according objects).

Fact 3.2. (1) The ord-collapse of a countable elementary submodel ofH(χ∗) is a nice candidate.(2) Unions, intersections etc. are generally not absolute for candidates. For example, letx ∈ M \Ord.

In M we can construct a sety such thatM |= y = ω1 ∪ {x}. Theny is not an ordinal and therefore asubset ofM, and in particulary is countable andy , ω1 ∪ {x}.

(3) Let j : M → M′ be the transitive collapse of a candidateM, and f : ω1 ∩ M′ → Ord the inverse(restricted to the ordinals). ObviouslyM′ is a countable transitive model of ZFC∗; moreoverMis characterized by the pair (M′, f ) (we call such a pair a “labeled transitive model”). Note that fsatisfiesf (α + 1) = f (α) + 1, f (α) = α for α ∈ ω ∪ {ω}. M |= (α is a limit) iff f (α) is a limit.M |= cf(α) = ω iff cf( f (α)) = ω, and in that casef [α] is cofinal in f (α). On the other hand, givena transitive countable modelM′ of ZFC∗ and an f as above, then we can construct a (unique)candidateM corresponding to (M′, f ).

(4) All candidatesM with M ∩ Ord ⊆ ω1 are hereditarily countable, so their number is at most 2ℵ0.Similarly, the cardinality of HCONα is at most continuum wheneverα < ω2.

(5) If M is a candidate, and ifHM is PM-generic overM, thenM[HM] is a candidate as well and anend-extension ofM such thatM ∩ Ord = M[HM] ∩ Ord. If M is nice and (M thinks that)PM isproper, thenM[HM] is nice as well.

(6) Forcing extensions commute with the transitive collapse j: If M corresponds to (M′, f ), thenHM ⊆ PM is PM-generic overM iff H′ ≔ j[HM] is P′ ≔ j(PM)-generic overM′, and in that caseM[HM] corresponds to (M′[H′], f ). In particular, the forcing extensionM[HM] of M satisfies theforcing theorem (everything that is forced is true, and everything true is forced).

(7) For elementary submodels, forcing extensions commute with ord-collapses: LetN be a countableelementary submodel ofH(χ∗), P ∈ N, k : N → M the ord-collapse (soM is a candidate), and letH beP-generic overV. ThenH is P-generic overN iff HM

≔ k[H] is PM≔ k(P)-generic overM;

and in that case the ord-collapse ofN[H] is M[HM].


Assume that a nice candidateM thinks that (PM, QM) is a forcing iteration of lengthω2V (we will

usually writeω2 for the length of the iteration, by this we will always meanω2V and not the possibly

differentω2M). In this section, we will construct an iteration (P, Q) in V, also of lengthω2, such that each

PMα canonically andM-completely embeds intoPα for all α ∈ ω2 ∩ M. Once we know (by induction) that

PMα M-completely embeds intoPα, we know that aPα-generic filterHα induces aPM

α -generic (overM)filter which we callHM

α . ThenM[HMα ] is a candidate, but nice only ifPM

α is proper. We will not need thatM[HM

α ] is nice, actually we will only investigate sets of reals (orelements ofH(ℵ1)) in M[HMα ], so it does

not make any difference whether we useM[HMα ] or its transitive collapse.

Remark 3.3. In the discussion so far we omitted some details regarding the theory ZFC∗ (that a candidatehas to satisfy). The following “fine print” hopefully absolves us from any liability. (It is entirely irrelevantfor the understanding of the paper.)

We have to guarantee that eachM[HMα ] that we consider satisfies enough of ZFC to make our arguments

work (for example, the definitions and basic properties of ultralaver and Janus forcings should work). Thisturns out to be easy, since (as usual) we do not need the full power set axiom for these arguments (just theexistence of, say,i5). So it is enough that eachM[HM

α ] satisfies some fixed finite subset of ZFC minuspower set, which we call ZFC∗.

Of course we can also find a bigger (still finite) set ZFC∗∗ that implies:i10 exists, and each forcingextension of the universe with a forcing of size≤ i4 satisfies ZFC∗. And it is provable (in ZFC) that eachH(χ) satisfies ZFC∗∗ for sufficiently large regularχ.

We define “candidate” using the weaker theory ZFC∗, and require that nice candidates satisfy thestronger theory ZFC∗∗. This guarantees that all forcing extensions (by small forcings) of nice candidateswill be candidates (in particular, satisfy enough of ZFC such that our arguments about Janus or ultralaverforcings work). Also, every ord-collapse of a countable elementary submodelN of H(χ) will be a nicecandidate.

3.B. Partial countable support iterations. We introduce the notion of “partial countable support limit”:a subset of the countable support (CS) limit containing the union (i.e., the direct limit) and satisfying somenatural requirements.

Let us first describe what we mean by “forcing iteration”. They have to satisfy the following require-ments:

• A “ topless forcing iteration” ( Pα,Qα)α<ε is a sequence of forcing notionsPα andPα-namesQαof quasiorders with a weakest element 1Qα . A “ topped iteration” additionally has a final limitPε.EachPα is a set of partial functions onα (as, e.g., in [Gol93]). More specifically, ifα < β ≤ εand p ∈ Pβ, thenp↾α ∈ Pα. Also, p↾β Pβ p(β) ∈ Qβ for all β ∈ dom(p). The order onPβ willalways be the “natural” one:q ≤ p iff q↾α forces (inPα) thatqtot(α) ≤ ptot(α) for all α < β, wherer tot(α) = r(α) for all α ∈ dom(r) and 1Qα otherwise.Pα+1 consists ofall p with p↾α ∈ Pα andp↾α ptot(α) ∈ Qα, so it is forcing equivalent toPα ∗ Qα.

• Pα ⊆ Pβ wheneverα < β ≤ ε. (In particular, the empty condition is an element of eachPβ.)• For anyp ∈ Pε and anyq ∈ Pα (α < ε) with q ≤ p↾α, the partial functionq∧ p≔ q∪ p↾[α, ε) is a

condition inPε as well (so in particular,p↾α is a reduction ofp, hencePα is a complete subforcingof Pε; andq∧ p is the weakest condition inPε stronger than bothq andp).• Abusing notation, we usually just writeP for an iteration (be it topless or topped).• We usually writeHβ for the generic filter onPβ (which inducesPα-generic filters calledHα forα ≤ β). For topped iterations we call the filter on the final limit sometimes justH instead ofHε.

We use the following notation for quotients of iterations:

• For α < β, in the Pα-extensionV[Hα], we let Pβ/Hα be the set of allp ∈ Pβ with p↾α ∈ Hα(ordered as inPβ). We may occasionally writePβ/Pα for thePα-name ofPβ/Hα.• SincePα is a complete subforcing ofPβ, this is a quotient with the usual properties, in particular

Pβ is equivalent toPα ∗ (Pβ/Hα).

Remark 3.4. It is well known that quotients of proper countable support iterations are naturally equivalentto (names of) countable support iterations. In this paper, we can restrict our attention to proper forcings, butwe do not really have countable support iterations. It turnsout that it is not necessary to investigate whether


our quotients can naturally be seen as iterations of any kind, so to avoid the subtle problems involved wewill not consider the quotient as an iteration by itself.

Definition 3.5. Let P be a (topless) iteration of limit lengthε. We define three limits ofP:

• The “direct limit” is the union of thePα (for α < ε). So this is the smallest possible limit of theiteration.• The “inverse limit” consists ofall partial functionsp with domain⊆ ε such thatp↾α ∈ Pα for allα < ε. This is the largest possible limit of the iteration.• The “full countable support limit PCS

ε ” of P is the inverse limit if cf(ε) = ω and the direct limitotherwise.

We say thatPε is a “partial CS limit”, if Pε is a subset of the full CS limit and the sequence (Pα)α≤ε is atopped iteration. In particular, this means thatPε contains the direct limit, and satisfies the following foreachα < ε: Pε is closed underp 7→ p↾α, and wheneverp ∈ Pε, q ∈ Pα, q ≤ p↾α, then also the partialfunctionq∧ p is in Pε.

So for a given toplessP there is a well-defined inverse, direct and full CS limit. If cf(ε) > ω, then thedirect and the full CS limit coincide. If cf(ε) = ω, then the direct limit and the full CS limit (=inverse limit)differ. Both of them are partial CS limits, but there are many morepossibilities for partial CS limits. Bydefinition, all of them will yield iterations.

Note that the name “CS limit” is slightly inappropriate, as the size of supports of conditions is not partof the definition. To give a more specific example: Consider a topped iterationP of lengthω+ω wherePωis the direct limit andPω+ω is the full CS limit. Letp be any element of the full CS limit ofP↾ω which isnot in Pω; thenp is not inPω+ω either. So not every countable subset ofω+ω can appear as the support ofa condition.

Definition 3.6. A forcing iterationP is called a “partial CS iteration”, if

• every limit is a partial CS limit, and• everyQα is (forced to be) separative.27

The following fact can easily be proved by transfinite induction:

Fact 3.7. Let P be a partial CS iteration. Then for allα the forcing notionPα is separative.

From now on, all iterations we consider will be partial CS iterations. In this paper, we will only beinterested in proper partial CS iterations, but propernessis not part of the definition of partial CS iteration.(The reader may safely assume that all iterations are proper.)

Note that separativity of theQα implies that all partial CS iterations satisfy the following (triviallyequivalent) properties:

Fact 3.8. Let P be a topped partial CS iteration of lengthε. Then:

(1) Let H bePε-generic. Thenp ∈ H iff p↾α ∈ Hα for all α < ε.(2) For allq, p ∈ Pε: If q↾α ≤∗ p↾α for eachα < ε, thenq ≤∗ p.(3) For allq, p ∈ Pε: If q↾α ≤∗ p↾α for eachα < ε, thenq ‖ p.

We will be concerned with the following situation:Assume thatM is a nice candidate,PM is (in M) a topped partial CS iteration of lengthε (a limit ordinal

in M), andP is (in V) a topless partial CS iteration of lengthε′ ≔ sup(ε ∩ M). (Recall that “cf(ε) = ω”is absolute betweenM andV, and that cf(ε) = ω impliesε′ = ε.) Moreover, assume that we already havea system ofM-complete coherent28 embeddingsiβ : PM

β→ Pβ for β ∈ ε′ ∩ M = ε ∩ M. (Recall that any

potential partial CS limit ofP is a subforcing of the full CS limitPCSε′

.) It is easy to see that there is only onepossibility for an embeddingj : PM

ε → PCSε′ (in fact, into any potential partial CS limit ofP) that extends

the iβ’s naturally:

27The reason for this requirement is briefly discussed in Section 6. Separativity, as well as the relations≤∗ and=∗, are defined onpage 3.

28I.e., they commute with the restriction maps:iα(p↾α) = iβ(p)↾α for α < β andp ∈ PMβ

.


Definition 3.9. For a topped partial CS iterationPM in M of lengthε and a topless oneP in V of lengthε′ ≔ sup(ε∩M) together with coherent embeddingsiβ, we definej : PM

ε → PCSε′

, the “canonical extension”,in the obvious way: Givenp ∈ PM

ε , take the sequence of restrictions toM-ordinals, apply the functionsiβ,and let j(p) be the union of the resulting coherent sequence.

We do not claim thatj : PMε → PCS

ε′ is M-complete.29 In the following, we will construct partial CSlimits Pε′ such thatj : PM

ε → Pε′ is M-complete. (Obviously, one requirement for such a limit is thatj[PMε ] ⊆ Pε′ .) We will actually define two versions: The almost FS (“almost finite support”) and the almost

CS (“almost countable support”) limit.Note that there is only one effect that the “top” ofPM (i.e., the forcingPM

ε ) has on the canonical exten-sion j: It determines the domain ofj. In particular it will generally depend onPM

ε whether j is completeor not. Apart from that, the value of any givenj(p) does not depend onPM

ε .Instead of arbitrary systems of embeddingsiα, we will only be interested in “canonical” ones. We as-

sume for notational convenience thatQMα is a subset ofQα (this will naturally be the case in our application

anyway).

Definition 3.10 (The canonical embedding). Let P be a partial CS iteration inV and PM a partial CSiteration in M, both topped and of lengthε ∈ M. We construct by induction onα ∈ (ε + 1) ∩ M thecanonicalM-complete embeddingsiα : PM

α → Pα. More precisely: We try to construct them, but itis possible that the construction fails. If the construction succeeds, then we say that “PM (canonically)embeds intoP”, or “ the canonical embeddings work”, or just: “P is overPM”, or “ over PM

ε ”.

• Let α = β + 1. By induction hypothesis,iβ is M-complete, so aV-generic filterHβ ⊆ Pβ inducesanM-generic filterHM

β≔ i−1

β[Hβ] ⊆ PM

β. We require that (in theHβ extension) the setQM

β[HMβ

] isanM[HM

β]-complete subforcing ofQβ[Hβ]. In this case, we defineiα in the obvious way.

• Forα limit, let iα be the canonical extension of the family (iβ)β∈α∩M. We require thatPα contains therange ofiα, and thatiα is M-complete; otherwise the construction fails. (Ifα′ ≔ sup(α ∩ M) < α,theniα will actually be anM-complete map intoPα′ , assuming that the requirement is fulfilled.)

In this section we try to construct a partial CS iterationP (over a givenPM) satisfying additional prop-erties.

Remark 3.11. What is the role ofε′ ≔ sup(ε ∩ M)? When our inductive construction ofP arrives atPεwhereε′ < ε, it would be too late30 to take care ofM-completeness ofiε at this stage, even if alliαwork nicely for α ∈ ε ∩ M. Note thatε′ < ε implies thatε is uncountable inM, and that thereforePMε =⋃

α∈ε∩M PMα . So the natural extensionj of the embeddings (iα)α∈ε∩M has range inPε′ , which will be

a complete subforcing ofPε. So we have to ensureM-completeness already in the construction ofPε′ .

For now we just record:

Lemma 3.12. Assume that we have topped iterationsPM (in M) of lengthε andP (in V) of lengthε′ ≔sup(ε ∩ M), and that for allα ∈ ε ∩ M the canonical embedding iα : PM

α → Pα works. Let iε : PMε → PCS

ε′

be the canonical extension.

(1) If PMε is (in M) a direct limit (which is always the case ifε has uncountable cofinality) then iε

(might not work, but at least) has range in Pε′ and preserves incompatibility.(2) If iε has a range contained in Pε′ and maps predense sets D⊆ PM

ε in M to predense sets iε[D] ⊆Pε′ , then iε preserves incompatibility (and therefore works).

29 For example, ifε = ε′ = ω and if PMω is the finite support limit of a nontrivial iteration, thenj : PM

ω → PCSω is not complete:

For notational simplicity, assume that allQMn are (forced to be) Boolean algebras. InM, let cn be (aPM

n -name for) a nontrivialelement ofQM

n (so¬cn, the Boolean complement, is also nontrivial). Letpn be thePMn -condition (c0, . . . , cn−1), i.e., the truth value

of “cm ∈ H(m) for all m< n”. Let qn be thePMn+1-condition (c0, . . . , cn−1,¬cn), i.e., the truth value of “n is minimal withcn < H(n)”.

In M, the setA = {qn : n ∈ ω} is a maximal antichain inPMω . Moreover, the sequence (pn)n∈ω is a decreasing coherent sequence,

thereforein(pn) defines an elementpω in PCSω , which is clearly incompatible with allj(qn), hencej[A] is not maximal.

30 For example: Letε = ω1 andε′ = ω1 ∩ M. Assume thatPMω1

is (in M) a (or: the unique) partial CS limit of a nontrivialiteration. Assume that we have a topless iterationP of lengthε′ in V such that the canonical embeddings work for allα ∈ ω1 ∩ M.If we setPε′ to be the full CS limit, then we cannot further extend it to anyiteration of lengthω1 such that the canonical embeddingiω1 works: Letpα andqα be as in footnote 29. InM, the setA = {qα : α ∈ ω1} is a maximal antichain, and the sequence (pα)α∈ω1 isa decreasing coherent sequence. But inV there is an elementpε′ ∈ PCS

ε′with pε′↾α = j(pα) for all α ∈ ε ∩ M. This conditionpε′ is

clearly incompatible with all elements ofj[A] = { j(qα) : α ∈ ε ∩ M}. Hencej[A] is not maximal.


Proof. (1) SincePMε is a direct limit, the canonical extensioniε has range in

⋃

α<ε′ Pα, which is subset ofany partial CS limitPε′ . Incompatibility inPM

ε is the same as incompatibility inPMα for sufficiently large

α ∈ ε ∩ M, so by assumption it is preserved byiα and hence also byiε.(2) Fix p1, p2 ∈ PM

ε , and assume that their images are compatible inPε′ ; we have to show that they arecompatible inPM

ε . So fix a generic filterH ⊆ Pε′ containingiε(p1) andiε(p2).In M, we define the following setD:

D ≔ {q ∈ PMε : (q ≤ p1 ∧ q ≤ p2) or (∃α < ε : q↾α ⊥PM

αp1↾α) or (∃α < ε : q↾α ⊥PM

αp2↾α)}.

Using Fact 3.8(3) it is easy to check thatD is dense. Sinceiε preserves predensity, there isq ∈ Dsuch thatiε(q) ∈ H. We claim thatq is stronger thanp1 and p2. Otherwise we would have without lossof generalityq↾α ⊥PM

αp1↾α for someα < ε. But the filterH↾α contains bothiα(q↾α) and iα(p1↾α),

contradicting the assumption thatiα preserves incompatibility. �

3.C. Almost finite support iterations. Recall Definition 3.9 (of the canonical extension) and the setupthat was described there: We have to find a subsetPε′ of PCS

ε′such that the canonical extensionj : PM

ε → Pε′is M-complete.

We now define the almost finite support limit. (The direct limit will in general not do, as it may notcontain the rangej[PM

ε ]. The almost finite support limit is the obvious modificationof the direct limit, andit is the smallest partial CS limitPε′ such thatj[PM

ε ] ⊆ Pε′ , and it indeed turns out to beM-complete aswell.)

Definition 3.13. Let ε be a limit ordinal inM, and letε′ ≔ sup(ε∩M). Let PM be a topped iteration inMof lengthε, and letP be a topless iteration inV of lengthε′. Assume that the canonical embeddingsiαwork for all α ∈ ε ∩ M = ε′ ∩ M. Let iε be the canonical extension. We define thealmost finite supportlimit of P overPM (or: almost FS limit) as the following subforcingPε′ of PCS

ε′:

Pε′ ≔ {q∧ iε(p) ∈ PCSε′ : p ∈ PM

ε andq ∈ Pα for someα ∈ ε ∩ M such thatq ≤Pα iα(p↾α) }.

Note that for cf(ε) > ω, the almost FS limit is equal to the direct limit, as eachp ∈ PMε is in fact inPM

α

for someα ∈ ε ∩ M, soiε(p) = iα(p) ∈ Pα.

Lemma 3.14. Assume thatP andPM are as above and let Pε′ be the almost FS limit. ThenP⌢Pε′ is apartial CS iteration, and iε works, i.e., iε is an M-complete embedding from PM

ε to Pε′ . (As Pε′ is a completesubforcing of Pε, this also implies that iε is M-complete from PMε to Pε.)

Proof. It is easy to see thatPε′ is a partial CS limit and contains the rangeiε[PMε ]. We now show preserva-

tion of predensity; this impliesM-completeness by Lemma 3.12.Let (p j) j∈J ∈ M be a maximal antichain inPM

ε . (SincePMε does not have to be ccc inM, J can have any

cardinality inM.) Let q∧ iε(p) be a condition inPε′ . (If ε′ < ε, i.e., if cf(ε) > ω, then we can choosep tobe the empty condition.) Fixα ∈ ε∩M be such thatq ∈ Pα. Let Hα bePα-generic and containq, sop↾α isin HM

α . Now in M[HMα ] the set{p j : j ∈ J, p j ∈ PM

ε /HMα } is predense inPM

ε /HMα (since this is forced by the

empty condition inPMα ). In particular,p is compatible with somep j , witnessed byp′ ≤ p, p j in PM

ε /HMα .

We can findq′ ≤Pα q deciding j andp′; since certainlyq′ ≤∗ iα(p′↾α), we may assume even≤ withoutloss of generality. Nowq′ ∧ iε(p′) ≤ q ∧ iε(p) (sinceq′ ≤ q andp′ ≤ p), andq′ ∧ iε(p′) ≤ iε(p j) (sincep′ ≤ p j). �

Definition and Claim 3.15. Let PM be a topped partial CS iteration inM of lengthε. We can construct byinduction onβ ∈ ε + 1 analmost finite support iterationP overPM (or: almost FS iteration) as follows:

(1) As induction hypothesis we assume that the canonical embeddingiα works for allα ∈ β ∩ M. (Sothe notationM[HM

α ] makes sense.)(2) Let β = α + 1. If α ∈ M, then we can use anyQα provided that (it is forced that)QM

α is anM[HM

α ]-complete subforcing ofQα. (If α < M, then there is no restriction onQα.)(3) Letβ ∈ M and cf(β) = ω. ThenPβ is the almost FS limit of (Pα,Qα)α<β overPM

β.

(4) Let β ∈ M and cf(β) > ω. ThenPβ is again the almost FS limit of (Pα,Qα)α<β overPMβ

(whichalso happens to be the direct limit).

(5) For limit ordinals not inM, Pβ is the direct limit.


So the claim includes that the resultingP is a (topped) partial CS iteration of lengthε over PM (i.e.,the canonical embeddingsiα work for all α ∈ (ε + 1)∩ M), where we only assume that theQα satisfy theobvious requirement given in (2). (Note that we can always find some suitableQα for α ∈ M, for examplewe can just takeQM

α itself.)

Proof. We have to show (by induction) that the resulting sequenceP is a partial CS iteration, and thatPM

embeds intoP. For successor cases, there is nothing to do. So assume thatα is a limit. If Pα is a directlimit, it is trivially a partial CS limit; if Pα is an almost FS limit, then the easy part of Lemma 3.14 showsthat it is a partial CS limit.

So it remains to show that for a limitα ∈ M, the (naturally defined) embeddingiα : PMα → Pα is

M-complete. This was the main claim in Lemma 3.14. �

The following lemma is natural and easy.

Lemma 3.16. Assume that we construct an almost FS iterationP overPM where each Qα is (forced to be)ccc. Then Pε is ccc (and in particular proper).

Proof. We show thatPα is ccc by induction onα ≤ ε. For successors, we use thatQα is ccc. Forα ofuncountable cofinality, we know that we took the direct limitcoboundedly often (and allPβ are ccc forβ < α), so by a result of SolovayPα is again ccc. Forα a limit of countable cofinality not inM, just usethat all Pβ are ccc forβ < α, and the fact thatPα is the direct limit. This leaves the case thatα ∈ M hascountable cofinality, i.e., thePα is the almost FS limit. LetA ⊆ Pα be uncountable. Eacha ∈ A has theform q∧ iα(p) for p ∈ PM

α andq ∈⋃

γ<α Pγ. We can thin out the setA such thatp are the same and allqare in the samePγ. So there have to be compatible elements inA. �

All almost FS iterations that we consider in this paper will satisfy the countable chain condition (andhence in particular be proper).

We will need a variant of this lemma forσ-centered forcing notions.

Lemma 3.17. Assume that we construct an almost FS iterationP overPM where only countably many Qαare nontrivial (e.g., only those withα ∈ M) and where each Qα is (forced to be)σ-centered. Then Pε isσ-centered as well.

Proof. By induction: The direct limit of countably manyσ-centered forcings isσ-centered, as is the almostFS limit ofσ-centered forcings (to colorq∧ iα(p), usep itself together with the color ofq). �

We will actually need two variants of the almost FS construction: Countably many modelsMn; andstarting the almost FS iteration with someα0.

Firstly, we can construct an almost FS iteration not just over one iterationPM, but over an increasingchain of iterations. Analogously to Definition 3.13 and Lemma 3.14, we can show:

Lemma 3.18. For each n∈ ω, let Mn be a nice candidate, and letPn be a topped partial CS iterationin Mn of length31 ε ∈ M0 of countable cofinality, such that Mm ∈ Mn and Mn thinks thatPm canonicallyembeds intoPn, for all m < n. LetP be a topless iteration of lengthε into which allPn canonically embed.

Then we can define the almost FS limit Pε over (Pn)n∈ω as follows: Conditions in Pε are of the formq∧ inε(p) where n∈ ω, p ∈ Pn

ε, and q∈ Pα for someα ∈ Mn ∩ ε with q≤ inα(p↾α). Then Pε is a partial CSlimit over eachPn.

As before, we get the following corollary:

Corollary 3.19. Given Mn and Pn as above, we can construct a topped partial CS iterationP such thateachPn embeds Mn-completely into it; we can choose Qα as we wish (subject to the obvious restrictionthat each Qn

α is an Mn[Hnα]-complete subforcing). If we always choose Qα to be ccc, thenP is ccc; this is

the case if we set Qα to be the union of the (countable) sets Qnα.

Proof. We can definePα by induction. Ifα ∈⋃

n∈ω Mn has countable cofinality, then we use the almostFS limit as in Lemma 3.18. Otherwise we use the direct limit. If α ∈ Mn has uncountable cofinality,thenα′ ≔ sup(α ∩ M) is an element ofMn+1. In our induction we have already consideredα′ and have

31Or only: ε ∈ Mn0 for somen0.


definedPα′ by Lemma 3.18 (applied to the sequence (Pn+1, Pn+2, . . .)). This is sufficient to show thatinα : Pn

α → Pα′ ⋖ Pα is Mn-complete. �

Secondly, we can start the almost FS iteration after someα0 (i.e., P is already given up toα0, and wecan continue it as an almost FS iteration up toε), and get the same properties that we previously showedfor the almost FS iteration, but this time for the quotientPε/Pα0. In more detail:

Lemma 3.20. Assume thatPM is in M a (topped) partial CS iteration of lengthε, and thatP is in V atopped partial CS iteration of lengthα0 over PM↾α0 for someα0 ∈ ε ∩ M. Then we can extendP to a(topped) partial CS iteration of lengthε over PM, as in the almost FS iteration (i.e., using the almost FSlimit at limit pointsβ > α0 with β ∈ M of countable cofinality; and the direct limit everywhere else). Wecan use any Qα for α ≥ α0 (provided QM

α is an M[HMα ]-complete subforcing of Qα). If all Qα are ccc, then

Pα0 forces that Pε/Hα0 is ccc (in particular proper); if moreover all Qα areσ-centered and only countablymany are nontrivial, then Pα0 forces that Pε/Hα0 isσ-centered.

3.D. Almost countable support iterations. “Almost countable support iterationsP” (over a given itera-tion PM in a candidateM) will have the following two crucial properties: There is a canonicalM-completeembedding ofPM into P, and P preserves a given random real (similar to the usual countable supportiterations).

Definition and Claim 3.21. Let PM be a topped partial CS iteration inM of lengthε. We can construct byinduction onβ ∈ ε + 1 thealmost countable support iterationP overPM (or: almost CS iteration):

(1) As induction hypothesis, we assume that the canonical embeddingiα works for everyα ∈ β ∩ M.We set32

(3.22) δ ≔ min(M \ β), δ′ ≔ sup(α + 1 : α ∈ δ ∩ M).

Note thatδ′ ≤ β ≤ δ.(2) Letβ = α + 1. We can choose any desired forcingQα; if β ∈ M we of course require that

(3.23) QMα is anM[HM

α ]-complete subforcing ofQα.

This definesPβ.(3) Let cf(β) > ω. ThenPβ is the direct limit.(4) Let cf(β) = ω and assume thatβ ∈ M (soM ∩ β is cofinal inβ andδ′ = β = δ). We definePβ = Pδ

as the union of the following two sets:• The almost FS limit of (Pα,Qα)α<δ, see Definition 3.13.• The setPgen

δof M-generic conditionsq ∈ PCS

δ, i.e., those which satisfy

q PCSδ

i−1δ [HPCS

δ] ⊆ PM

δ is M-generic.

(5) Let cf(β) = ω and assume thatβ < M but M ∩ β is cofinal inβ, soδ′ = β < δ. We definePβ = Pδ′as the union of the following two sets:• The direct limit of (Pα,Qα)α<δ′ .• The setPgen

δ′of M-generic conditionsq ∈ PCS

δ′, i.e., those which satisfy

q PCSδ′

i−1δ [HPCS

δ′] ⊆ PM

δ is M-generic.

(Note that theM-generic conditions form an open subset ofPCSβ= PCS

δ′.)

(6) Let cf(β) = ω andM ∩ β not cofinal inβ (soβ < M). ThenPβ is the full CS limit of (Pα,Qα)α<β(see Definition 3.5).

So the claim is that for every choice ofQα (with the obvious restriction (3.23)), this construction alwaysresults in a partial CS iterationP overPM. The proof is a bit cumbersome; it is a variant of the usual proofthat properness is preserved in countable support iterations (see e.g. [Gol93]).

We will use the following fact inM (for the iterationPM):

(3.24)

Let P be a topped iteration of lengthε. Let α1 ≤ α2 ≤ β ≤ ε. Let p1 be aPα1-namefor a condition inPε, and letD be an open dense set ofPβ. Then there is aPα2-namep2 for a condition inD such that the empty condition ofPα2 forces: p2 ≤ p1↾β and:if p1 is in Pε/Hα2, then the conditionp2 is as well.

32 So for successorsβ ∈ M, we haveδ′ = β = δ. Forβ ∈ M limit, β = δ andδ′ is as in Definition 3.9.


(Proof: Work in thePα2-extension. We know thatp′ ≔ p1 ↾ β is aPβ-condition. We now definep2 asfollows: If p′ < Pβ/Hα2 (which is equivalent top1 < Pǫ/Hα2), then we choose anyp2 ≤ p′ in D (which isdense inPβ). Otherwise (using thatD∩Pβ/Hα2 is dense inPβ/Hα2) we can choosep2 ≤ p′ in D∩Pβ/Hα2.)

The following easy fact will also be useful:

(3.25)Let P be a subforcing ofQ. We defineP↾p ≔ {r ∈ P : r ≤ p}. Assume thatp ∈ PandP↾p = Q↾p.Then for anyP-name

˜x and any formulaϕ(x) we have:p P ϕ(

˜x) iff p Q ϕ(

˜x).

We now prove by induction onβ ≤ ε the following statement (which includes that the DefinitionandClaim 3.21 works up toβ). Let δ, δ′ be as in (3.22).

Lemma 3.26. (a) The topped iterationP of lengthβ is a partial CS iteration.(b) The canonical embedding iδ : PM

δ→ Pδ′ works, hence also iδ : PM

δ→ Pδ works.

(c) Moreover, assume that• α ∈ M ∩ δ,•

˜p ∈ M is a PM

α -name of a PMδ

-condition,• q ∈ Pα forces (in Pα) that

˜p↾α[HM

α ] is in HMα .

Then there is a q+ ∈ Pδ′ (and therefore in Pβ) extending q and forcing that˜p[HM

α ] is in HMδ

.

Proof. First let us deal with the trivial cases. It is clear that we always get a partial CS iteration.

• Assume thatβ = β0 + 1 ∈ M, i.e.,δ = δ′ = β. It is clear thatiβ works. To getq+, first extendq tosomeq′ ∈ Pβ0 (by induction hypothesis), then defineq+ extendingq′ by q+(β0) ≔

˜p(β0).

• If β = β0 + 1 < M, there is nothing to do.• Assume that cf(β) > ω (whetherβ ∈ M or not). Thenδ′ < β. Soiδ : PM

δ→ Pδ′ works by induction,

and similarly (c) follows from the inductive assumption. (Use the inductive assumption forβ = δ′;theδ that we got at that stage is the same as the currentδ, and theq+ we obtained at that stage willstill satisfy all requirements at the current stage.)• Assume that cf(β) = ω and thatM ∩ β is bounded inβ. Then the proof is the same as in the

previous case.

We are left with the cases corresponding to (4) and (5) of Definition 3.21: cf(β) = ω andM∩β is cofinalin β. So eitherβ ∈ M, thenδ′ = β = δ, or β < M, thenδ′ = β < δ and cf(δ) > ω.

We leave it to the reader to check thatPβ is indeed a partial CS limit. The main point is to see thatfor all p, q ∈ Pβ the conditionq ∧ p is in Pβ as well, providedq ∈ Pα andq ≤ p↾α for someα < β. Ifp ∈ Pgen

β, then this follows becausePgen

βis open inPCS

β; the other cases are immediate from the definition

(by induction).We now turn to claim (c). Assumeq ∈ Pα and

˜p ∈ M are given,α ∈ M ∩ δ.

Let (Dn)n∈ω enumerate all dense sets ofPMδ

which lie in M, and let (αn)n∈ω be a sequence of ordinals inM which is cofinal inβ, whereα0 = α.

Using (3.24) inM, we can find a sequence (˜pn)n∈ω satisfying the following inM, for all n > 0:

•˜p0 =

˜p.

•˜pn ∈ M is aPM

αn-name of aPM

δ-condition inDn.

• PMαn ˜

pn ≤PMδ ˜

pn−1.• PM

αnIf

˜pn−1↾αn ∈ HM

αn, then

˜pn↾αn ∈ HM

αnas well.

Using the inductive assumption for theαn’s, we can now find a sequence (qn)n∈ω of conditions satisfyingthe following:

• q0 = q, qn ∈ Pαn.• qn↾αn−1 = qn−1.• qn Pαn

˜pn−1↾αn ∈ HM

αn, so also

˜pn↾αn ∈ HM

αn.

Let q+ ∈ PCSβ

be the union of theqn. Then for alln:

(1) qn PCSβ ˜

pn↾αn ∈ HMαn

, so alsoq+ forces this.(Using induction onn.)

(2) For alln and allm≥ n: q+ PCSβ ˜

pm↾αm ∈ HMαm

, so also˜pn↾αm ∈ HM

αm.

(As˜pm ≤

˜pn.)


(3) q+ PCSβ ˜

pn ∈ HMδ

.

(Recall thatPCSβ

is separative, see Fact 3.7. Soiδ(˜pn) ∈ Hδ iff iαn(

˜p↾αm) ∈ Hαm for all largem.)

As q+ PCSβ ˜

pn ∈ Dn∩HMδ

, we conclude thatq+ ∈ Pgenβ

(using Lemma 3.12, applied toPCSβ

). In particular,

Pgenβ

is dense inPβ: Let q∧ iδ(p) be an element of the almost FS limit; soq ∈ Pα for someα < β. Now finda genericq+ extendingq and stronger thaniδ(p), thenq+ ≤ q∧ iδ(p).

It remains to show thatiδ is M-complete. LetA ∈ M be a maximal antichain ofPMδ

, andp ∈ Pβ. Assumetowards a contradiction thatp forces inPβ that i−1

δ[Hβ] does not intersectA in exactly one point.

SincePgenβ

is dense inPβ, we can find someq ≤ p in Pgenβ

. Let

P′ ≔ {r ∈ PCSβ : r ≤ q} = {r ∈ Pβ : r ≤ q},

where the equality holds becausePgenβ

is open inPCSβ

.

Let Γ be the canonical name for aP′-generic filter, i.e.:Γ ≔ {(r, r) : r ∈ P′}. Let R be eitherPCSβ

or Pβ.We write〈Γ〉R for the filter generated byΓ in R, i.e.,〈Γ〉R≔ {r ∈ R : (∃r ′ ∈ Γ) r ′ ≤ r}. So

(3.27) q R HR = 〈Γ〉R.

We now see that the following hold:

– q Pβ i−1δ

[HPβ ] does not intersectA in exactly one point. (By assumption.)– q Pβ i−1

δ[〈Γ〉Pβ ] does not intersectA in exactly one point. (By (3.27).)

– q PCSβ

i−1δ

[〈Γ〉Pβ ] does not intersectA in exactly one point. (By (3.25).)

– q PCSβ

i−1δ

[〈Γ〉PCSβ

] does not intersectA in exactly one point. (Becauseiδ mapsA into Pβ ⊆ PCSβ

, so

A∩ i−1δ

[〈Y〉Pβ ] = A∩ i−1δ

[〈Y〉PCSβ

] for all Y.)

– q PCSβ

i−1δ

[HPCSβ

] does not intersectA in exactly one point. (Again by (3.27).)

But this, according to the definition ofPgenβ

, impliesq < Pgenβ

, a contradiction. �

We can also show that the almost CS iteration of proper forcingsQα is proper. (We do not really needthis fact, as we could allow non-proper iterations in our preparatory forcing, see Section 6.A(4). In somesense,M-completeness replaces properness, so the proof ofM-completeness was similar to the “usual”proof of properness.)

Lemma 3.28. Assume that in Definition 3.21, every Qα is (forced to be) proper. Then also each Pδ isproper.

Proof. By induction onδ ≤ ε we prove that for allα < δ the quotientPδ/Hα is (forced to be) proper. Weuse the following facts about properness:

(3.29) If P is proper andP forces thatQ is proper, thenP ∗ Q is proper.

(3.30)If P is an iteration of lengthω and if eachQn is forced to be proper, then the inverselimit Pω is proper, as are all quotientsPω/Hn.

(3.31)If P is an iteration of lengthδ with cf(δ) > ω, and if all quotientsPβ/Hα (for α <β < δ) are forced to be proper, then the direct limitPδ is proper, as are all quotientsPδ/Hα.

If δ is a successor, then our inductive claim easily follows fromthe inductive assumption togetherwith (3.29).

Let δ be a limit of countable cofinality, sayδ = supn δn. Define an iterationP′ of lengthω withQ′n ≔ Pδn+1/Hδn. (EachQ′n is proper, by inductive assumption.) There is a natural forcing equivalencebetweenPCS

δandP′CS

ω , the full CS limit of P′.Let N ≺ H(χ∗) containP,Pδ, P′,M, PM. Let p ∈ Pδ ∩ N. Without loss of generalityp ∈ Pgen

δ. So below

p we can identifyPδ with PCSδ

and hence withP′CSω ; now apply (3.30).

The case of uncountable cofinality is similar, using (3.31) instead. �


Recall the definition of⊏n and⊏ from Definition 1.26, the notion of (quick) interpretationZ∗ (of a name

˜Z of a code for a null set) and the definition of local preservation of randoms from Definition 1.50. Recallthat we have seen in Corollaries 1.51 and 2.22:

Lemma 3.32. • If QM is an ultralaver forcing in M and r a real, then there is an ultralaver forcingQ over QM locally preserving randomness of r over M.

• If QM is a Janus forcing in M and r a real, then there is a Janus forcing Q over QM locallypreserving randomness of r over M.

We will prove the following preservation theorem:

Lemma 3.33. Let P be an almost CS iteration (of lengthε) over PM, r random over M, and p∈ PMε .

Assume that each Pα forces that Qα locally preserves randomness of r over M[HMα ]. Then there is some

q ≤ p in Pε forcing that r is random over M[HMε ].

What we will actually need is the following variant:

Lemma 3.34. Assume thatPM is in M a topped partial CS iteration of lengthε, and we already have sometopped partial CS iterationP overPM↾α0 of lengthα0 ∈ M ∩ ε. Let

˜r be a Pα0-name of a random real

over M[HMα0

]. Assume that we extendP to lengthε as an almost CS iteration33 using forcings Qα which

locally preserve the randomness of˜r over M[HM

α ], witnessed by a sequence(DQMα

k )k∈ω. Let p∈ PMε . Then

we can find a q≤ p in Pε forcing that˜r is random over M[HM

ε ].

Actually, we will only prove the two previous lemmas under the following additional assumption (whichis enough for our application, and saves some unpleasant work). This additional assumption is not reallynecessary; without it, we could use the method of [GK06] for the proof.

Assumption 3.35. • For eachα ∈ M ∩ ε, (PMα forces that)QM

α is either trivial34 or adds a newω-sequence of ordinals. Note that in the latter case we can assume without loss of generality that⋂

n∈ω DQMα

n = ∅ (and, of course, that theDQMα

n are decreasing).• Moreover, we assume that already inM there is a setT ⊆ ε such thatPM

α forces:QMα is trivial iff

α ∈ T. (So whetherQMα is trivial or not does not depend on the generic filter belowα, it is already

decided in the ground model.)

The result will follow as a special case of the following lemma, which we prove by induction onβ.(Note that this is a refined version of the proof of Lemma 3.26 and similar to the proof of the preservationtheorem in [Gol93, 5.13].)

Definition 3.36. Under the assumptions of Lemma 3.34 and Assumption 3.35, let˜Z be aPδ-name,α0 ≤

α < δ, and letp = (pk)k∈ω be a sequence ofPα-names of conditions inPδ/Hα. Let Z∗ be aPα-name.We say that ( ¯p,Z∗) is a quick interpretation of

˜Z if p interprets

˜Z asZ∗ (i.e., Pα forces thatpk forces

˜Z↾k = Z∗↾k for all k), and moreover:

Lettingβ ≥ α be minimal withQMβ

nontrivial (if suchβ exists):Pβ forces that the sequence

(pk(β))k∈ω is quick inQMβ

, i.e., pk(β) ∈ DQMβ

k for all k.

It is easy to see that:

(3.37) For every name˜Z there is a quick interpretation ( ¯p,Z∗).

Lemma 3.38. Under the same assumptions as above, letβ, δ, δ′ be as in(3.22)(so in particular we haveδ′ ≤ β ≤ δ ≤ ε).Assume that

• α ∈ M ∩ δ (= M ∩ β) andα ≥ α0 (soα < δ′),• p ∈ M is a PM

α -name of a PMδ

-condition,•

˜Z ∈ M is a PM

δ-name of a code for null set,

33Of course our official definition of almost CS iteration assumes that we start the construction at 0, so we modify this definitionin the obvious way.

34More specifically,QMα = {∅}.


• Z∗ ∈ M is a PMα -name of a code for a null set,

• PMα forces: p = (pk)k∈ω ∈ M is a quick sequence in PM

δ/HMα interpreting

˜Z as Z∗ (as in Defini-

tion 3.36),• PM

α forces: if p↾α ∈ HMα , then p0 ≤ p,

• q ∈ Pα forces p↾α ∈ HMα ,

• q forces that r is random over M[HMα ], so in particular there is (in V) a Pα-name

˜c0 below q for

the minimal c with Z∗ ⊏c r.

Then there is a condition q+ ∈ Pδ′ , extending q, and forcing the following:

• p ∈ HMδ

,• r is random over M[HM

δ],

•˜Z ⊏

˜c0 r.

We actually claim a slightly stronger version, where instead of Z∗ and˜Z we have finitely many codes

for null sets and names of codes for null sets, respectively.We will use this stronger claim as inductiveassumption, but for notational simplicity we only prove theweaker version; it is easy to see that the weakerversion implies the stronger version.

Proof. The nontrivial successor case: β = γ + 1 ∈ M.If QM

γ is trivial, there is nothing to do.Now letγ0 ≥ α be minimal withQM

γ0nontrivial. We will distinguish two cases:γ = γ0 andγ > γ0.

Consider first the case thatγ = γ0. Work in V[Hγ] whereq ∈ Hγ. Note thatM[HMγ ] = M[HM

α ]. So r israndom overM[HM

γ ], and (pk(γ))k∈ω quickly interprets˜Z asZ∗ in QM

γ . Now letq+↾γ = q, and use the factthatQγ locally preserves randomness to findq+(γ) ≤ p0(γ).

Next consider the case thatQMγ is nontrivial andγ ≥ γ0+1. Again work inV[Hγ]. Let k∗ be maximal with

pk∗↾γ ∈ HMγ . (Thisk∗ exists, since the sequence (pk)k∈ω was quick, so there is even ak with pk↾(γ0 + 1) <

HMγ0+1.) Consider

˜Z as aQM

γ -name, and (using (3.37)) find a quick interpretationZ′ of˜Z witnessed by a

sequence starting withpk∗ (γ). In M[HMα ], Z′ is now aPM

γ /HMα -name. Clearly, the sequence (pk↾γ)k∈ω is a

quick sequence interpretingZ′ asZ∗. (Use the fact thatpk↾γ forcesk∗ ≥ k.)Using the induction hypothesis, we can first extendq to a conditionq′ ∈ Pγ and then (again by ourassumption thatQγ locally preserves randomness) to a conditionq+ ∈ Pγ+1.

The nontrivial limit case: M ∩ β unbounded inβ, i.e., δ′ = β. (This deals with cases (4) and (5) inDefinition 3.21. In case (4) we haveβ ∈ M, i.e.,β = δ; in case (5) we haveβ < M andβ < δ.)

Let α = δ0 < δ1 < · · · be a sequence ofM-ordinals cofinal inM ∩ δ′ = M ∩ δ. We may assume35 thateachQM

δnis nontrivial.

Let (˜Zn)n∈ω be a list of allPM

δ-names inM of codes for null sets (starting with our given null set

˜Z =

˜Z0).

Let (En)n∈ω enumerate all open dense sets ofPMδ

from M, without loss of generality36 we can assume that:

(3.39) En decides˜Z0↾n, . . . ,

˜Zn↾n.

We write pk0 for pk, andZ0,0 for Z∗; as mentioned above,

˜Z =

˜Z0.

By induction onn we can now find a sequence ¯pn = (pkn)k∈ω and PM

δn-namesZi,n for i ∈ {0, . . . , n}

satisfying the following:

(1) PMδn

forces thatp0n ≤ pk

n−1 wheneverpkn−1 ∈ PM

δ/HMδn

.(2) PM

δnforces thatp0

n ∈ En. (ClearlyEn ∩ PMδ/HMδn

is a dense set.)(3) pn ∈ M is aPM

δn-name for a quick sequence interpreting (

˜Z0, . . . ,

˜Zn) as (Z0,n, . . . ,Zn,n) (in PM

δ/HMδn

),soZi,n is aPM

δn-name of a code for a null set, for 0≤ i ≤ n.

Note that this implies that the sequence (pkn−1↾δn) is (forced to be) a quick sequence interpreting (Z0,n, . . . ,Zn−1,n)

as (Z0,n−1, . . . ,Zn−1,n−1).Using the induction hypothesis, we now define a sequence (qn)n∈ω of conditionsqn ∈ Pδn and a sequence

(cn)n∈ω (wherecn is aPδn-name) such that (forn > 0) qn extendsqn−1 and forces the following:

35If from someγ on all QMζ

are trivial, thenPMδ= PM

γ , so by induction there is nothing to do. IfQMα itself is trivial, then we let

δ0 ≔ min{ζ : QMζ

nontrivial} instead.36well, if we just enumerate a basis of the open sets instead of all of them. . .


• p0n−1↾δn ∈ HM

δn.

• Therefore,p0n ≤ p0

n−1.• r is random overM[HM

δn].

• Let cn be the leastc such thatZn,n ⊏c r.• Zi,n ⊏ci r for i = 0, . . . , n− 1.

Now let q =⋃

n qn ∈ PCSδ′

. As in Lemma 3.26 it is easy to see thatq ∈ Pgenδ′⊆ Pδ′ . Moreover, by (3.39) we

get thatq forces that˜Zi = limn Zi,n. Since each setCc,r ≔ {x : x ⊏c r} is closed, this implies thatq forces

˜Zi ⊏ci r, in particular

˜Z =

˜Z0 ⊏c0 r.

The trivial cases: In all other cases,M ∩ β is bounded inβ, so we already dealt with everything at stageβ0 ≔ sup(β ∩ M). Note thatδ′0 andδ0 used at stageβ0 are the same as the currentδ′ andδ. �

4. The forcing construction

In this section we describe aσ-closed “preparatory” forcing notionR; the generic filter will define a“generic” forcing iterationP, so elements ofR will be approximations to such an iteration. In Section 5 wewill show that the forcingR ∗ Pω2 forces BC and dBC.

From now on, we assume CH in the ground model.

4.A. Alternating iterations, canonical embeddings and the preparatory forcing R. The preparatoryforcingR will consist of pairs (M, P), whereM is a countable model andP ∈ M is an iteration of ultralaverand Janus forcings.

Definition 4.1. An alternating iteration37 is a topped partial CS iterationP of lengthω2 satisfying thefollowing:

• EachPα is proper.38

• For α even, either bothQα and Qα+1 are (forced by the empty condition to be) trivial,39 or Pαforces thatQα is an ultralaver forcing adding the generic realℓα, andPα+1 forces thatQα+1 is aJanus forcing based onℓ∗α (whereℓ∗ is defined fromℓ as in Lemma 1.23).

We will call an even index an “ultralaver position” and an oddone a “Janus position”.As in any partial CS iteration, eachPδ for cf(δ) > ω (and in particularPω2) is a direct limit.Recall that in Definition 3.10 we have defined the notion “PM canonically embeds intoP” for nice

candidatesM and iterationsP ∈ V andPM ∈ M. Since our iterations now have lengthω2, this means thatthe canonical embedding works up to and including40ω2.

In the following, we will use pairsx = (Mx, Px) as conditions in a forcing, wherePx is an alternatingiteration in the nice candidateMx. We will adapt our notation accordingly: Instead of writingM, PM, PM

α

HMα (the induced filter),QM

α , etc., we will writeMx, Px, Pxα, Hx

α, Qxα, etc. Instead of “Px canonically embeds

into P” we will say41 “ x canonically embeds intoP” or “( Mx, Px) canonically embeds intoP” (which is amore exact notation anyway, since the test whether the embedding is Mx-complete uses bothMx andPx,not justPx).

The following rephrases Definition 3.10 of a canonical embedding in our new notation, taking intoaccount that:

LDx is anMx-complete subforcing ofLD iff D extendsDx

(see Lemma 1.5).

Fact 4.2. x = (Mx, Px) canonically embeds intoP, if (inductively) for allβ ∈ ω2∩Mx∪{ω2} the followingholds:

37See Section 6 for possible variants of this definition.38This does not seem to be necessary, see Section 6, but it is easy to ensure and might be comforting to some of the readers and/or

authors.39For definiteness, let us agree that the trivial forcing is thesingleton{∅}.40This is stronger than to require that the canonical embedding works for everyα ∈ ω2 ∩ M, even though bothPω2 andPM

ω2are

just direct limits; see footnote 30.41Note the linguistic asymmetry here: A symmetric and more verbose variant would say “x = (Mx, Px) canonically embeds into

(V, P)”.


• Let β = α+1 forα even (i.e., an ultralaver position). Then eitherQxα is trivial (andQα can be trivial

or not), or we require that (Pα forces that) theV[Hα]-ultrafilter systemD used forQα extends theMx[Hx

α]-ultrafilter systemDx used forQxα.

• Let β = α + 1 for α odd (i.e., a Janus position). Then eitherQxα is trivial, or we require that (Pα

forces that) the Janus forcingQxα is anMx[Hx

α]-complete subforcing of the Janus forcingQα.• Let β be a limit. Then the canonical extensioniβ : Px

β→ Pβ is Mx-complete. (The canonical

extension was defined in Definition 3.9.)

Fix a sufficiently large regular cardinalχ∗ (see Remark 3.3).

Definition 4.3. The “preparatory forcing” R consists of pairsx = (Mx, Px) such thatMx ∈ H(χ∗) is a nicecandidate (containingω2), andPx is in Mx an alternating iteration (in particular topped and of lengthω2).We definey to be stronger thanx (in symbols:y ≤R x), if the following holds: eitherx = y, or:

• Mx ∈ My andMx is countable inMy.• My thinks that (Mx, Px) canonically embeds intoPy.

Note that this order onR is transitive.We will sometimes writeix,y for the canonical embedding (inMy) from Px

ω2to Py

ω2.

There are several variants of this definition which result inequivalent forcing notions. We will brieflycome back to this in Section 6.

The following is trivial by elementarity:

Fact 4.4. Assume thatP is an alternating iteration (inV), thatx = (Mx, Px) ∈ R canonically embeds intoP, and thatN ≺ H(χ∗) containsx andP. Let y = (My, Py) be the ord-collapse of (N, P). Theny ∈ R andy ≤ x.

This fact will be used, for example, to get from the followingLemma 4.5 to Corollary 4.6.

Lemma 4.5. Given x∈ R, there is an alternating iterationP such that x canonically embeds intoP.

Proof. For the proof, we use either of the partial CS constructions introduced in the previous chapter (i.e.,an almost CS iteration or an almost FS iteration overPx). The only thing we have to check is that we canindeed chooseQα that satisfy the definition of an alternating iteration (i.e., as ultralaver or Janus forcings)and such thatQx

α is Mx-complete inQα.In the ultralaver case we arbitrarily extendDx to an ultrafilter systemD, which is justified by Lemma 1.5.In the Janus case, we takeQα ≔ Qx

α (this works by Fact 2.8). Alternatively, we could extendQxα to a

random forcing (using Lemma 2.20). �

Corollary 4.6. Given x∈ R and an HCON object b∈ H(χ∗) (e.g., a real or an ordinal), there is a y≤ xsuch that b∈ My.

What we will actually need are the following three variants:

Lemma 4.7. (1) Given x∈ R there is aσ-centered alternating iterationP above x.(2) Given a decreasing sequencex = (xn)n∈ω in R, there is an alternating iterationP such that each

xn embeds intoP. Moreover, we can assume that for all Janus positionsβ, the Janus42 forcing Qβis (forced to be) the union of the Qxn

β, and that for all limitsα, the forcing Pα is the almost FS limit

over(xn)n∈ω (as in Corollary 3.19).(3) Let x, y ∈ R. Let jx be the transitive collapse of Mx, and define jy analogously. Assume that

jx[Mx] = jy[My], that jx(Px) = jy(Py) and that there areα0 ≤ α1 < ω2 such that:• Mx ∩ α0 = My ∩ α0 (and thus jx↾α0 = jy↾α0).• Mx ∩ [α0, ω2) ⊆ [α0, α1).• My ∩ [α0, ω2) ⊆ [α1, ω2).

Then there is an alternating iterationP such that both x and y canonically embed into it.

Proof. For (1), use an almost FS iteration. We only use the coordinates in Mx, and use the (countable!)Janus forcingsQα ≔ Qx

α for all Janus positionsα ∈ Mx (see Fact 2.8). Ultralaver forcings areσ-centeredanyway, soPε will be σ-centered, by Lemma 3.17.

42If all Qxnβ

are trivial, then we may also setQβ to be the trivial forcing, which is formally not a Janus forcing.


For (2), use the almost FS iteration over the sequence (xn)n∈ω as in Corollary 3.19, and at Janus positionsα setQα to be the union of theQxn

α . (By Fact 2.8,Qxnα is Mxn-complete inQα, so Corollary 3.19 can be

applied here.)For (3), we again use an almost FS construction. This time we start with an almost FS construction over

x up toα1, and then continue with an almost FS construction overy. �

As above, Fact 4.4 gives us the following consequences:

Corollary 4.8. (1) R is σ-closed. HenceR does not add new HCON objects (and in particular: nonew reals).

(2) R forces that the generic filter G⊆ R isσ-directed, i.e., for every countable subset B of G there isa y ∈ G stronger than each element of B.

(3) R forces CH. (Since we assume CH in V.)(4) Given a decreasing sequencex = (xn)n∈ω in R and any HCON object b∈ H(χ∗), there is a y∈ R

such that• y ≤ xn for all n,• My contains b and the sequencex,• for all Janus positionsβ, My thinks that the Janus forcing Qy

βis (forced to be) the union of

the Qxnβ

,

• for all limits α, My thinks that Pyα is the almost FS limit43 over(xn)n∈ω (of (Pyβ)β<α).

Proof. Item (4) directly follows from Lemma 4.7(2) and Fact 4.4. Item (1) is a special case of (4), and (2)and (3) are trivial consequences of (1). �

Another consequence of Lemma 4.7 is:

Lemma 4.9. The forcing notionR is ℵ2-cc.

Proof. Recall that we assume thatV (and henceV[G]) satisfies CH.Assume towards a contradiction that (xi : i < ω2) is an antichain. Using CH we may without loss of

generality assume that for eachi ∈ ω2 the transitive collapse of (Mxi , Pxi ) is the same. SetLi ≔ Mxi ∩ ω2.Using the∆-lemma we find some uncountableI ⊆ ω2 such that theLi for i ∈ I form a∆-system with rootL.Setα0 = sup(L) + 3. Moreover, we may assume sup(Li) < min(L j \ α0) for all i < j.

Now take anyi, j ∈ I , set x ≔ xi andy ≔ x j , and use Lemma 4.7(3). Finally, use Fact 4.4 to findz≤ xi , x j. �

4.B. The generic forcing P′. Let G be R-generic. ObviouslyG is a ≤R-directed system. Using thecanonical embeddings, we can construct inV[G] a direct limit P′ω2

of the directed systemG: Formally,we set

P′ω2≔ {(x, p) : x ∈ G andp ∈ Px

ω2},

and we set (y, q) ≤ (x, p) if y ≤R x andq is (in y) stronger thanix,y(p) (whereix,y : Pxω2→ Py

ω2is the

canonical embedding). Similarly, we define for eachα

P′α ≔ {(x, p) : x ∈ G, α ∈ Mx andp ∈ Pxα}

with the same order.To summarize:

Definition 4.10. Forα ≤ ω2, the direct limit of thePxα with x ∈ G is calledP′α.

Formally, elements ofP′ω2are defined as pairs (x, p). However, thex does not really contribute any

information. In particular:

Fact 4.11. (1) Assume that (x, px) and (y, py) are inP′ω2, thaty ≤ x, and that the canonical embedding

ix,y witnessingy ≤ x mapspx to py. Then (x, px) =∗ (y, py).(2) (y, q) is in P′ω2

stronger than (x, p) iff for some (or equivalently: for any)z ≤ x, y in G the canoni-cally embeddedq is in Pz

ω2stronger than the canonically embeddedp. The same holds if “stronger

than” is replaced by “compatible with” or by “incompatible with”.

43constructed in Lemma 3.18


(3) If (x, p) ∈ P′α, and ify is such thatMy = Mx andPy↾α = Px↾α, then (y, p) =∗ (x, p).

In the following, we will therefore often abuse notation andjust write p instead of (x, p) for an elementof P′α.

We can define a natural restriction map fromP′ω2to P′α, by mapping (x, p) to (x, p↾α). Note that by the

fact above, we can assume without loss of generality thatα ∈ Mx. More exactly: There is ay ≤ x in Gsuch thatα ∈ My (according to Corollary 4.6). Then inP′ω2

we have (x, p) =∗ (y, p).

Fact 4.12. The following is forced byR:

• P′β

is completely embedded intoP′α for β < α ≤ ω2 (witnessed by the natural restriction map).• If x ∈ G, thenPx

α is Mx-completely embedded intoP′α for α ≤ ω2 (by the identity mapp 7→ (x, p)).• If cf(α) > ω, thenP′α is the union of theP′

βfor β < α.

• By definition,P′ω2is a subset ofV.

G will always denote anR-generic filter, while theP′ω2-generic filter overV[G] will be denoted byH′ω2

(and the inducedP′α-generic byH′α). Recall that for eachx ∈ G, the mapp 7→ (x, p) is anMx-completeembedding ofPx

ω2into P′ω2

(and ofPxα into P′α). This wayH′α ⊆ P′α induces anMx-generic filterHx

α ⊆ Pxα.

Sox ∈ R forces thatP′α is approximated byPxα. In particular we get:

Lemma 4.13. Assume that x∈ R, that α ≤ ω2 in Mx, that p ∈ Pxα, that ϕ(t) is a first order formula

of the language{∈} with one free variable t and thatτ is a Pxα-name in Mx. Then Mx |= p Px

αϕ(τ) iff

x R (x, p) P′α Mx[Hxα] |= ϕ(τ[H

xα]).

Proof. “⇒” is clear. So assume thatϕ(τ) is not forced inMx. Then someq ≤Pxα

p forces the negation. Nowx forces that (x, q) ≤ (x, p) in P′α; but the conditions (x, p) and (x, q) force contradictory statements. �

4.C. The inductive proof of ccc. We will now prove by induction onα thatP′α is (forced to be) ccc and(equivalent to) an alternating iteration. Once we know this, we can prove Lemma 4.28, which easily impliesall the lemmas in this section. So in particular these lemmaswill only be needed to prove ccc and not foranything else (and they will probably not aid the understanding of the construction).

In this section, we try to stick to the following notation:R-names are denoted with a tilde underneath(e.g.,

˜τ), while Px

α-names orP′α-names (for anyα ≤ ω2) are denoted with a dot accent (e.g., ˙τ). We useboth accents when we deal withR-names forP′α-names (e.g.,

˜τ).

We first prove a few lemmas that are easy generalizations of the following straightforward observation:Assume thatx R (

˜z,

˜p) ∈ P′α. In particular,x

˜z ∈ G. We first strengthenx to somex1 that decides

˜z and

˜p to bez∗ and p∗. Thenx1 ≤

∗ z∗ (the order≤∗ is defined on page 3), so we can further strengthenx1 to somey ≤ z∗. By definition, this means thatz∗ is canonically embedded intoPy; so (by Fact 4.11)the Pz∗

α -conditionp∗ can be interpreted as aPyα-condition as well. So we end up with somey ≤ x and a

Pyα-conditionp∗ such thaty R (

˜z,

˜p) =∗ (y, p∗).

SinceR isσ-closed, we can immediately generalize this to countably many (R-names for)P′α-conditions:

Fact 4.14. Assume thatx R˜pn ∈ P′α for all n ∈ ω. Then there is ay ≤ x and there arep∗n ∈ Py

α such thaty R

˜pn =

∗ p∗n for all n ∈ ω.

Recall that more formally we should write:x R (˜zn,

˜pn) ∈ P′α; andy R (

˜zn,

˜pn) =∗ (y, p∗n).

We will need a variant of the previous fact:

Lemma 4.15. Assume thatP′β

is forced to be ccc, and assume that x forces (inR) that˜rn is a P′

β-name for

a real (or an HCON object) for every n∈ ω. Then there is a y≤ x and there are Pyβ-namesr∗n in My such

that y R ( P′β ˜

rn = r∗n) for all n.

(Of course, we mean:˜rn is evaluated byG ∗ H′

β, while r∗n is evaluated byHy

β.)

Proof. The proof is an obvious consequence of the previous fact, since names of reals in a ccc forcing canbe viewed as a countable sequence of conditions.

In more detail: For notational simplicity assume all˜rn are names for elements of 2ω. Working inV, we

can find for eachn,m∈ ω names for a maximal antichainÃn,m and for a function

˜fn,m :

Ãn,m→ 2 such that

x forces that (P′β

forces that)˜rn(m) =

˜fn,m(a) for the uniquea ∈

Ãn,m ∩ H′

β. SinceP′

βis ccc, each

Ãn,m is

countable, and sinceR is σ-closed, it is forced that the sequence˜Ξ = (

Ãn,m,

˜fn,m)n,m∈ω is in V.


In V, we strengthenx to x1 to decide˜Ξ to be someΞ∗. We can also assume thatΞ∗ ∈ Mx1 (see

Corollary 4.6). EachA∗n,m consists of countably manya such thatx1 forcesa ∈ P′β. Using Fact 4.14

iteratively (and again the fact thatR is σ-closed) we get somey ≤ x1 such that each sucha is actually anelement ofPy

β. So inMy, we can use (A∗n,m, f

∗n,m)n,m∈ω to constructPy

β-names ˙r∗n in the obvious way.

Now assume thaty ∈ G and thatH′β

is P′β-generic overV[G]. Fix anya ∈ A∗n,m = ˜

An,m. Sincea ∈ Pyβ, we

geta ∈ Hyβ

iff a ∈ H′β. So there is a unique elementa of A∗n,m∩Hy

β, and ˙r∗n(m) = f ∗n,m(a) =

˜fn,m(a) =

˜rn(m). �

We will also need the following modification:

Lemma 4.16. (Same assumptions as in the previous lemma.) In V[G][H′β], let Qβ be the union of Qz

β[Hzβ]

for all z ∈ G. In V, assume that x forces that each˜rn is a name for an element ofQβ. Then there is a y≤ x

and there is in My a sequence(r∗n)n∈ω of Pyβ-names for elements of Qy

βsuch that y forces

˜rn = r∗n for all n.

So the difference to the previous lemma is: We additionally assume that˜rn is in

⋃

z∈G Qzβ, and we

additionally get that ˙r∗n is a name for an element ofQyβ.

Proof. Assumex ∈ G and work inV[G]. Fix n. P′β

forces that there is someyn ∈ G and somePyn

β-name

τn ∈ Myn of an element ofQyn

βsuch that

˜rn (evaluated byH′

β) is the same asτn (evaluated byHyn

β). Since

we assume thatP′β

is ccc, we can find a countable setYn ⊆ G of the possibleyn, i.e., the empty conditionof P′

βforcesyn ∈ Yn. (AsR isσ-closed andYn ⊆ R ⊆ V, we must haveYn ∈ V.)

So in V, there is (for eachn) anR-name˜Yn for this countable set. SinceR is σ-closed, we can find

somez0 ≤ x deciding each˜Yn to be some countable setY∗n ⊆ R. In particular, for eachy ∈ Y∗n we know

thatz0 R y ∈ G, i.e.,z0 ≤∗ y; so using once again thatR is σ-closed we can find somez stronger thanz0

and all they ∈⋃

n∈ω Y∗n. Let X contain allτ ∈ My such that for somey ∈⋃

n∈ω Y∗n, τ is a Pyβ-name for a

Qyβ-element. Sincez≤ y, eachτ ∈ X is actually44 a Pz

β-name for an element ofQz

β.

SoX is a set ofPzβ-names forQz

β-elements; we can assume thatX ∈ Mz. Also, z forces that

˜rn ∈ X for

all n. Using Lemma 4.15, we can additionally assume that there arenamesPzβ-name ˙r∗n in Mz such thatz

forces that˜rn = r∗n is forced for eachn. By Lemma 4.13, we know thatMz thinks thatPz

βforces that ˙r∗n ∈ X.

Therefore ˙r∗n is aPzβ-name for aQz

β-element. �

We now prove by induction onα thatP′α is equivalent to a ccc alternating iteration:

Lemma 4.17. The following holds in V[G] for α < ω2:

(1) P′α is equivalent to an alternating iteration. More formally: There is an iteration(Pβ,Qβ)β<αwith limit Pα that satisfies the definition of alternating iteration (up toα), and there is a naturallydefined dense embedding jα : P′α → Pα, such that forβ < α we have jβ ⊆ jα, and the embeddingscommute with the restrictions.45 EachQα is the union of all Qx

α with x ∈ G. For x∈ G withα ∈ Mx,the function ix,α : Px

α → Pα that maps p to jα(x, p) is the canonical Mx-complete embedding.(2) In particular, aP′α-generic filter H′α can be translated into aPα-generic filter which we call Hα

(and vice versa).(3) Pα has a dense subset of sizeℵ1.(4) Pα is ccc.(5) Pα forces CH.

Proof. α = 0 is trivial (sinceP0 andP′0 both are trivial: P0 is a singleton, andP′0 consists of pairwisecompatible elements).

So assume that all items hold for allβ < α.

Proof of (1).Ultralaver successor case: Letα = β+1 with β an ultralaver position. LetHβ bePβ-generic overV[G].

Work in V[G][Hβ]. By induction, for everyx ∈ G the canonical embeddingix,β defines aPxβ-generic filter

overMx calledHxβ.

44Here we use two consequences ofz ≤ y: Every Pyβ-name inMy can be canonically interpreted as aPz

β-name inMz, andQy

βis

(forced to be) a subset ofQzβ.

45I.e., jβ(x, p↾β) = jα(x, p↾β) = jα(x, p)↾β.


Definition ofQβ (and thus ofPα): In Mx[Hxβ], the forcing notionQx

βis defined asLDx for some system

of ultrafiltersDx in Mx[Hxβ]. Fix somes ∈ ω<ω. If y ≤ x in G, thenDy

s extendsDxs. Let Ds be the union

of all Dxs with x ∈ G. SoDs is a proper filter. It is even an ultrafilter: Letr be aPβ-name for a real. Using

Lemma 4.15, we know that there is somey ∈ G and somePyβ-name

˜ry ∈ My such that (inV[G][Hβ]) we

have˜ry[Hy

β] = r. Sor ∈ My[Hy

β], hence eitherr or its complement is inDy

s and therefore inDs. So all filtersin the familyD = (Ds)s∈ω<ω are ultrafilters.

Now work again inV[G]. We setQβ to be thePβ-name forLD. (Note thatPβ forces thatQβ literally isthe union of theQx

β[Hxβ] for x ∈ G, again by Lemma 4.15.)

Definition of jα: Let (x, p) be inP′α. If p ∈ Pxβ, then we setjα(x, p) = jβ(x, p), i.e., jα will extend jβ. If

p = (p↾β, p(β)) is in Pxα but not inPx

β, we setjα(x, p) = (r, s) ∈ Pβ ∗ Qβ wherer = jβ(x, p↾β) ands is the

(Pα-name for)p(β) as evaluated inMx[Hxβ]. FromQβ =

⋃

x∈G Qxβ[Hxβ] we conclude that this embedding is

dense.The canonical embedding:By induction we know thatix,β which mapsp ∈ Px

βto jβ(x, p) is (the restric-

tion to Pxβ

of) the canonical embedding ofx into Pω2. So we have to extend the canonical embedding toix,α : Px

α → Pα. By definition of “canonical embedding”,ix,α mapsp ∈ Pxα to the pair (ix,β(p↾β), p(β)). This

is the same asjα(x, p). We already know thatDxs is (forced to be) anMx[Hx

β]-ultrafilter that is extended

by Ds.Janus successor case: This is similar, but simpler than the previous case: Here,Qβ is just defined as

the union of allQxβ[Hxβ] for x ∈ G. We will show below that this union satisfies the ccc; just as in Fact 2.8,

it is then easy to see that this union is again a Janus forcing.In particular,Qβ consists of hereditarily countable objects (since it is theunion of Janus forcings, which

by definition consist of hereditarily countable objects). So sincePβ forces CH,Qβ is forced to have sizeℵ1.Also note that since all Janus forcings involved are separative, the union (which is a limit of an incom-patibility-preserving directed system) is trivially separative as well.

Limit case: Let α be a limit ordinal.Definition ofPα and jα: First we definejα : P′α → PCS

α : For each (x, p) ∈ P′α, let jα(x, p) ∈ PCSα be

the union of all jβ(x, p↾β) (for β ∈ α ∩ Mx). (Note thatβ1 < β2 implies that jβ1(x, p↾β1) is a restriction ofjβ2(x, p↾β2), so this union is indeed an element ofPCS

α .)Pα is the set of allq∧ p, wherep ∈ jα[P′α], q ∈ Pβ for someβ < α, andq ≤ p↾β.It is easy to check thatPα actually is a partial countable support limit, and thatjα is dense. We will

show below thatPα satisfies the ccc, so in particular it is proper.The canonical embedding:To see thatix,α is the (restriction of the) canonical embedding, we just have

to check thatix,α is Mx-complete. This is the case sinceP′α is the direct limit of allPyα for y ∈ G (without

loss of generalityy ≤ x), and eachix,y is Mx-complete (see Fact 4.12).

Proof of (3).Recall that we assume CH in the ground model.The successor case,α = β + 1, follows easily from (3)–(5) forPβ (sincePβ forces thatQβ has size

2ℵ0 = ℵ1 = ℵV1 ).

If cf(α) > ω, thenPα =⋃

β<α Pβ, so the proof is easy.So let cf(α) = ω. The following straightforward argument works for any ccc partial CS iteration where

all iterandsQβ are of size≤ ℵ1.For notational simplicity we assume Pβ Qβ ⊆ ω1 for all β < α (this is justified by inductive assump-

tion (5)). By induction, we can assume that for allβ < α there is a denseP∗β⊆ Pβ of sizeℵ1 and that every

P∗β

is ccc. For eachp ∈ Pα and allβ ∈ dom(p) we can find a maximal antichainApβ⊆ P∗

βsuch that each

elementa ∈ Apβ

decides the value ofp(β), saya Pβ p(β) = γpβ(a). Writing46 p ∼ q if p ≤ q andq ≤ p, the

map p 7→ (Apβ, γ

pβ)β∈dom(p) is 1-1 modulo∼. Since eachAp

βis countable, there are onlyℵ1 many possible

values, therefore there are onlyℵ1 many∼-equivalence classes. Any set of representatives will be dense.Alternatively, we can prove (3) directly forP′α. I.e., we can find a≤∗-dense subsetP′′ ⊆ P′α of cardinal-

ity ℵ1. Note that a condition (x, p) ∈ P′α essentially depends only onp (cf. Fact 4.11). More specifically,

46Since≤ is separative,p ∼ q iff p =∗ q, but this fact is not used here.


given (x, p) we can “transitively47collapsex aboveα”, resulting in a=∗-equivalent condition (x′, p′). Since|α| = ℵ1, there are onlyℵ1

ℵ0 = 2ℵ0 many such candidatesx′ and since eachx′ is countable andp′ ∈ x′,there are only 2ℵ0 many pairs (x′, p′).

Proof of (4).Ultralaver successor case: Let α = β + 1 with β an ultralaver position. We already know thatPα =

Pβ ∗Qβ whereQβ is an ultralaver forcing, which in particular is ccc, so by inductionPα is ccc.Janus successor case: As above it suffices to show thatQβ, the union of the Janus forcingsQx

β[Hxβ] for

x ∈ G, is (forced to be) ccc.Assume towards a contradiction that this is not the case, i.e., that we have an uncountable antichain

in Qβ. We already know thatQβ has sizeℵ1 and therefore the uncountable antichain has sizeℵ1. So,working inV, we assume towards a contradiction that

(4.18) x0 R p0 Pβ {ãi : i ∈ ω1} is a maximal (uncountable) antichain inQβ.

We construct by induction onn ∈ ω a decreasing sequence of conditions such thatxn+1 satisfies thefollowing:

(i) For all i ∈ ω1 ∩ Mxn there is (inMxn+1) a Pxn+1β

-namea∗i for a Qxn+1β

-condition such that

xn+1 R p0 Pβ ãi = a∗i .

Why can we get that? Just use Lemma 4.16.(ii) If τ is in Mxn a Pxn

β-name for an element ofQxn

β, then there isk∗(τ) ∈ ω1 such that

xn+1 R p0 Pβ (∃i < k∗(τ))ãi ‖Qβ τ.

Also, all thesek∗(τ) are inMxn+1.Why can we get that? First note thatxn p0 (∃i ∈ ω1)

ãi ‖ τ. SincePβ is ccc,xn forces that there

is some bound˜k(τ) for i. So it suffices thatxn+1 determines

˜k(τ) to bek∗(τ) (for all the countably

manyτ).

Setδ∗ ≔ ω1 ∩⋃

n∈ω Mxn. By Corollary 4.8(4), there is somey such that

• y ≤ xn for all n ∈ ω,• (xn)n∈ω and (a∗i )i∈δ∗ are inMy,• (My thinks that)Py

βforces thatQy

βis the union ofQxn

β, i.e., as a formula:My |= Py

β Qy

β=

⋃

n∈ω Qxnβ

.

Let G beR-generic (overV) containingy, and letHβ bePβ-generic (overV[G]) containingp0.SetA∗ ≔ {a∗i [H

yβ] : i < δ∗}. Note thatA∗ is in My[Hy

β]. We claim

(4.19) A∗ ⊆ Qyβ[Hyβ] is predense.

Pick anyq0 ∈ Qyβ. So there is somen ∈ ω and someτ which is inMxn a Pxn

β-name of aQxn

β-condition, such

thatq0 = τ[Hxnβ

]. By (ii) above,xn+1 and thereforey forces (inR) that for somei < k∗(τ) (and thereforesomei < δ∗) the conditionp0 forces the following (inPβ):

The conditionsãi andτ are compatible inQβ. Also,

ãi = a∗i andτ both are inQy

β, andQy

β

is an incompatibility-preserving subforcing ofQβ. ThereforeMy[Hyβ] thinks thata∗i andτ

are compatible.

This proves (4.19).SinceQy

β[Hyβ] is My[Hy

β]-complete inQβ[Hβ], and sinceA∗ ∈ My[Hy

β], this implies (as ˙a∗i [H

yβ] =

ãi [G ∗

Hβ] for all i < δ∗) that{ãi [G ∗ Hβ] : i < δ∗} already is predense, a contradiction to (4.18).

Limit case: We work withP′α, which by definition only contains HCON objects.Assume towards a contradiction thatP′α has an uncountable antichain. We already know thatP′α has a

dense subset of sizeℵ1 (modulo=∗), so the antichain has sizeℵ1.

47In more detail: We define a functionf : Mx → V by induction as follows: Ifβ ∈ Mx ∩ α + 1 or if β = ω2, then f (β) = β.Otherwise, ifβ ∈ Mx ∩ Ord, then f (β) is the smallest ordinal abovef [β]. If a ∈ Mx \ Ord, then f (a) = { f (b) : b ∈ a ∩ Mx}. Itis easy to see thatf is an isomorphism fromMx to Mx′

≔ f [Mx] and thatMx′ is a candidate. Moreover, the ordinals that occur inMx′ are subsets ofα + ω1 together with the interval [ω2, ω2 + ω1]; i.e., there areℵ1 many ordinals that can possibly occur inMx′ ,and therefore there are 2ℵ0 many possible such candidates. Moreover, settingp′ ≔ f (p), it is easy to check that (x, p) =∗ (x′ , p′)(similarly to Fact 4.11).


Again, work inV. We assume towards a contradiction that

(4.20) x0 R {ãi : i ∈ ω1} is a maximal (uncountable) antichain inP′α.

So eachãi is anR-name for an HCON object (x, p) in V.

To lighten the notation we will abbreviate elements (x, p) ∈ P′α by p; this is justified by Fact 4.11.Fix any HCON objectp andβ < α. We will now define the (R ∗ P′

β)-names

˜ι(β, p) and

˜r(β, p): Let G

beR-generic and containingx0, andH′β

beP′β-generic. LetR be the quotientP′α/H

′β. If p is not inR, set

˜ι(β, p) =

˜r(β, p) = 0. Otherwise, let

˜ι(β, p) be the minimali such that

ãi ∈ R and

ãi andp are compatible

(in R), and set˜r(β, p) ∈ R to be a witness of this compatibility. SinceP′

βis (forced to be) ccc, we can

find (in V[G]) a countable set˜Xι(β, p) ⊆ ω1 containing all possibilities for

˜ι(β, p) and similarly

˜Xr(β, p)

consisting of HCON objects for˜r(β, p).

To summarize: For everyβ < α and every HCON objectp, we can define (inV) theR-names˜Xι(β, p)

and˜Xr(β, p) such that

(4.21) x0 R P′β

(

p ∈ P′α/H′β → (∃i ∈

˜Xι(β, p)) (∃r ∈

˜Xr(β, p)) r ≤P′α/H′β p,

ãi

)

.

Similarly to the Janus successor case, we define by inductionon n ∈ ω a decreasing sequence of con-ditions such thatxn+1 satisfies the following: For allβ ∈ α ∩ Mxn and p ∈ Pxn

α , xn+1 decides˜Xι(β, p) and

˜Xr (β, p) to be someXι∗(β, p) andXr∗(β, p). For all i ∈ ω1 ∩ Mxn, xn+1 decides

ãi to be somea∗i ∈ Pxn+1

α .Moreover, each suchXι∗ andXr∗ is in Mxn+1, and everyr ∈ Xr∗(β, p) is in Pxn+1

α . (For this, we just useFact 4.14 and Lemma 4.15.)

Setδ∗ ≔ ω1 ∩⋃

n∈ω Mxn, and setA∗ ≔ {a∗i : i ∈ δ∗}. By Corollary 4.8(4), there is somey such that

y ≤ xn for all n ∈ ω,(4.22)

x≔ (xn)n∈ω andA∗ are inMy,(4.23)

(My thinks that)Pyα is defined as the almost FS limit over ¯x.(4.24)

We claim thaty forces

(4.25) A∗ is predense inPyα.

SincePyα is My-completely embedded intoP′α, and sinceA∗ ∈ My (and since

ãi = a∗i for all i ∈ δ∗) we get

that{ãi : i ∈ δ∗} is predense, a contradiction to (4.20).

So it remains to show (4.25). LetG beR-generic containingy. Let r be a condition inPyα; we will find

i < δ∗ such thatr is compatible witha∗i . SincePyα is the almost FS limit over ¯x, there is somen ∈ ω and

β ∈ α ∩ Mxn such thatr has the formq∧ p with p in Pxnα , q ∈ Py

βandq ≤ p↾β.

Now let H′β

beP′β-generic containingq. Work in V[G][H′

β]. Sinceq ≤ p↾β, we getp ∈ P′α/H

′β. Let ι∗

be the evaluation byG ∗ H′β

of˜ι(β, p), and letr∗ be the evaluation of

˜r(β, p). Note thatι∗ < δ∗ andr∗ ∈ Py

α.So we know thata∗ι∗ andp are compatible inP′α/H

′β

witnessed byr∗. Findq′ ∈ H′β

forcing r∗ ≤P′α/H′β p, a∗ι∗.

We may findq′ ≤ q. Now q′ ∧ r∗ witnesses thatq∧ p anda∗ι∗ are compatible inPyα.

To summarize: The crucial point in proving the ccc is that “densely” we choose (a variant of) a finitesupport iteration, see (4.24). Still, it is a bit surprisingthat we get the ccc, since we can also argue thatdensely we use (a variant of) a countable support iteration.But this does not prevent the ccc, it onlyprevents the generic iteration from having direct limits instages of countable cofinality.48

Proof of (5).This follows from (3) and (4). �

4.D. The generic alternating iteration P. In Lemma 4.17 we have seen:

Corollary 4.26. Let G beR-generic. Then we can construct49 (in V[G]) an alternating iterationP suchthat the following holds:

• P is ccc.

48Assume thatx forces thatP′α is the union of theP′β

for β < α; then we can find a strongery that uses an almost CS iterationoverx. This almost CS iteration contains a conditionp with unbounded support. (Take any condition in the generic part of the almostCS limit; if this condition has bounded domain, we can extendit to have unbounded domain, see Definition 3.21.) Nowp will be inP′α and have unbounded domain.

49in an “absolute way”: GivenG, we first defineP′ω2to be the direct limit ofG, and then inductively construct thePα ’s from P′ω2

.


• If x ∈ G, then x canonically embeds intoP. (In particular, a Pω2-generic filter Hω2 induces aPxω2

-generic filter over Mx, called Hxω2

.)• EachQα is the union of all Qx

α[Hxα] with x ∈ G.

• Pω2 is equivalent to the direct limitP′ω2of G: There is a dense embedding j: P′ω2

→ Pω2, and foreach x∈ G the function p7→ j(x, p) is the canonical embedding.

Lemma 4.27. Let x∈ R. ThenR forces the following: x∈ G iff x canonically embeds intoP.

Proof. If x ∈ G, then we already know thatx canonically embeds intoP.So assume (towards a contradiction) thaty forces thatx embeds, buty x < G. Work in V[G] where

y ∈ G. Both x (by assumption) andy ∈ G canonically embed intoP. Let N be an elementary submodelof HV[G] (χ∗) containingx, y, P; let z = (Mz, Pz) be the ord-collapse of (N, P). Thenz ∈ V (asR is σ-closed) andz ∈ R, and (by elementarity)z ≤ x, y. This shows thatx ‖R y, i.e., y cannot forcex < G, acontradiction. �

Using ccc, we can now prove a lemma that is in fact stronger than the lemmas in the previous Sec-tion 4.C:

Lemma 4.28. The following is forced byR: Let N ≺ HV[G] (χ∗) be countable, and let y be the ord-collapseof (N, P). Then y∈ G. Moreover, if x∈ G∩ N, then y≤ x.

Proof. Work in V[G] with x ∈ G. Pick an elementary submodelN containingx andP. Let y be the ord-collapse of (N, P) via a collapsing mapk. As above, it is clear thaty ∈ R andy ≤ x. To showy ∈ G, itis (by the previous lemma) enough to show thaty canonically embeds. We claim thatk−1 is the canonicalembedding ofy into P. The crucial point is to showMy-completeness. LetB ∈ My be a maximal antichainof Py

ω2, sayB = k(A) whereA ∈ N is a maximal antichain ofPω2. So (by ccc)A is countable, henceA ⊆ N.

So not onlyA = k−1(B) but evenA = k−1[B]. Hencek−1 is anMy-complete embedding. �

Remark 4.29. We used the ccc ofPω2 to prove Lemma 4.28; this use was essential in the sense that wecan in turn easily prove the ccc ofPω2 if we assume that Lemma 4.28 holds. In fact Lemma 4.28 easilyimplies all other lemmas in Section 4.C as well.

5. The proof of BC+dBC

We first50 prove that no uncountableX in V will be smz or sm in the final extensionV[G ∗ H]. Then weshow how to modify the argument to work for all uncountable sets inV[G ∗ H].

5.A. BC+dBC for ground model sets.

Lemma 5.1. Let X∈ V be an uncountable set of reals. ThenR ∗ Pω2 forces that X is not smz.

Proof.

(1) Fix any evenα < ω2 (i.e., an ultralaver position) in our iteration. The ultralaver forcingQα adds a(canonically defined code for a) closed null setF constructed from the ultralaver realℓα. (RecallCorollary 1.21.) In the following, when we consider variousultralaver forcingsQα, Qα, Qx

α, wetreatF not as an actual name, but rather as a definition which dependson the forcing used.

(2) According to Theorem 0.2, it is enough to show thatX+ F is non-null in theR ∗Pω2-extension, orequivalently, in everyR∗Pβ-extension (α < β < ω2). So assume towards a contradiction that thereis aβ > α and anR ∗ Pβ-name

˜Z of a (code for a) Borel null set such that some (x, p) ∈ R ∗ Pω2

forces thatX + F ⊆˜Z.

(3) Using the dense embeddingjω2 : P′ω2→ Pω2, we may replace (x, p) by a condition (x, p′) ∈ R∗P′ω2

.According to Fact 4.14 (recall that we now know thatPω2 satisfies ccc) and Lemma 4.15 we canassume thatp′ is already aPx

β-conditionpx and that

˜Z is (forced byx to be the same as) aPx

β-name

Zx in Mx.(4) We construct (inV) an iterationP in the following way:

50Note that for this weak version, it would be enough to producea generic iteration of length 2 only, i.e.,Q0 ∗Q1, whereQ0 is anultralaver forcing andQ1 a corresponding Janus forcing.


(a) Up toα, we take an arbitrary alternating iteration into whichx embeds. In particular,Pα willbe proper and hence force thatX is still uncountable.

(b) Let Qα be any ultralaver forcing (overQxα in caseα ∈ Mx). So according to Corollary 1.21,

we know thatQα forces thatX + F is not null.Therefore we can pick (inV[Hα+1]) some ˙r in X + F which is random over (the countablemodel)Mx[Hx

α+1], whereHxα+1 is induced byHα+1.

(c) In the rest of the construction, we preserve randomness of r overMx[Hxζ] for eachζ ≤ ω2. We

can do this using an almost CS iteration overx where at each Janus position we use a randomversion of Janus forcing and at each ultralaver position we use a suitable ultralaver forcing;this is possible by Lemma 3.32. By Lemma 3.34, this iterationwill preserve the randomnessof r.

(d) So we getP overx (with canonical embeddingix) andq ≤Pω2ix(px) such thatq↾β forces (in

Pβ) that r is random overMx[Hxβ], in particular that ˙r < Zx.

We now pick a countableN ≺ H(χ∗) containing everything and ord-collapse (N, P) to y ≤ x. (SeeFact 4.4.) SetXy

≔ X ∩ My (the image ofX under the collapse). By elementarity,My thinks that(a)–(d) above holds forPy and thatXy is uncountable. Note thatXy ⊆ X.

(5) This gives a contradiction in the obvious way: LetG beR-generic overV and containy, and letHβbePβ-generic overV[G] and containq↾β. So My[Hy

β] thinks thatr < Zx (which is absolute) and

thatr = x+ f for somex ∈ Xy ⊆ X and f ∈ F (actually even inF as evaluated inMy[Hyα+1]). So

in V[G][Hβ], r is the sum of an element ofX and an element ofF. So (y, q) ≤ (x, p′) forces thatr ∈ (X + F) \

˜Z, a contradiction to (2). �

Of course, we need this result not just for ground model setsX, but forR ∗ Pω2-names˜X = (

˜xi : i ∈ ω1)

of uncountable sets. It is easy to see that it is enough to dealwith R ∗ Pβ-names for (all)β < ω2. So given

˜X, we can (in the proof) pickα such that

˜X is actually anR∗Pα-name. We can try to repeat the same proof;

however, the problem is the following: When constructingP in (4), it is not clear how to simultaneouslymake all the uncountably many names (

˜xi) into P-names in a sufficiently “absolute” way. In other words:

It is not clear how to end up with someMy and Xy uncountable inMy such that it is guaranteed thatXy

(evaluated inMy[Hyα]) will be a subset of

˜X (evaluated inV[G][Hα]). We will solve this problem in the

next section by factoringR.Let us now give the proof of the corresponding weak version ofdBC:

Lemma 5.2. Let X∈ V be an uncountable set of reals. ThenR ∗ Pω2 forces that X is not strongly meager.

Proof. The proof is parallel to the previous one:

(1) Fix any evenα < ω2 (i.e., an ultralaver position) in our iteration. The Janus forcingQα+1 adds a(canonically defined code for a) null setZ∇. (See Definition 2.6 and Fact 2.7.)

(2) According to (0.1), it is enough to show thatX+ Z∇ = 2ω in theR ∗Pω2-extension, or equivalently,in everyR ∗ Pβ-extension (α < β < ω2). (For every realr, the statementr ∈ X + Z∇, i.e.,(∃x ∈ X) x+ r ∈ Z∇, is absolute.) So assume towards a contradiction that thereis aβ > α and anR ∗ Pβ-name

˜r of a real such that some (x, p) ∈ R ∗ Pω2 forces that

˜r < X + Z∇.

(3) Again, we can assume that˜r is aPx

β-name ˙r x in Mx.

(4) We construct (inV) an iterationP in the following way:(a) Up toα, we take an arbitrary alternating iteration into whichx embeds. In particular,Pα again

forces thatX is still uncountable.(b1) Let Qα be any ultralaver forcing (overQx

α). ThenQα forces thatX is not thin (see Corol-lary 1.24).

(b2) LetQα+1 be a countable Janus forcing. SoQα+1 forcesX + Z∇ = 2ω. (See Lemma 2.9.)(c) We continue the iteration in aσ-centered way. I.e., we use an almost FS iteration overx

of ultralaver forcings and countable Janus forcings, usingtrivial Qζ for all ζ < Mx; seeLemma 3.17.

(d) SoPβ still forces thatX+ Z∇ = 2ω, and in particular that ˙r x ∈ X+ Z∇. (Again by Lemma 2.9.)Again, by collapsing someN as in the previous proof, we gety ≤ x andXy ⊆ X.


(5) This again gives the obvious contradiction: LetG beR-generic overV and containy, and letHβbePβ-generic overV[G] and containp. SoMy[Hy

β] thinks thatr = x+ z for somex ∈ Xy ⊆ X and

z ∈ Z∇ (this time,Z∇ is evaluated inMy[Hyβ]), contradicting (2). �

5.B. A factor lemma. We can restrictR to anyα∗ < ω2 in the obvious way: Conditions are pairsx =(Mx, Px) of nice candidatesMx (containingα∗) and alternating iterationsPx, but nowMx thinks thatPx haslengthα∗ (and notω2). We call this variantR↾α∗.

Note that all results of Section 4 aboutR are still true forR↾α∗. In particular, wheneverG ⊆ R↾α∗ isgeneric, it will define a direct limit (which we callP′∗), and an alternating iteration of lengthα∗ (calledP∗);again we will have thatx ∈ G iff x canonically embeds intoP∗.

There is a natural projection map fromR (more exactly: from the dense subset of thosex which satisfyα∗ ∈ Mx) into R↾α∗, mappingx = (Mx, Px) to x↾α∗ ≔ (Mx, Px↾α∗). (It is obvious that this projection isdense and preserves≤.)

There is also a natural embeddingϕ from R↾α∗ to R: We can just continue an alternating iteration oflengthα∗ by appending trivial forcings.ϕ is complete: It preserves≤ and⊥. (Assume thatz≤ ϕ(x), ϕ(y). Thenz↾α∗ ≤ x, y.) Also, the projection

is a reduction: Ify ≤ x↾α∗ in R↾α∗, then letMz be a model containing bothx andy. In Mz, we can firstconstruct an alternating iteration of lengthα∗ overy (using almost FS overy, or almost CS — this doesnot matter here). We then continue this iterationPz using almost FS or almost CS overx. Sox andy bothembed intoPz, hencez= (Mz, Pz) ≤ x, y.

So according to the general factor lemma of forcing theory, we know thatR is forcing equivalent toR↾α∗ ∗ (R/R↾α∗), whereR/R↾α∗ is the quotient ofR andR↾α∗, i.e., the (R↾α∗-name for the) set ofx ∈ Rwhich are compatible (inR) with all ϕ(y) for y ∈ G↾α∗ (the generic filter forR↾α∗), or equivalently, the setof x ∈ R such thatx↾α∗ ∈ G↾α∗. So Lemma 4.27 (relativized toR↾α∗) implies:

(5.3) R/R↾α∗ is the set ofx ∈ R that canonically embed (up toα∗) into Pα∗ .

Setup. Fix someα∗ < ω2 of uncountable cofinality.51 Let G↾α∗ beR↾α∗-generic over V and work inV∗ ≔ V[G↾α∗]. SetP∗ = (P∗

β)β<α∗ , the generic alternating iteration added byR↾α∗. LetR∗ be the quotient

R/R↾α∗.

We claim thatR∗ satisfies (inV∗) all the properties that we proved in Section 4 forR (in V), with theobvious modifications. In particular:

(A)α∗ R∗ is ℵ2-cc, since it is the quotient of anℵ2-cc forcing.(B)α∗ R∗ does not add new reals (and more generally, no new HCON objects), since it is the quotient of a

σ-closed forcing.52

(C)α∗ Let G∗ beR∗-generic overV∗. ThenG∗ is R-generic overV, and therefore Corollary 4.26 holdsfor G∗. (Note thatP′ω2

and thenPω2 is constructed fromG∗.) Moreover, it is easy to see53 that Pstarts withP∗.

(D)α∗ In particular, we get a variant of Lemma 4.28: The following is forced byR∗: Let N ≺ HV[G∗ ](χ∗)be countable, and lety be the ord-collapse of (N, P). Theny ∈ G∗. Moreover: Ifx ∈ G∗ ∩ N, theny ≤ x.

We can use the last item to prove theR∗-version of Fact 4.14:

Corollary 5.4. In V∗, the following holds:

(1) Assume that x∈ R∗ forces that p∈ Pω2. Then there is a y≤ x and a py ∈ Pyω2

such that y forcespy =∗ p.

(2) Assume that x∈ R∗ forces that˜r is a Pω2-name of a real. Then there is a y≤ x and a Py

ω2-namery

such that y forces thatry and˜r are equivalent asPω2-names.

51Probably the cofinality is completely irrelevant, but the picture is clearer this way.52It is easy to see thatR∗ is evenσ-closed, by “relativizing” the proof forR, but we will not need this.53Forβ ≤ α∗, let P′∗

βbe the direct limit of (G↾α∗)↾β andP′

βthe direct limit ofG∗↾β. The functionkβ : P′∗

β→ P′

βthat maps (x, p)

to (ϕ(x), p) preserves≤ and⊥ and is surjective modulo=∗, see Fact 4.11(3). So it is clear that definingP∗↾β by induction fromP′∗β

yields the same result as definingP↾β from P′β.


Proof. We only prove (1), the proof of (2) is similar.Let G∗ containx. In V[G∗], pick an elementary submodelN containingx, p, P and let (Mz, Pz, pz) be

the ord-collapse of (N, P, p). Thenz ∈ G∗. This whole situation is forced by somey ≤ z ≤ x ∈ G∗. Soyandpy is as required, wherepy ∈ Py

ω2is the canonical image ofpz. �

We also get the following analogue of Fact 4.4:

(5.5)In V∗ we have: Letx ∈ R∗. Assume thatP is an alternating iteration that extendsP↾α∗

and thatx = (Mx, Px) ∈ R canonically embeds intoP, and thatN ≺ H(χ∗) containsxandP. Let y = (My, Py) be the ord-collapse of (N, P). Theny ∈ R∗ andy ≤ x.

We now claim thatR∗Pω2 forces BC+dBC. We know thatR is forcing equivalent toR↾α∗∗R∗. Obviouslywe have

R ∗ Pω2 = R↾α∗ ∗ R∗ ∗ Pα∗ ∗ Pα∗ , ω2

(wherePα∗ , ω2 is the quotient ofPω2 andPα∗ ). Note thatPα∗ is already determined byR↾α∗, soR∗ ∗ Pα∗ is(forced byR↾α∗ to be) a productR∗ × Pα∗ = Pα∗ × R∗.

But note that this is not the same asPα∗ ∗R∗, where we evaluate the definition ofR∗ in thePα∗ -extensionof V[G↾α∗]: We would get new candidates and therefore new conditions in R∗ after forcing withPα∗ . Inother words, we cannot just argue as follows:

Wrong argument. R ∗ Pω2 is the same as (R↾α∗ ∗ Pα∗ ) ∗ (R∗ ∗ Pα∗ ,ω2); so given anR ∗ Pω2-nameX of aset of reals of sizeℵ1, we can chooseα∗ large enough so thatX is an (R↾α∗ ∗Pα∗ )-name. Then, working inthe (R↾α∗ ∗ Pα∗ )-extension, we just apply Lemmas 5.1 and 5.2.

So what do we do instead? Assume that˜X = {

˜ξi : i ∈ ω1} is anR ∗ Pω2-name for a set of reals of

sizeℵ1. So there is aβ < ω2 such that˜X is added byR ∗Pβ. In theR-extension,Pβ is ccc, therefore we can

assume that each˜ξi is a system of countably many countable antichains

Ãm

i of Pβ, together with functions

˜f mi :

Ãm

i → {0, 1}. For the following argument, we prefer to work with the equivalentP′β

instead ofPβ. Wecan assume that each of the sequencesBi ≔ (

Ãm

i ,˜f mi )m∈ω is an element ofV (sinceP′

βis a subset ofV and

sinceR is σ-closed). So eachBi is decided by a maximal antichainZi of R. SinceR is ℵ2-cc, theseℵ1

many antichains all are contained in someR↾α∗ with α∗ ≥ β.So in theR↾α∗-extensionV∗ we have the following situation: Eachξi is a very “absolute54” R∗ ∗ Pα∗ -

name (or equivalently,R∗ ×Pα∗-name), in fact they are already determined by antichains that are inPα∗ anddo not depend onR∗. So we can interpret them asPα∗ -names.

Note that:

(5.6) Theξi are forced (byR∗ ∗ Pα∗ ) to be pairwise different, and therefore already byPα∗ .

Now we are finally ready to prove thatR ∗Pω2 forces that every uncountableX is neither smz nor sm. Itis enough to show that for every name

˜X of an uncountable set of reals of sizeℵ1 the forcingR ∗Pω2 forces

that˜X is neither smz nor sm. For the rest of the proof we fix such a name

˜X, the corresponding

˜ξi ’s (for

i ∈ ω1), and the appropriateα∗ as above. From now on, we work in theR↾α∗-extensionV∗.So we have to show thatR∗ ∗ Pω2 forces that

˜X is neither smz nor sm.

After all our preparations, we can just repeat the proofs of BC (Lemma 5.1) and dBC (Lemma 5.2) ofSection 5.A, with the following modifications. The modifications are the same for both proofs; for betterreadability we describe the results of the change only for the proof of dBC.

(1) Change: Instead of an arbitrary ultralaver positionα < ω2, we obviously have to chooseα ≥ α∗.For the dBC: We chooseα ≥ α∗ an arbitrary ultralaver position. The Janus forcingQα+1 adds a(canonically defined code for a) null setZ∇.

(2) Change: No change here. (Of course we now have anR∗ ∗ Pα∗ -name˜X instead of a ground model

set.)For the dBC: It is enough to show that

˜X+ Z∇ = 2ω in theR∗ ∗Pω2-extension ofV∗, or equivalently,

in everyR∗ ∗ Pβ-extension (α < β < ω2). So assume towards a contradiction that there is aβ > α

and anR∗ ∗ Pβ-name˜r of a real such that some (x, p) ∈ R∗ ∗ Pω2 forces that

˜r <

˜X + Z∇.

54or: “nice” in the sense of [Kun80, 5.11]


(3) Change: no change. (But we use Corollary 5.4 instead of Lemma 4.15.)For dBC: Using Corollary 5.4(2), without loss of generalityx forcespx =∗ p and there is aPx

β-

name ˙r x in Mx such that ˙r x =˜r is forced.

(4) Change: The iteration obviously has to start with theR↾α∗-generic iterationP∗ (which is ccc), therest is the same.For dBC: InV∗ we construct an iterationP in the following way:(a1) Up toα∗, we use the iterationP∗ (which already lives in our current universeV∗). As explained

above in the paragraph preceding (5.6),˜X can be interpreted as aPα∗ -nameX, and by (5.6),X

is forced to be uncountable.(a2) We continue the iteration fromα∗ to α in a way that embedsx and such thatPα is proper. So

Pα will force that X is still uncountable.(b1) LetQα be any ultralaver forcing (overQx

α). ThenQα forces thatX is not thin.(b2) LetQα+1 be a countable Janus forcing. SoQα+1 forcesX + Z∇ = 2ω.(c) We continue the iteration in aσ-centered way. I.e., we use an almost FS iteration overx of

ultralaver forcings and countable Janus forcings, using trivial Qζ for all ζ < Mx.(d) SoPβ still forces thatX + Z∇ = 2ω, and in particular that ˙r x ∈ X + Z∇.

We now pick (inV∗) a countableN ≺ H(χ∗) containing everything and ord-collapse (N, P) toy ≤ x, by (5.5). The HCON objecty is of course inV (and even inR), but we can say more: Sincethe iterationP starts with the (R↾α∗)-generic iterationP∗, the conditiony will be in the quotientforcingR∗.SetXy

≔ X∩My (which is the image ofX under the collapse, since we viewX as a set of HCON-names). By elementarity,My thinks that (a)–(d) above holds forPy and thatXy is forced to beuncountable. Note thatXy ⊆ X in the following sense: WheneverG∗ ∗ H is R∗ ∗ Pω2-generic overV∗, andy ∈ G∗, then the evaluation ofXy in My[Hy] is a subset of the evaluation ofX in V∗[G∗∗H].

(5) Change: No change here.For dBC: We get our desired contradiction as follows:Let G∗ beR∗-generic overV∗ and containy. Let Hβ bePβ-generic overV∗[G∗] and containp. SoMy[Hy

β] thinks thatr = x+ z for somex ∈ Xy ⊆ X and55 z ∈ Z∇, contradicting (2).

6. A word on variants of the definitions

The following is not needed for understanding the paper, we just briefly comment on alternative wayssome notions could be defined.

6.A. Regarding “alternating iterations”. We call the set ofα ∈ ω2 such thatQα is (forced to be) nontriv-ial the “true domain” of P (we use this notation in this remark only). ObviouslyP is naturally isomorphicto an iteration whose length is the order type of its true domain. In Definitions 4.1 and 4.3, we could haveimposed the following additional requirements. All these variants lead to equivalent forcing notions.

(1) Mx is (an ord-collapse of) anelementarysubmodel ofH(χ∗).This is equivalent, as conditions coming from elementary submodels are dense in ourR, byFact 4.4.While this definition looks much simpler and therefore nicer(we could replace ord-transitive mod-els by the better understood elementary models), it would not make things easier and just “hides”the point of the construction: For example, we use modelsMx that are (an ord-collapse of) anelementary submodel ofHV′(χ∗) for some forcing extensionV′ of V.

(2) Require that (Mx thinks that) the true domain ofPx isω2.This is equivalent for the same reason as (1) (and this requirement is compatible with (1)).This definition would allow to drop the “trivial” option fromthe definition. The whole proof would

55Note that we get the same Borel code, whether we evaluateZ∇ in My[Hyβ] or in V∗[G∗ ∗ Hβ]. Accordingly, the actual Borel set

of reals coded byZ∇ in the smaller universe is a subset of the corresponding Borel set in the larger universe.


still work with minor modifications — in particular, becauseof the following fact:56

(6.1) The finite support iteration ofσ-centered forcing notions of length< (2ℵ0)+ is againσ-centered.

We chose our version for two reasons: first, it seems more flexible, and second, we were initiallynot aware of (6.1).

(3) Alternatively, require that (Mx thinks that) the true domain ofPx is countable.Again, equivalence can be seen as in (1), again (3) is compatible with (1) but obviously not with (2).This requirement would not make the definition easier, so there is no reason to adopt it. It wouldhave the slight inconvenience that instead of using ord-collapses as in Fact 4.4, we would have toput another model on top to make the iteration countable. Also, it would have the (purely aesthetic)disadvantage that the generic iteration itself does not satisfy this requirement.

(4) Also, we could have dropped the requirement that the iteration is proper. It is never directly used,and “densely”P is proper anyway. (E.g., in Lemma 5.1(4)(a), we would just constructP up toαto be proper or even ccc, so thatX remains uncountable.)

6.B. Regarding “almost CS iterations and separative iterands”. Recall that in Definition 3.6 we requiredthat each iterandQα in a partial CS iteration is separative. This implies the property (actually: the threeequivalent properties) from Fact 3.8. Let us call this property “suitability” for now. Suitability is a prop-erty of the limit Pε of P. Suitability always holds for finite support iterations andfor countable supportiterations. However, if we do not assume that eachQα is separative, then suitability may fail for partial CSiterations. We could drop the separativity assumption, andinstead add suitability as an additional naturalrequirement to the definition of partial CS limit.

The disadvantage of this approach is that we would have to check in all constructions of partial CSiterations that suitability is indeed satisfied (which we found to be straightforward but rather cumbersome,in particular in the case of the almost CS iteration).

In contrast, the disadvantage of assuming thatQα is separative is minimal and purely cosmetic: It iswell known that every quasiorderQ can be made into a separative one which is forcing equivalentto theoriginalQ (e.g., by just redefining the order to be≤∗Q).

6.C. Regarding “preservation of random and quick sequences”. Recall Definition 1.50 of local preser-vation of random reals and Lemma 3.32.

In some respect the dense setsDn are unnecessary. For ultralaver forcingLD, the notion of a “quick”sequence refers to the setsDn of conditions with stem of length at leastn.

We could define a new partial order onLD as follows:

q ≤′ p ⇔ (q = p) or (q ≤ p and the stem ofq is strictly longer than the stem ofp).

Then (LD,≤) and (LD,≤′) are forcing equivalent, and any≤′-interpretation of a new real will automatically

be quick.Note however that (LD,≤

′) is now not separative any more. Therefore we chose not to take this approach,since losing separativity causes technical inconvenience, as described in 6.B.

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Institut fur Diskrete Mathematik und Geometrie, Technische UniversitatWien, Wiedner Hauptstraße 8–10/104, 1040 Wien,Austria

E-mail address: [email protected]: http://www.tuwien.ac.at/goldstern/

KurtGodel Research Center forMathematical Logic, UniversitatWien, Wahringer Straße 25, 1090 Wien, AustriaE-mail address: [email protected]: http://www.logic.univie.ac.at/∼kellner/

Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The Hebrew University of Jerusalem, Jerusalem,91904, Israel, and Department of Mathematics, Rutgers University, New Brunswick, NJ 08854, USA

E-mail address: [email protected]: http://shelah.logic.at/

Institut fur Diskrete Mathematik und Geometrie, Technische UniversitatWien, Wiedner Hauptstraße 8–10/104, 1040 Wien,Austria

E-mail address: [email protected]: http://www.wohofsky.eu/math/

http://www.arxiv.org/abs/math/0910.2132

arxiv:1105.0823v4 [math.lo] 21 feb 2013 · arxiv:1105.0823v4 [math.lo] 21 feb 2013 borel conjecture...

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