saharon shelah- vive la difference iii

8/3/2019 Saharon Shelah- Vive La Difference III

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VIVE LA DIFFERENCE III

SAHARON SHELAH

Abstract. We show that, consistently, there is an ultrafilter F on such that if Nn = (P

n Q

n, P

n, Q

n, R

n) (for = 1, 2, n < ), P

n

Qn , andQ

n


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Theorem 0.1 (See [Sh 405]). It is consistent with the axioms of set the-ory that there is a non-principal ultrafilter F on such that for any two

sequences of discrete rank 1 valuation rings (Rin)n=1,2,... (i = 1, 2) havingcountable residue fields, any isomorphism F :

n

R1n/F n

R2n/F is an

ultraproduct of isomorphisms Fn : R1n R

2n (for a set of ns contained in

F). In particular, for Fmajority of the n, the valuation rings R1n, R2n are

isomorphic.

In the case of the rings Fp[[t]] and Zp, we see that (Iso 2) fails. For thisour main work was to show the following statement which actually frommodel theoretic point of view is more basic and interesting.

Theorem 0.2 (See [Sh 405]). It is consistent with the axioms of set theorythat there is a non-principal ultrafilter F on such that for any two se-

quences of countable trees (Tin)n=1,2,... for i = 1, 2, with each tree Tin count-able with levels, and with each node having at least two immediate suc-

cessors, if Ti =n

Tin/F, then for any isomorphism F : T1 T2 there is

an element a T1 such that the restriction of F to the cone above a is therestriction of an ultraproduct of maps Fn : T

1n T

2n .

From a model theoretic point of view this still is not the right level ofgenerality for a problem of this type. There are two natural ways to posethe problem. From now on

Convention 0.3. In the rest of 0 and 2,3 models are countable withcountable vocabulary if not said otherwise, and we use M, N to denote

models. If we say a model may be uncountable we still assume its vocabularyis countable if not said otherwise.

Problem 1. Characterize the pairs of countable models M, N which arepseudo-isomorphic, where

Definition 0.4. We say that the countable models M, N are pseudo-isomorphicif:

(a) if F is a non principal ultrafilter over then M/F, N/F areisomorphic, and

(b) clause (a) continue to hold after forcing by any (set) forcing .

Of course this is not isomorphism (see below on models of a stable theory).

A related problem isProblem 2. Characterize the pairs of countable models M, N with non-isomorphic ultrapowers modulo any non-principal ultrafilter F, M/F, N/Fin some forcing extension. (I.e., the negation is: such that for every forc-ing extension there is a non-principal ultrafilter F on we have M/F N/F.)

There are two variants of the second problem: the ultrapowers may beformed either using one ultrafilter twice (called 2(A)), or may consider using


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VIVA III 3

any two ultrafilters (called 2(B)), but see below. As when the continuumhypothesis holds is too easy ask:

Problem 3. Characterize the pairs M, N of countable models such thatin some forcing extension failing in continuum hypothesis, for every non-principal ultrafilter F on , M/F = N/F

Problem 4. Let us write M N whenever in every forcing extension, ifF is an ultrafilter on such that N/F is saturated, then M/F is alsosaturated. Characterize this relation.

This is related to the Keisler order (see Keisler [Ke67], or [Sh:a], or [Sh:c,Chapter VI]), but does not depend on the fact that the ultrafilter is regular,so some of the results there apply to Problem 4, this in turn implies results onProblem 2(A). By [Sh:c, VI] we know the following. Let D be a non-principle

ultrafilter on , and M (countable) model (with countable vocabulary). IfTh(M) is stable then M/D is saturated. We can replace 0 here by anycardinal satisfying


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4 SAHARON SHELAH

T2 such that M1 M2. The present paper is dedicated to sheding somefurther light.

Problem 6. We may be more interested in the ultrafilter, so define the orderon the family of ultrafilters on but here our focus is on model theory. Morespecifically, we may ask to investigate uf where F1 uf F2 iff F1, F2 arenon-principal ultrafilter on such that for every countable model M, ifM/F1 is saturated then M

/F2 is saturated.Working on [Sh 405] we had hoped to continue it sometime. However, we

actually began only when Jarden asked:

() Suppose that Fn are finite fields (for n < , = 1, 2) satisfyingF1n F

2n . Can we have (a universe and) an ultrafilter F on such

that n


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VIVA III 5

such that the set of n for which Fn is an isomorphism from N1n onto

N2n belongs to the ultrafilter and n


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6 SAHARON SHELAH

model M or theory T with the obvious meaning. We will use theletters p, q (with sub/super-scripts) to denote types.

(4) The universe of a model M will be denoted |M|, but we will oftenabuse this notation and write, e.g., a M. The cardinality of a setA will be denoted A, and, for a model M, M will stand for thecardinality of its universe.

Comment: Why the 3 ? We like to have a preliminary forcing notion Appwhich for some , is -complete, +-c.c., A tp(a, , M) = tp(a, , N) }.

We call it the (A, M)place.(1) A local bigness notion for K (without parameters, in one variablex) is a function with domain K which for every model M K gives

M = (M) {(x, a) : L() & a M},

+M = +(M) = {(x, a) : L() & a M} \ M

such that(a) M is preserved by automorphisms of M,

(b) M is a proper ideal, i.e., +M = and


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() if M |= (x)((x, a) (x, b)) and (x, b) M, then(x, a) M,

() if 1(x, a1), 2(x, a2) M, then 1(x, a1) (x, a2) M.

Elements of M are called small inM, members of +M are big.

A local bigness notion for K with parameters1 from A is definedsimilarly but Dom() is an Aplace K in K and in clause (a) theautomorphisms are over A.

(2) We say that a local bigness notion is invariant for K (or for anA-place K) if for M N from K (or from the A-place K) we haveM

N and

+M

+N.

(3) A big type p(x) inM is a set of formulas (x, a) all of whose finiteconjunctions are big in M.

(4) A pre tbigness notion scheme is a sentence (in possibly in-finitary logic) in the vocabulary (t) {P}, where P is a unarypredicate, we may say using P.

(5) An interpretation with parameters of t in a model M K is =R(yR, aR) : R (t), where R L() and aR is a sequence ofappropriate length of elements of M. So a predicate R from (t) isinterpreted as

{b : M |= R(b, aR), lg(b) = lg(yR) (= the arity ofR) }.

The interpreted model is called M[] or M[] and we demand thatit is a model of t; so in particular M[] is a (t)-model and itsuniverse is {b M : M |= =(b,b, a=)} defined by =(x,y, a=)which we demand is an equivalence relation; here usually equalityon its domains, so we may write just =(x, a=) or just (x, a); ofcourse we could use k-tuples for elements and then lg(yR) = kn forR an n-place predicate from (t)

(6) For a pre tbigness notion scheme = and an interpretation of t in M K with parameters from A M, we define the derived local pre-bigness notion = , = [] with parameters

from A M (in the A-place KA,M) as follows:Given M KA,M, a formula (x, b) in L() (with parameters fromM of course) is []big in M

if for any quite saturated N,M N, letting

P = {a N[] : N |= [a, b]}

we have (N[], P) |= .In full we may write = (,t,) and even = (,t,,M,A).

(7) We say is a t-bigness notion (for T) if for every interpretation oft in some A-place K K, t,, is an invariant

2 local bigness notion

1Alternatively use the monster model.2the invariant really follows


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8 SAHARON SHELAH

for our fixed K. If there is no T mentioned or understood we meanfor every T. So it is enough in (6) above if we define M when

M M.Proposition 1.2. (1) If is a local bigness notion for K with parame-

ters in A, M KA,M andp(x) is a big type in M, then it can beextended to big type q in M which is a complete type over M.

(2) Assume t,, ,M,A are as in Definition 1.1(6). The truth value of(y, a) is (t,,)-big depends just on (M

, a, c)cA wheneverthe formulas in and belong to L().

Proposition 1.3. For T, K = KT and t as in 1.1,

() if N M are from K, and = R(yR, aR) : R (t) is aninterpretation of t in N, then is an interpretation of t in M (i.e.,M[] |= t) and moreover N[] M[].

The following definition is crucial in our application, the proofs give someamount of definability, a local version and we need to deduce from it aglobal one. This is a good property criterion for closing the gap which havein fact been used for tind, see more systematically in [Sh 800].

Definition 1.4. Let t be a first order theory in a vocabulary (t). Supposethat is a tbigness notion scheme, using P (t), a unary predicate,and (y, x) is a (t)formula. We say that is (2, 1)(P, )separative

whenever the following condition ()P, holds and for simplicity we assume

=(x,y, a=) is equality on its domain3.

()P, For every 2 compact4 model M and every interpretation =

R(yR, aR) : R (t) of t in M and a set X |M| of cardinalityat most 1, including all parameters of we have:

if N M, X |N|, N 1, and p(x) is a []big type overN, p(x) 1, and a1, a2 are distinct members of |M| \ |N|with (recalling 1.1(5))

M |= P[a1, aP] P[a2, aP]

then the type p(x) {(a1, x) (a2, x)} is []big.

We now define the main bigness notion used

Definition 1.5 (See [Sh:e, Def. 3.4, 3.5, Chapter XI]). (1) tind = tind0is the first order theory in vocabulary (tind) = {P,Q,R}, where

P, Q are unary predicates and R is a binary predicate, includingsentences

(x)(y)(x R y P(x) Q(y)), and(x)(P(x) Q(x))

3Otherwise we should inside ()P, , demands further that for any c N we haveM |= =(c, a1, a=) =(c, a2, a=) =(a1, a2, a=).

4A model M is called -compact if every type over it of cardinality < is realized; ifwe omit we mean the cardinality of the model


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and saying that for each n < and any pairwise distinct elementsa1, . . . , a2n P, there is c Q such that

ai R c if and only if i n.

tind1 is tind0 plus

(x)(y)(z)

Q(x) Q(y) x = y P(z) (z R x z R y)

.

(2) We define a pre tindbigness notion scheme ind as follows. Thesentence ind says that P Q and (P,Q,R,P) satisfies:

for every n < , there is a finite set A P such thatfor every distinct a1, . . . , a2n P \ A there is c P

satisfying

a R c for n, and (a R c) for n < 2n.

(So ind is not first order.)(3) We say that a first order theory T has the strong independence

property if some5 formula (x, y) defines a two place relation whichis a model oftind1 with P, Q chosen as the whole model i.e. for M |= Tdefine the tind1

-model M, |M| = |M| = PM

= QM

, RM

= {(a, b) :

M |= (a, b)}In such case we may also say (x, y) has the strong independence prop-

erties (for )

Plainly,Proposition 1.6. (1) For a model M of tind1 , an automorphism of

M is determined by PM (i.e., if 1, 2 Aut(M) are such that1 P

M = 2 PM, then 1 = 2).

(2) Moreover, if is an interpretation of tind1 in M, M = M[],

Aut(M) and PM is definable in M (with parameters inM), then so is .

Proposition 1.7. (See [Sh:e, Chapter XI, 3] and [Sh 107]) ind is a tindbigness notion scheme. It is (2, 1)(P, )separative where P (t

ind0 ) is

given and we choose (y, x) := y R x.

Definition 1.8. A mapping F : N1 N2 is a embedding from N1 toN2 whenever is a set of formulas in L,((N1) (N2)) and

if and N1 |= [a1, . . . , an],then N2 |= [F(a1), . . . , F (an)].

[of course, if is closed under negation, then we have if and only if.]

5of course (x, y), lg(x) = m = lg(y) can serve as well


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2. The forcing notion App

As explained in the introduction, we work in a Cohen generic extension ofa suitable ground model. In this section we present how that suitable groundmodel can be obtained: we start with V |= GCH and we force with theforcing notion App defined in 2.4 below , the App comes for approximations,as the members are approximations to a name for an ultrafilter as we desire.

Definition 2.1. (1) The Cohen forcing of adding 3 Cohen reals is de-noted by C3. Thus a condition p in C3 is a finite partial functionfrom 3 to , and the order ofC3 is the natural one. Thecanonical C3name for

th Cohen real will be called x.

(2) Let A 3. For a condition p C3, its restriction to A iscalled p A, and we let C3 A = CA = {p A : p C3}. Also,

we let A = ()

VC3

A

.(3) For a sequence An : n < of non-empty sets (and A 3), we

defineAn


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VIVA III 11

and() G(A, F

) is a non-empty set of triples (t,

,

), where7 t is a

(countable) first order theory (or just a CAname of a (count-able) first order theory),

is a CAname of tbigness notion

scheme, and

is (a CAname for) an interpretation of t inAn


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12 SAHARON SHELAH

(b) F

is a canonical CAname of a non-principal ultrafilter on ,such that for < 3 divisible by 2,

F (A )

def= F

{a

: a

is a CAname of a subset of }

is a CAname (of an ultrafilter on );Why canonical? for the same reasons as in 2.1(4)

(c)

= : A & cf() = 2, where each

is a local bigness

notion

[

] for some (t,

,

) G(A , F (A ));

(d) If cf() = 2, A, then it is forced (i.e., C3 equivalently

CA) that:

the type realized by the element x in the model

An


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Lemma 2.7. Suppose thatq App, Aq 3, andp

is aCAqname of a

type over the modelAq

n


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(b) p App and p

is as CAp-name of a finitely satisfiable set of

formulas in one free variable x overAp

n


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H1 CA1 is generic over V and H0 {p1} H

1, then in V[H1] we have

{n :

M

n

A1 |= (

y)

A

(y

) &

(y

)} F 1

[H

1]

(remember p

1 is a type overAq1n 0 (otherwise apply 2.6) and 2 > 0 (otherwiselet 2 = 1, 0 = 0, q

0 App 0 be above pi 0 for i < 1; so it just means

F

q0 is an ultrafilter extending F

pi0 for i < 1; now if = 0, then r = q

0is as required and otherwise we have reduced the case 2 = 0 to the case2 = 1).

We may assume that j = sup{ + 1 : Aqj} (for j < 2), and also

that the sequence j : j < 2 is strictly increasing. Let = supj


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The Subcase 1A: = 0 + 1 is a successorIn this case our inductive hypotheses applies to the pi 0, q

, and 0,

yielding r0 in App 0 with pi 0 r0 for i < 1 and q end r0.What remains to be done is an amalgamation of r0 with all of the pi, whereApi Ar0 {0}, and where one may as well suppose that 0 is in A

pi for alli. This is a slight variation on (1) or (2). For instance, suppose cf(0) = 2.We let

A2 =

i


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[Thus, at a successor stage j + 1, the inductive hypothesis is applied to pi j+1, rj , j , and j+1. At a limit stage j, we apply the inductive hypothesis

to pi j for i < 1, rj for j < j, j for j < j, and j .] Finally, we letr = (

j


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For this n we find p CApi witnessing that n c (i.e. p (Api j) H0

and p CApi

n a

) and next we let p = p p. Clearly p n a

b

,

a contradiction.4) Follows, i.e., it is the case 2 = 0 of part (3).

5) We choose qn Appn for n < such thatAqn := Ap { : < n}, p n qn and qn end qn+1 for n < and letA = {Aqn : n < }

This is possible for n = 0 let qn = p n+1, for n = k + 1, let qn App

be such that Aqn = Aqk {n} and qk end q

n, exists by 2.7, and then qn

as required exists by 2.8(1).Let x

be the following CA-name of an -sequence:

x

= xn(n) : n < .

Now we shall choose q such that Aq = A = {Aqn : n < } = Ap {n :n < }, n < qn end q and p q and CA x

realizes p

.Again the only problem is to find a CA- name of an ultrafilter on which

include

Fp

{F

qn : n < } { {n : M

nAp |= (x

(n))} : (x) p

}

As without loss of generality p

is closed under conjunction it is enough toshow that:

if a

is a CAp-name of a member of Fp, n < ,

b

is a CAqn -name of a member of F

qn

(x) is a CAp-name of a formula from p

then CA a b {n : MA

p

|= (x(n))} = . As in previous casesthis is easy.

Lemma 2.9. Assume V |= GCH. The forcing notion App satisfies the3chain condition, it is 2complete, App = 3 and App 2 forevery 3. Consequently, the forcing withApp does not collapse cardinalsnor changes cofinalities, andApp GCH.

Proof. The only perhaps unclear part is the chain condition. Suppose to-wards contradiction that we have an antichain {q : 3 & cf() = 2} App (the index is taken to vary over ordinals of cofinality 2 just for con-venience). An important point is that G can offer at most 2 candidates

for the bigness notion at < 3, cf() = 2, hence for each 3 therestricted forcing App has cardinality 2. Applying Fodors lemmatwice, we find a stationary set S { 3 : cf() = 2} and a conditionq App such that ( S)(q = q). Pick 1, 2 S such thatsup(Aq1 ) < 2; it follows from Lemma 2.8(3) that the conditions q1, q2are compatible, a contradiction.

Proposition 2.10. (1) For each p App and 3, there is a condi-tion q App stronger than p and such that Aq.


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(2) F

def=

{F

r : r GApp} is an App-name of a C3name for a

non-principal ultrafilter on . Also, for each r G

App we have:

F

P()(V[G App])CAr = F

r.

Proof. Should be clear (for (1) use 2.7 + 2.8(3); then (2) follows).

Definition 2.11. (1) Suppose GApp App is generic over V, V =V[GApp]. For 3 we let G = GApp (App ). It is a genericsubset ofApp ; let F

be the (App )-name of the C-name

{F

q : q G}. Note: F

q being a CAq -name is a C-name whenAq . So in V the sequence F

: < 3 is forced (i.e. C) to

be increasing , let F

= F3 so F

is the Cname for the restriction

F of the ultrafilter F

to the sets from the universe (V)C .

(2) We define an Appname

of a Cname as p for every p G

App

such that Ap. (So it is an App Cname.)Lemma 2.12. (1) Suppose that GApp App is generic over V, V =

V[GApp], and < 3, cf() = 2, and H C is generic over V.

Then, in V[GApp (App )][H ], we have9:

n


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20 SAHARON SHELAH

(ii) t

,

,

are CAnames as in 2.2(3),

L((t

)) is a CA-name,equality belongs to it, and

=

(t

,

,

,) is a bigness notion as

there, (t

) is countable; we can assume (t

) is an object (not aname) by adding for each m, 0 predicates with m places said(by t

) to be empty.

(iii) N

n, for n < and {1, 2}, are CAnames for countable

models of a (countable) theory tn, and the universes |N

n| are

subsets of and with vocabulary (t

).

Also it is forced (i.e., C3 ) that t Th

n


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(b) for every C3 name x

for an element ofn


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()2 the truth value of(a, b) R,m depends only onL,(A)type realized by (a, b) over MA in M,

()3 R,m is minimal such that ()1 + ()2 hold.(viii) The relations R,m mentioned above satisfy (i.e. (q A, p

)forces):()1 if a1, a2 are finite sequences of the same length m of mem-

bers of N1 , and p {N1 (x, a1),

N1 (x, a2)} is a big

type over M, and ,

[m], where N1 is as in-

terpreted in the interpretation 1,then (a1, F(a2)) / R,m.

()2 Above, we may replace , by any pair 0, 1 of contra-dictory formulas from

[m].

(ix) Note that also

()py

, p C3

the

type which y

realizes over N

2 = (

n


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() the types of (x, y

) and of (x, y

) overAqn


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24 SAHARON SHELAH

n


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() f

include the identity may onAqn


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26 SAHARON SHELAH

(b) a

is a CAq2 - name such that p Aq2 a

F

q2(c) b

is a CAs-name such that p As b

F

s

(d) c = {n : MnAr

|= [b0(n), . . . , bk1(n)]} where

is a CAq2 -name of a first order formula in the vocabulary Aq2 , withoutloss of generality a predicate as an atomic formula, such that

p CAr

Arn


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Let : < cf(2) be increasing continuous with limit 2 such that0 = 1, cf() < 2 and each is divisible by 2 and stipulate cf(2) = 2.

We choose (r, s, f) by induction on cf(2) such that

(a) + ,q2,r,s,f

holds.

(b) (r0, s0, f

0) = (r0, s , f

)

(c) r r(d) if < then r r, s s and CAr f

f

Clearly if we succeed we are done with case 4.For = 0 this is trivial.For = + 1 first find r App such that r

r and r

= r,

possibly by 2.8(3). Second apply the induction hypothesis with (, , q2 , q2 , r, s, f

, r) standing for (0, 2, q0, q2,r,s,f ,r

).

For limit of uncountable cofinality take the union (see 2.8(4)).For limit of countable cofinality, we first repeat the argument in case 2.Then use 2.8 and then 3.10. 3.5A

4. Back to Model Theory

In this section we present just enough to solve the problem on finite fields.

Definition 4.1. Let M be a model. Assume N1 = M[1] , N2 = M

[2]

are models of t0 interpreted in M by the sequences 1, 2 of formulas with

parameters from M, and they have the same vocabulary = (N1) =(N2). Furthermore, let be an invariant bigness notion in M (over some

set A0 of < parameters, more exactly in K(M,A0)), and L,((N1))and > 0 (for simplicity) and for a formula (x) let (x) be the

result of substituting in so N |= [a] iff a lgx(N) and M |= [a].

(1) We say that (N1, N2) is (, , )complicated in M when:for every embedding F of N1 into N2, and for every big typep0(x) inside M of cardinality < such that p0(M) N1, there is abig type p1(x) inside M of cardinality < which includes p0(x)and such that, letting (p1) (M) consist of those predicates andfunction symbols mentioned in p1(x) (so |(p1)| < ) and A M bethe set of parameters of p0 union with A0 so |A| < and A0 A,we have

()p1(x) letting

Rmdef= {(a, b) : a m(N1), b

m(N2) and for some c m(N1) we have

tpL,((p))(ab ,A,M) = tpL,((p))(c

F(c), A , M ) }

the parallel of 3.6(vii)+(viii) holds, so()1 if a1, a2 are finite sequences of the same length m of members

of N1, and p1 {N11

(x, a1), N11

(x, a2)} is a big type over

M, and , , then (a1, F(a2)) / Rm.


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()2 Moreover, in 1 we can replace , by any pair 0, 1 ofcontradictory formulas from .

(2) In part (1):(i) We do not mention if it is the set of quantifier free formulas

(ofL,((N1))).(ii) We replace by (t, ) if we mean for all bigness notions of the

form = (t,,), where is an interpretation of t in M with< parameters and |t| < , L, (i.e., L+, for some < and in the vocabulary (t) {P}).

(iii) We omit if we mean for all s as in (ii).(iv) We say M is complicated (or: (, , )complicated) and

omit N1, N2 if this holds for all N1, N2 as in our assumptions,but with |(N1)| < .

Remark 4.2. More on the relation Rn etc., see [Sh 800].

Theorem 4.3. Let G be a full (3, 2)bigness guide (see 2.2; recall thereis one by 2.3). Assume that G AppG is generic over V and H C3is generic over V[G] and F = F

3[G][H], and let Mn = M

n3

: n < be a sequence of models as in 2.1(4), that is each with a countable universebeing the set of natural numbers for simplicity, all with the same vocabularysuch that for every k and a sequence Rn : n < with Rn being a kplace relation on Mn there is a k-place predicate in the common vocabularysatisfying RMn = Rn for each n. Then

(1) in V[G][H] the model M =n


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(6) If Nn are finite fields (for = 1, 2 and n < ), andn


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Proposition 4.4. Assume thatM is acomplicatedcompact model. LetN1, N2 be interpretations of t

ind1 in M. Then for any isomorphism from

N1 onto N2, the function is definable in M by a first order formula (withparameters).

Proof. Let N = M[] (so has parameters in M) for = 1, 2 and let F

be an isomorphism from N1 onto N2.Let be the bigness notion (tind,ind,1,) (so

ind L1,). Let p0(x) bethe type just saying x QN1 , and let p1 be the type guaranteed to exists inDefinition 4.1(1), without loss of generality closed under conjunctions. LetA M, |A| < and M, |

| < be given by the definition of beingcomplicated (applied to F). [Without loss of generality, A includes theparameters of 1, 2 and is closed under F and F1, and for every n andfor every formula (x) p1, A includes the finite set mentioned in 1.5(2).]

Let R1 be as in 4.1(1). Clearly, recalling Definition 1.5(2), there are nodistinct a1, a2 P

N1 \ A and b N2 such that (a1, b), (a2, b) R1, buta PN1 (a, F(a)) R1. Hence

{(b, a) : (a, b) R1 and a PN1 }

is the graph of a partial function from PN2 into PN1 which includes the graphof F1 PN2. But F is one-to-one and onto. Therefore, R1 (P

N1 PN2)is the graph of F PN1. But R1 P

N1 is definable in (M , c)cA bya formula from L,, so also F PN1 is, and thus if N1, N2 are models oftind1 also F is (by 1.6). Applying [Sh 72, 1.9] (or [Sh:e, Ch XI]) we concludethat it is definable by a first order formula with parameters from M, as

required. Similarly we can show the following.

Proposition 4.5. Assume that is a (2, 1)(P, )separative bignessnotion, see Definition 1.4. Suppose that N1, N2 are interpretations of tin M, and M is compact complicated (or just complicated for ), > 0.

(1) If F is an isomorphism from N1 onto N2, then()1 F P

N1 is definable in (M , c)cA by a formula fromL,,recalling M, || < , A M, |A| < .

(2) If F is an embedding of N1 into N2, then()2 there is a partial function f from P

N2 into PN1 which extends

F1 and is definable in (M , c)cA by a formula fromL,,where , A are as above.

Remark 4.6. (1) The proposition 4.5 should be the beginning of an anal-ysis of first order theories T. For more in this direction see [Sh 503],[Sh 800].

(2) As stated in the introduction, we may avoid the preliminary forcingwith App and construct the name F

in the ground model V, provided

V is somewhat Llike. Assuming {


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may also use the weaker principle from [HLSh 162] and [Sh 405,Appendix].

(3) We may vary the cardinals, e.g., we may replace 2, 3 by , ,respectively, provided = +, =


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References

[Du80] Jean-Louis Duret. Les corps faiblement algebriquement clos non

separablement clos ont la propriete dindependence. In Model theory ofalgebra and arithmetic (Proc. Conf., Karpacz, 1979), volume 834 of LectureNotes in Math., pages 136162. Springer, Berlin New York, 1980.

[HLSh 162] Bradd Hart, Claude Laflamme, and Saharon Shelah. Models with second orderproperties, V: A General principle. Annals of Pure and Applied Logic, 64:169194, 1993. math.LO/9311211.

[Ho93] Wilfrid Hodges. Model theory, volume 42 of Encyclopedia of Mathematics andits Applications. Cambridge University Press, Cambridge, 1993.

[J] Thomas Jech. Set theory. Springer Monographs in Mathematics. Springer-Verlag, Berlin, 2003. The third millennium edition, revised and expanded.

[Ke67] H. Jerome Keisler. Ultraproducts which are not saturated.Journal of SymbolicLogic, 32:2346, 1967.

[Sh 384] Saharon Shelah. Compact logics in ZFC : Complete embeddings of atomless

Boolean rings. In Non structure theory, Ch X.[Sh 482] Saharon Shelah. Compactness in ZFC of the Quantifier on Complete embed-ding of BAs. In .

[Sh:e] Saharon Shelah. Nonstructure theory, accepted. Oxford University Press.[Sh 800] Saharon Shelah. On complicated models. Preprint.[Sh 13] Saharon Shelah. Every two elementarily equivalent models have isomorphic

ultrapowers. Israel Journal of Mathematics, 10:224233, 1971.[Sh:a] Saharon Shelah. Classification theory and the number of nonisomorphic mod-

els, volume 92 ofStudies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam-New York, xvi+544 pp, $62.25, 1978.

[Sh 72] Saharon Shelah. Models with second-order properties. I. Boolean algebraswith no definable automorphisms. Annals of Mathematical Logic, 14:5772,1978.

[Sh 107] Saharon Shelah. Models with second order properties. IV. A general method

and eliminating diamonds. Annals of Pure and Applied Logic, 25:183212,1983.

[Sh:c] Saharon Shelah. Classification theory and the number of nonisomorphic mod-els, volume 92 ofStudies in Logic and the Foundations of Mathematics. North-Holland Publishing Co., Amsterdam, xxxiv+705 pp, 1990.

[Sh 326] Saharon Shelah. Vive la difference I: Nonisomorphism of ultrapowers of count-able models. In Set Theory of the Continuum, volume 26 of Mathematical Sci-ences Research Institute Publications, pages 357405. Springer Verlag, 1992.

[Sh 503] Saharon Shelah. The number of independent elements in the prod-uct of interval Boolean algebras. Mathematica Japonica, 39:15, 1994.math.LO/9312212.

[Sh 405] Saharon Shelah. Vive la difference II. The Ax-Kochen isomorphism theorem.Israel Journal of Mathematics, 85:351390, 1994.

Institute of Mathematics, 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://www.math.rutgers.edu/ shelah

saharon shelah- vive la difference iii

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