john t. baldwin and saharon shelah- a hanf number for saturation and omission

8/3/2019 John T. Baldwin and Saharon Shelah- A Hanf number for saturation and omission

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A Hanf number for saturation and omission

John T. Baldwin

University of Illinois at Chicago

Saharon Shelah

Hebrew University of Jerusalem

Rutgers University

May 23, 2011

Abstract

Suppose t = (T,T1, p) is a triple of two countable theories T T1 in vocab-ularies 1 and a 1-type p over the empty set. We show the Hanf number for

the property: There is a model M1 ofT1 which omits p, but M1 is saturated

is essentially equal to the Lowenheim number of second order logic. In Section 4

we make exact computations of these Hanf numbers and note some distinctions

between first order and second order quantification. In particular, we show that

if is uncountable, h3(L,(Q), ) = h3(L1,, ), where h

3 is the normal

notion of Hanf function (Definition 4.12.)

Newelski asked in [New] whether it is possible to calculate the Hanf number of thefollowing property PN. In a sense made precise in Theorem 0.2, we show the answer

is no. In accordance with the original question, we focus on countable vocabularies for

the first three sections. We deal with extensions to larger vocabularies in Section 4.

Definition 0.1 We say M1 |= t where t = (T, T1, p) is a triple of two theories invocabularies 1 , respectively, T T1 andp is a 1-type over the empty set if M1is a model of T1 which omits p , butM1 is saturated. LetKt denote the class of

models M1 which satisfy t.

ForK = Kt, with t in a countable vocabulary, let PcN(Kt, ) hold if |1| 0

and for some M1 with |M1| = , M1 |= t. PfN(Kt, ) is the same property restricted

to triples where T1 and T are finitely axiomatizable in finite vocabularies and p is

definable in second order logic.

Recall Hanfs observation [Han60] that for any such property P(K, ), where Kranges over a setof classes of models, there is a cardinal = H(P) such that is theleast cardinal satisfying: if P(K, ) holds for some then P(K, ) holds for

We give special thanks to the Mittag-Leffler Institute where this research was conducted. This is pa-

per 958 in Shelahs bibliography. Baldwin was partially supported by NSF-0500841. Shelah thanks the

Binational Science Foundation for partial support of this research.

1


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arbitrarily large . H(P) is called the Hanf number ofP. E.g. P(K, ) might be the

property thatK

has a model of power . Similarly the Lowenheim number (P) of aset P of classes is the least cardinal such that any class K P that has a model hasone of cardinality .

Theorem 0.2 Assume the collection of with


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Newelskis question arose in the study of the model theory of groups and the ex-

istence of groups with bounded orbits. The authors acknowledge very fruitful discus-sions with Jouko Vaananen and Tapani Hyttinen concerning the material of this paper.

We also thank the referee for an unusually detailed and helpful series of reports.

1 Some Second Order Logic

By (pure) second order logic, LII, we mean the logic with individual variables and

variables for relations of all arities but no non-logical constants. The atomic formulas

are equalities between variables and expressions X(x) where X is an n-ary relationvariable and x is an n-tuple of individual variables. Note that a structure A for this

logic is simply a set so is determined entirely by its cardinality. But we use the full

semantics: the n-ary relation variables range over all n-ary relations on A.

We explain the connection of our restriction to with =


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(b) Ifspec2() is bounded and nonempty there is a 4 LII with spec2(4)

bounded and nonempty andsup(spec

2

(4)) < sup(spec

2

(4)).

These transformations imply:

Fact 1.4 1. H1(LII), H2(LII), 1(LII), 2(LII) are strong limit cardinals.

2. There is no sentence attaining any of these values exactly. (E.g., there is no

LII with sup(spec()) = H1(LII).)

3. For either spectrum, i(LII) = sup{min{speci()} : LII has a model}and similarly Hi(LII) = sup{sup{speci()} : LII is bounded}.

Note that any logic satisfying Fact 1.3 will also satisfy Fact 1.4. We use this obser-

vation without comment in studying infinitary second order logics in Section 4.

Using Assumption 0.3 we can show:

Lemma 1.5 H(LII) = H2(LII), (LII) = 2(LII)

Proof. One direction is easy. For every sentence of second order logic, there is a

sentence such that:

spec2() = spec1().

just expresses the conjunction of with with


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Definition 2.3 A type p in a vocabulary satisfying Convention 2.2 is definable in sec-

ond order logic if we can code the type as a subset Ap of so that in the vocabularywith constant symbol 0 and relation symbol S there is a second order sentence andsecond order formula (x) satisfying the following condition. For the first cardinal which satisfies if M is (, 0, S), with 0 interpreted as 0 and S as successor on thenatural numbers, then Ap = {n : M |= (n)}.

Now for convenience we restrict our triples to those satisfying the convention. For-

mally:

Notation 2.4 Tf denotes the set of triples t such that T is finitely axiomatizable andp is second order definable.

Theorem 2.5 For every second order sentence , there is a triple t Tf such that if

2(LII). Then there is a triplet Tf such that sup(spec(t)) 2(LII) .

Let C = { : =


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then some N |= t has cardinality (take a saturated elementary submodel). Thus,

|= t

if and only if every C

belongs to spec(t

).Now spec(t) is exactly { : t}, whence spec(t) is { : > t}. So theLowenheim number oft is (t)+ > 2(LII) and this contradiction completes theproof.

2.8

Lemma 2.9 H(PfN) 2(LII) where LII denotes second order logic.

Proof. Suppose for contradiction that there is a second order sentence such that

0 = min(spec2()) H(PfN). By the definition of spec

2,


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of the proof sketch. Below we say certain conditions hold to mean they hold in any

model of T. We first describe and T. In particular, contains unary predicatesQ1, Q2 that partition the universe.

There is a binary relation


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Fact 3.4 If M |= T, a Q2(M) and each subset of Ua is coded by an element of

Q2(M), then M |=

(a) if and only Ua(M) |= .

Add the following axiom to T1 to obtain the theory T:

(u)(w)[((z)R(z, w) z


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simplified the proof of Theorem 2.5 since we could have worked with countably many

constants and omitted the function g. Similarly the arguments of Sections 2 and 3 ex-tend from finitely axiomatizable to arithmetic by coding a model of arithmetic in the

second order sentence. And it is easy to see that the theory constructed in Theorem 2.5

is recursive. This observation is generalized in Theorem 4.11 to remove the restric-

tions on axiomatizability. The key idea is to see that we can use the same ideas as in

Section 3 to code the syntax of infinitary second order logic by a triple t.

We extend our notion of second order logic in two ways. First we allow infinite

conjunctions and strings of quantifiers. Secondly we now allow some relation constants

instead of dealing with pure second order logic.

Definition 4.1 1. L denotes the stage in the construction of the inner model L.

2. LetL+,(II) denote second order logic allowing strings of second order quan-

tifiers of cardinality < +

and conjunctions and disjunctions of cardinality .

Remark 4.2 Again using Assumption 0.3, note that as in Fact 1.3 the Lowenheim num-

ber of L+,(II) is a strong limit cardinal of cofinality > and is an accumulationpoint of{ : =


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f1, f2, we see g = f12 f1 is a bijection from X1 onto X2. But then g is an isomorphism

with respect to as it is easy to check (from the definition of coding) that for anyy1, z1 X1 with y2 = g(y1) and z2 = g(z1), y1 z1 if and only if y2 z2. That is,y1 z1 if and only pr(f1(y1), f1(z1)) A if and only pr(f2(y2), f2(z2)) A if andonly y2 z2. But then since X1 X2, -induction yields that X1 = X2. 4.6

We first define certain set of ordinals A, in V that codes and L+,(II)(). Here tc(X) denotes the transitive closure of X.

Definition 4.7 For every vocabulary L and every sentence L+,(II)()we define A, to be a set which codes tc({, }) {, }) in the sense of Defini-tion 4.5.

Such a code exists since the standard construction of L+,(II)() yields

that each formula is in H(+

), each subformula of tc(), and has at most subformulas.

Now we define the sentence L(I I , ), which does not depend on or , sothat the interpretation of a predicate Q as A, in a model M of will identify as the

sentence under consideration. The function G in Definition 4.8.2c simply formalizes

the normal definition of truth.

Definition 4.8 We define a sentence L(I I , ) M |= if and only ifM satisfies:

1. (M, RM1 ) (H(), ) for some with


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Lemma 4.9 Fix + , a vocabulary L and L+,(II)(). Choose A,

and L(I I , ) satisfying Definitions 4.7 and 4.8, respectively. For any cardinal = , the following are equivalent.

1. has no model of cardinality < .

2. There is a model (M, RM1 , QM, cM , c

M , c

M , c

M ) with cardinality of the sen-

tence defined in Definition 4.8 such that letting PM denote

{b : M |= b is an ordinal bRM1 c}

andPM1 denote

{b : M |= b is an ordinal bRM1 c};

(a) (PM, RM1 ) has order type ;

(b) (PM1 , RM1 ) has order type ;

(c)

A, = { < : for some a QM, = otp({bR1c : bR

M1 a}, R

M1 )}.

Proof. Suppose 2). Without loss of generality, we identify (M, RM1 ) with H(, ). H() since L. Then |M| = , (PM, RM1 ) has order type , and A, is theimage ofQM under an isomorphism from (PM, RM1 ) to . By the choice ofA, , themodel M correctly recognizes the vocabulary and the formula . The function GM

correctly represents truth in M by Definition 4.8.3 and Definition 4.8.4. So fails on

all subsets ofM with cardinality < by Definition 4.8 2d. Thus 2) implies 1). Clearly

if 1) holds we can construct a model M satisfying 2).

4.9We continue to use the conventions regarding PM, PM1 from the proof of

Lemma 4.9.

Definition 4.10 For as in Definition 4.9, spec( , ,,A,) is the set of the cardi-nalities of models M of with (PM, PM1 , Q , R

M1 ) ( ,,A, ,


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(x)

p so the worries in the first sections about the second order definability of p

disappear. Clearly 2 3.With from Lemma 4.9 and applying that lemma with as {=} and with = +)yields:

{min(spec2()) : L+,+} {sup(spec2( , ,A,)) : L+,+ is bounded}.

(We can replace by a whose only model is the model of with minimum cardi-

nality to guarantee the containment.) Thus, 3 4.The proof that 4 1 is obtained by modifying the proof of Theorem 2.5.

Add to the vocabulary in the T from the proof in section 3 of Theorem 2.5, sym-

bols P, P1, Q , R1 and symbols c for each A, and use the same coding ideasto guarantee that P1, Q are contained in P and all three are well-ordered by R1.

Thus, for L+,+(II) we can construct t, encoding the second order sen-tence L(I I , ) defined in Definition 4.8 and where the type p also codes thatQM A, so that the two spectra are related as in Theorem 2.5. The type p is just{Q(x)} {x = c : A,}. This yields 4 1 by slightly modifying theargument for Lemma 2.9.

4.11

Our discussion of the Hanf and Lowenheim numbers of second order logic fo-

cused on two vocabularies: {=} and . In contrast, in many studies of the Hanf andLowenheim numbers of logics the number is taken as the supremum for a given logic

over all vocabularies of a bounded cardinality. That is, a Lowenheim or Hanf function

(with argument the cardinality of the vocabulary) is defined:

Definition 4.12 For any logic L and any cardinal .

1. Let the Lowenheim function 3(L, ) be the least cardinal such that for anyvocabulary of cardinality and any L(), if has a model it has oneof cardinality less than or equal .

2. Let the Hanf function h3(L, ) be the least cardinal such that for any vocabu-lary of cardinality and any L(), if has a model of cardinality , ithas arbitrarily large models.

The cardinalities of the vocabularies play a significant role. A trivial example is

that both the Lowenheim function and Hanf function in the sense of Definition 4.12

of first order logic map the cardinality of the vocabulary to itself. A more interesting

example is that for the logic L(Q) (with Q interpreted as there exists uncountablymany), the Lowenheim number in = {=} is 1, while for an arbitrary vocabulary it is 1 + ||. The Hanf number ofL(Q) for countable vocabularies is (See e.g.

3.3.12 of [Sch85]). Thus,

= h3(L,(Q), 0) < h

3(L1,, 0) = 1 .

Computing the Hanf function of L(Q) for vocabularies of cardinality 1 isconsiderably more complicated. The key point is the following observation which

does not depend on Assumption 0.3. Since we have not been able to find it in print we

describe the key innovation here.

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Theorem 4.13 If is uncountable, h3(L,(Q), ) = h3(L1,, ).

Proof. Lopez-Escobar and Chang (e.g. [Cha68] showed how to code sentences of

L1, as first order theories omitting types. Since each sentence will have only count-

ably many subformulas, each sentence regardless of the cardinality of the vocabulary

can be coded by a first order sentence omitting countably many types. And it is known

(e.g. by the proof of Theorem 5.1.4 of [She78]) that this omitting types problem can be

reduced to omitting one type p of the form P(x) {x = cn : n < }. More precisely,we can find for any pair (T, ) in a vocabulary (where is a countable collection oftypes) a pair (T1, {p}) in an expanded vocabulary 1 such that each -structure omitting can be expanded to a 1-structure omitting p and each model ofT1 omitting p alsoomits each type in . This translation can be done for of any cardinality. The cruxof the current extension is that with an uncountable language one can further reduce

the omission of the type p to the assertion (Qx)P(x). For this, add to the language

unary function symbols F for < 1 and a binary relation symbol


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We arrived at this result using our analysis of the Hanf number ofPN. But a variant

of Corollary 4.15 can be obtained, using the same ideas of coding syntax, withoutthe detour through PN. It shows that with only a finite vocabulary one can code any

reasonable similarity type.

Notation 4.16 is the vocabulary containing one unary predicate P , one binary

relation symbol < and a ternary predicate R.

Theorem 4.17 3(L+,(II), ) = 2(L+,(II)()).

Proof. Instead of considering an arbitrary vocabulary of size we can consider

a universal vocabulary of binary relations. (It is easy to code the first in the

second.) So the claim is that for any sentence L+,(II)() there is a sentence

ofL+,(II)() such that spec() = spec(). (The converse is obvious.)

Let

assert

1. (P,


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[New] Ludomir Newelski. Bounded orbits and measures on a group. preprint.

[Sch85] J. Schmerl. Transfer theorems and their applications to logics. In J. Barwiseand S. Feferman, editors, Model-Theoretic Logics, pages 177209. Springer-

Verlag, 1985.

[She78] S. Shelah. Classification Theory and the Number of Nonisomorphic Models.

North-Holland, 1978.

[Vaa79] Jouko Vaananen. On the Hanf numbers of unbounded logics. In B.Mayoh

F.Jensen and K.Moller, editors, Proceedings from 5th Scandinavian Logic

Symposium, pages 309328. Aalborg University Press, 1979.

[Vaa82] Jouko Vaananen. Abstract logic and set theory II: Large cardinals. Journal

of Symbolic Logic, 47:335346, 1982.

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john t. baldwin and saharon shelah- a hanf number for saturation and omission

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