saharon shelah- the theorems of beth and craig in abstract model theory,iii: delta-logics and...

8/3/2019 Saharon Shelah- The Theorems of Beth and Craig in Abstract Model Theory,III: Delta-Logics and Infinitary Logics

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THE THEOREMS OF BETH AND CRAIGIN ABSTRACT MODEL THEORY,III:

-LOGICS AND INFINITARY LOGICSSH113

Saharon Shelah

Institute of MathematicsThe Hebrew University

Jerusalem, IsraelRutgers University

Mathematics DepartmentNew Brunswick, NJ USA

Abstract. We give a general technique on how to produce counterexamples toBeths denability (and weak denability) theorem. The method is then applied forvarious innitary, cardinality quantier logics and -closure of such logics.

I would like to thank Alice Leonhardt for the beautiful typing.AMS-Classication: 03C95, 03C20, 03E55, 03C55, 03C30.The author would like to thank the United States Israel Binational Science Foundation for partiallysupporting this research.First Typed - 1975Latest Revision - 01/Mar/19

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2 SAHARON SHELAH

1 Introduction

For unexplained terms, background and history see the excellent representationin Makowsky [Ma]. We show that in many cases, Beth theorem and weak den-ability fail. This may seem less obviously true, just need proof now then in 1975,when this was done and should have been written. The reason is that then it wasnot unreasonable to think the subject will continue to produce counterexamplesonly. But, by Mekler Shelah [MkSh 166] it is consistent that the logic L( 1 ) hasW.(= weak ) Beth. By [Sh 199] INT L(Qcf j ) , L(aa ) ) holds and the Beth closure

of L(Qcf 2 0 ) (conality at most continuum) is compact.In 2 we concentrate on counterexamples to W. Beth, and in 3 on counterex-

amples to Beth for -closure of L , .

In 2 we gave sufficient conditions for the failure of Beth and Weak Beth (in 2.1-2.7). This involves P r xL 1 , L 2 (, , ) (see Denition 2.1). So we deal with ndinginstances of those properties for logics of the form L , (in 2.8) then discuss whywe cannot have some desirable cases (see 2.9-2.12); we state our conclusion on thefailure of Beth and Weak Beth for logics of the form L , in 2.13. In 2.14-2.18we look at properties of logics related to the properties existence of models withautomorphisms.

In 3 we show -closures do not satisfy Beth. In 3.2-3.4 we give a sufficient(quite general) condition for the failure of Beth (or Weak Beth) for the -closureof L , which include using a counterexample to interpolation ( 1 , 2).

In 3.3 we get specialized conclusion Beth theorem fail for ( L , ( L , )) if inL we can, essentially, have a pseudo elementary classes separating two regularcardinals as conality. For this we rely on the abstract theorem 3.4, but the mainwork is verifying the condition from there which is done in 3.13. Before this 3.6,give specic conclusion (L 2 , and L( )( = cf > 0) does not satisfy Betheven when we look for the explicit denition in ( L , ). This is an example forthe abstract conclusion: -interpolation does not imply Beth.

In the end 3.15, we use a forcing of Gitik to derive a universe where for regular < even in ( L , ) we cannot nd interpolants for L( ) (or any L , L( ) ( L )) .

The main work as mentioned above is in 3.13; which is a kind of generalization

of Morley omitting type theorem, this time controlling conalities of many orders(many a set of linear orders indexed by sets which have to be large themselves);[for few orders see [Sh 18].

I thank Janos Makowsky for his many contributions to this) paper. The readerwould do better to have a copy of [Ma]; (for Theorem 2.8 - [Sh 189]), for 2.10 - [Sh133], [Sh 228], for 2.11(1) - [Sh 129], for 2.12 - [Sh 125], for 3.13 stage [G] : [\Sha ,VII,5], for 3.15 - Gitik [G].


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THEOREMS OF BETH AND CRAIG 3

A logic L is a function such that for any vocabulary , L ( ) is a set of sentences(so for M a -model and L ( ), M |= or M |= , this is preserved byisomorphism. If not said otherwise, L is closed under the obvious operations: ,

, x, and substitution. Of course, 1 2 L [ 1] L [ 2] and L commutewith renaming relation and function symbols.

1.1 Open Problems: 1) W. Beth ( L( ) (i.e. provably in ZFC)?2) for strong limit singular, (or weakly compact) is there L + , all whosemodels have cardinality and are L , -equivalent (but there are at least two)?(See [Sh 228]).3) Is L(Qec ) (equal conality quantier) compact?(Note: 3.13 is an approximation: if we restrict ourselves to a suitable class of cardinals).

Recall:

1.2 Denition. : 1) M + N or [M, N ] is the disjoint sum of M, N (e.g. considerthem as structures of different sorts).2) For logics L 1 , L 2 let P P P (L 1 , L 2)(= pair preservation properties) means, thatfor models M, N , T hL 1 (M + N ) is determined by T hL 1 (M ), T hL 2 (N ) .3) For logics L 1 , L 2INT (L 1 , L 2) means: if 1 L 1( 1), 3 L 1( 2) and |= 1

3 then for some 2 L 2( 1 2) :|= 1 2 and |= 2 3 .4) = (P, R) L ({P }) R ) is a Beth denition (W. Beth denition) if for everyR-model M there is at most one [exactly one] P M | M | , (M, P ) |= .

5) For logicsL

1 ,L

2 , Beth(L

1 ,L

2) [or W. Beth (L

1 ,L

2)] means that: if (P,R)L 1({P }) R}) is a Beth denition (W. Beth denition) of P , then some (x, R)

L 2(R) is an explicit denition of P , i.e.

M = ( A, RM , P M ) |= P M = {a : M |= [a, R]}.

6) INT (L ) means INT (L , L ) and Beth( L ) means Beth( L , L ) and W.Beth( L )means W.Beth( L ).7) L 1 L 2 if for every vocabulary , L 1( ) L 2( ).

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4 SAHARON SHELAH

2 Sufficient Conditions for Beth Failure and Applications

2.1 Denition. 1) P raL 1 , L 2 (, ,) [

L a logic, , are innite cardinals, acardinal] means: (compare with 1.2)

( ) there is a sentence L 1 such that:(i) has only rigid models

(ii ) has exactly models up to isomorphism

(iii ) every model of has cardinality , (remember 0)

(iv)a there is M |= , and a, b M, a = b such that ( M, a ) L 2 (M, b)

(v) the vocabulary of has cardinality < .

2) P r bL 1 , L 2 (, ,) means as in (1) replacing ( iv)a by:

(iv)b there are models M 0 , M 1 of not isomorphic but L 2-equivalent (so > 1).

3) P r cL 1 , L 2 (, ,) means as in (1) when we replace ( iv)a by:

(iv)c if M 1 , M 2 are models of then M 1 L 2 M 2 .

4) P r dL 1 , L 2 (, ,) means as in (1) but instead ( iv)a :

(iv)d we have: (iv)c and ( iv)a for every M |= .

2.2 Observation. 1) There is obvious monotonicity (in L 1 , L 2 , ,); i.e. if L 1 L 1 , L 2 L 2 , , , and x { a,b,c,d} then P r xL 1 , L 2 (, ,) impliesP r xL

1 , L

2(, , ).

2) P r aL 1 , L 2 (, ) implies P rbL 1 , L 2 (

, ) for some , + .

3) If > 1, P rcL 1 , L 2 (, ,) implies P r

bL 1 , L 2 (, ,).

4) P r dL 1 , L 2 (, ,) implies P rxL 1 , L 2 (, , ) for x = a, c.

Proof. 1) - 4). Use the same example.


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2.3 Theorem. 1) If P r bL 1 , L 3 (, ,) and P P P (L 2 , L 3) and L, L 1 L 2 L 3 then Beth (L 1 , L 2) fail.2) Suppose P r aL 1 ,L 2 (1, , ), L, L 1 L 2 L 3 and P P P (L 2 , L 3) then W.

Beth (L 1 , L 2) fail.

Proof. 1) Let M be a model of L 1[ ]. For simplicity assume has no functionsymbols. We construct a { P }-structure B in the following way: B = M { 0}M { 1}, i.e. B is the disjoint union of two copies of M.

For each n-ary relation symbol R let RM be its interpretation in M . Nowwe put R i = {(( a1 , i), (an , i)) : (a1 , , a n ) RM } and R B = R0 R1 .P is a unary predicate and P B = M { 0}. (Alternatively, use multi-sorted models).Let F be a binary relation symbol not in { P }. Let be a { P, F }-sentenceof L 1 expressing that:

(1) Both, the relativized structure on P and on P are models of ;(2) F is a -isomorphisms from P to P ;(3) F is of order two, i.e. F 2 is the identity on P .

Clearly, denes F implicitly, since has only rigid models, by (i) of 2.1. Soassume, for contradiction, that Beth( L 1 , L 2) holds. Let L 2 be an explicitdenition of F . By 2.1(2), have non-isomorphic models M, N that M L 3 N .We use P P P (L

2, L

3) to conclude that

( ) M + N L 2 M + M

where the rst sort represents the interpretation of P and the second sort theinterpretation of P . Clearly, M + M has an expansion satisfying and therefore denes a -isomorphisms between the two sorts. So ( ) says that also denesa -isomorphism in M + N . This is expressed by a sentence in L, (as > | |)hence in L . Therefore M = N contradicting the choice of M, N .2) Like part (1) only in the end we use ( M, a ) L 2 (M, b), a = b where M is (the)model of .

2.4 Remark. The closure properties of the (set of sentences of the logics) L 1 arevery weak: we use a conjunction of a sentence from L, (the isomorphism) andtwo copies of .

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6 SAHARON SHELAH

2.5 Theorem. If P r bL 1 , L 3 (, ,) + + , + + + , L + , = L 0 L 1 L 2

L 3 , P P P (L 2 , L 3) then Weak Beth for (L 1 , L 2) fail.

2.6 Remark. 1) We assume (implicitly) the vocabulary consist of predicates andfunction symbols with nite arity.If we want to delete this assumption we should demand in 2.1, for every model M of :

M + { RM :R a predicate of M } +f

|{(a, b) : F M (a) = b}| :

F M a function of M is .

2) We choose below a proof which does not require that L i are closed under inni-tary operations just include L + , and closure under: relativization for L 1 (deni-tion of 1) and nitary operations for L 2 (see end of argument).3) I do not see many closure requirements on L (except isomorphism of vocabu-laries); only substitute in

xone sort

ysecond sort

[ (x, y ) ( x )rst sort

(x x )]?

Proof of 2.5. Let exemplify Pr bL 1 , L 2 (, , ).Let M i (i < ) be a complete list up to isomorphism with no repetition of

models of , and each has vocabulary , so | | (as | | < , + ) andwithout loss of generality M 0 L 3 M 1 . Without loss of generality the universe of each M i is { : i : i :< a 1 , . . . ,a n (P ) > P M i

(c) (similarly for every function symbol of F )

(d) < M is {< , > : < < ( )}.

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Let be the vocabulary of M . Let 0 L + , ( ) characterize M up toisomorphism. Let [ 1 , 2] be the vocabulary of [N 1 , N 2] when N has vocabulary (e.g. each N nitely sorted). Let 1 L 1([ , ]) be such that:

[N 1 , N 2] |= 1 (N 1 |= 0 and N 2 |= ) .

Let F be a new unary function, and let 2 , 3 L | | + , ([ , ] + {F }) say:

[N 1 , N 2] |= 2 iff F as a one to one function fromN 2 into N 1 and there is x N 1 such that :Rang F = {y : N 1 |= R[x, y ]} and for every predicateP of , and a1 , . . . ,a n (P ) N 2 ,

N 2 |= P (a1 , . . . ,a (P )) iff N 1 |= QP (x, F (a1),. . . ,F (an (P )))(similarly for function symbols) .

[N 1 , N 2] |= 3 iff F is as above, but the

x there is < N 1 -rst .

Now every model of 1 can be expanded uniquely to a model of 2 (equivalentlyto a model of 1 2):however, [M , M 0] L 1 [M , M 1] as M 0 L 3 M 1 (by ( iv)b of Denition 2.1(2) andP P P (L 1 , L 3)). If some (x, y ) L 2[ , ] dene F for models of 1 (as wouldbe the case if W. Beth ( L 1 , L 2) holds), then using (x, y ) we easily distinguishbetween [M , M 0] and [M , M 1]. Now expanded by the function it dened.

So one is a model of 3 the other not. So W. Beth( L 1 , L 2) fail as required.

2.7 Conclusion. If

(a) L, L 1 L 2 L 3(b) L 1[ ] has a unique model M up to isomorphism, | | < , M is rigid,

M + | | and for some a = b from M , (M, a ) L 3 (M, b)(c) P P P (L 2 , L 3)

then W. Beth( L 1 , L 2) fail.

Proof. By 2.5.

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8 SAHARON SHELAH

2.8 Theorem. 1) If 1 < = 0 , = cf () > 0 then P r cL + , ,L , (,,+ )

(hence P r bL + , ,L , (, ,+ )) .

2) If = cf () > 0 then for some , , P r bL + , ,L ,

(, ,+ ).3) Really P r xL + , ,L + , (,, 0) hold for x = a,b,c,d provided that [x = b, d |= >1] (with little effort, but this gives little more than the previous parts).

Proof. Essentially from [Sh 189].1) By [Sh 189, Fact 3.10,p.46] there is a smooth ( -system) A , A = = , withE (A ) = every h i,j is onto G i ( = cf( ) > 0 is understood).

Now look at the proof of [Sh 189, Fact 3.11,p.47], it proves all we want.We dene models M a for a Gr (A ); note |vocabulary ( M a )| .

By [Sh 189, 3.12,p.47]

M a = M b iff a b Fact( A ).

By [Sh 189, 3.13,p.48] any two M a , M b are L , -equivalent. Let be the sentenceexpressing ( ) of [Sh 189, 495]. By [Sh 189, 447 - 4413 ] {M a : a Gr (A )} is theclass of models of up to isomorphism. As

E (A ) = Gr (A )/ Fact( A )

we nish.

2) Like above using [Sh 189, 3.8] instead 3.15.3) Not used, and is easy, so we left it to the reader.

2.9 Denition. For a model M .1) For logic L

noL (M ) = {N/ = : N = L M , N = M } .2) When L = L , M we omit it.

2.10 Theorem. 1) If is a weakly compact cardinal or = 2 then P r aL + , ,L , (,, 0).

2) If is singular strong limit cardinal, < or = 2 then 1)s conclusion holds.

Proof. 1) By [Sh 133] there is a model M , M = | vocabulary( M )| andno(M ) = (see Denition 2.9) (decreasing the vocabulary is easy, see [Sh 189]).2) Use [Sh 228].

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2.11 Lemma. 1) Suppose V = L and is regular uncountable not weakly compact then P r bL , ,L , ( ,,

+ ) fail (hence by monotonicity P r xL + , ,L , ( ,,+ ).

2) If L 1 L 2 L 3 and

( ) there is no model M, M + |vocabulary (M )| = = M , 1 < noL 2 (M )

then P r xL 1 , L 3 ( ,,< ) fail for x = a,b,c,d .

Remark. The assumption V = L is necessary in 2.11, see 2.12.

Proof. 1) Follows from (2) as ( ) of part (2) of this lemma holds (for = 1 by

Palyutin, for 1 , by [Sh 129].2) By 2.2 without loss of generality x = b, and let exemplify P r bL 1 , L 2 ( ,,< ) and we shall eventually get a contradiction.

Let M 0 , M 1 be as in (iv)b of Denition 2.1. So in K = {M : M = , M L ,M 0} there are at least two non-isomorphic models. By ( ), in K there are at least+ non-isomorphic models, but as L 1 L 2 L 3 , all of them are models of , so (ii) of Denition 2.1 (applied to ) is contradicted.

2.12 Fact: In some generic extension of V :

(i) V |= G.C.H.(ii ) for some L 2 , , all models M of are L , 1 -equivalent, and no L , 1 (M ) =

0 .

Proof. By [Sh 125]. (Let G be a strongly 1-free abelian group of cardinality 1 .Let H = G Z , h the natural projection from H onto G. Considering H,G, Z asabelian groups, let M be [H,G, Z ] enriched by h and individual constant for everyc G Z ). Now as G is strongly 1-free, if h : H G is a homomorphism onto G,Ker h = Z then [H , G, Z ; h] L , 1 M . Easily (if you understand the denitions)

no(M ) = |Ext (G, Z ) |. But by [Sh 125], |Ext (G, Z )| can be 0 .

2.13 Conclusion. 1) If is regular, > 0 then W.Beth( L (2 ) + , , L , ) fail.2) If is weakly compact > 0 or strong limit, 0 < cf() < then W.Beth( L + , , L , )fail.3) If = cf > 0 then Beth( L + , , L , ) fail.

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10 SAHARON SHELAH

Proof. 1) Use 2.5 for L 1 = L (2 )+ , , L 2 = L 3 = L , ; = 2 = , = + (and there is 2 here) now we have to verify the hypothesis: P r bL 1 , L 3 (, ,) by 2.8(1);P P P (L 2 , L 3) is well known. (I think it is due to Malitz see [Ma]).

2) Like (1) but use 2.10 instead 2.8(1).3) Use 2.3(1) and 2.8(2).

Remark. 1) So by 2.11 in 2.8 we cannot replace = by . Can we replaceit by = + ? Note, that the proof of 2.11, use only the following consequences of V = L on

(a) every stationary S has a stationary subset S 1 which does not reect.Moreover, S 1 satises a square principle:

( )S 1 there is C : < , limit , C is a club of disjoint to S 1 ,

C & = sup( C ) C = C

(b) for every stationary S the weak diamond is satised (see Devlin Shelah[\DSh:65 ],[\Shb , Ch.XIV, 1].

Now those demands are not hard, e.g. dene

(a) there is a closed unbounded subset C of and a function h : C ,h() < such that < [h 1({}) does not reect and it satises ( )].

Now (a) (a) , (a ) holds in L, and if V |= ( a) then ( a) holds in anyextension of V (in which is still a regular cardinal).

Now we continue [Sh 199, 3].2.14 Denition. Let L 1 L 2 be logics.

(i) We say that the pair of logics L 1 , L 2 has the Local Homogeneity Property, if for every -structure M and c1 , c2 M such that < M, c 1 > L 2 < M, c 2 >and every T hL 1 (< M, c 1 , c2 > ) there is model < N, c N 1 , cN 2 > |= anda -automorphism g of N such that g(cN 1 ) = cN 2 . If L 1 = L 2 we just saythat L

1has the Local Homogeneity Property

(ii ) We say that L has the Local Automorphism Property, if for every -structure M and innite subset P M , every sentence of the theoryT hL (< M,P > ) has a model < N, P > which has an automorphism g of N such that g P = Id .

We now dene a Local Automorphism Property for pairs of logics.

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2.15 Denition. 1) We say that the pair of logics L 1 , L 2 has the Local Auto-morphism Property, if for every -structure M , every innite subset P M , andfor every a, b P , a = b such that < M,a > L 2 < M, b >, every sentence of the

theory T hL 1 (< M,P > ) has a model < N, P

> which has an automorphism g of N such that g P = Id.2) We dene similarly when ( L 1 , L 2) has the Local Homogeneity Property.

Note that we do not require in 2.14(i) that the automorphism interchanges aand b.

2.16 Theorem. Let L i , i = 1 , 2, 3 be three logics such that L 1 L 2 L 3 and that INT (L 1 , L 2) and P P P (L 2 , L 3) hold. Then the pair of logics (L 1 , L 3) hasthe Local Homogeneity Property.

Proof. Let M and c1 , c2 M be as in the hypothesis of the Local HomogeneityProperty (see Denition 2.14(1)). Let M , c1 , c2 be a disjoint copy. Put N =[M, M ]. Put T = T hL 2 (< N, c 1 , c2 , c1 > ) = T hL 2 (< N, c 1 , c2 , c2 > ). The equalityholds because of P P P (L 2 , L 3). Let c1 , c2 be constant symbols with interpretationsc1 , c2 and c be a constant symbol with interpretation c1 or c2 respectively. Let F 1 , F 2be two new function symbols. Let i , (i = 1 , 2) be the sentence which says that F iis a -isomorphism (modulo name changing) mapping the rst sort into the secondsort which maps ci into c.

Clearly, T { i } has a model for each i = 1 , 2. Let T L 1[ ]. If { , 1 , 2}

has a model [M 1 , M 1 , F 1 , F 2] we get the required automorphism in M 1 from thecomposition of F 1 and F 12 . So assume that { , 1 , 2} has no model. We now

apply INT (L 1 , L 2) and nd L 2[ ] such that 1 and 2 areboth valid. But T and T has models which allow expansions satisfying 1and also models which allow expansions satisfying 2 , a contradiction.

2.17 Theorem. Suppose

(a) L 1 L 2 L 3(b) P P P (L 2 , L 3)

(c) If L 2 ( { P, Q }) ( -vocabulary, P, Q predicates)

and: for every -model M

|{Q : for some P (M,Q,P ) |= }| 1

then for some (x) L 1( ) for every -model M


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12 SAHARON SHELAH

(M,Q,P ) |= |= Q = {a : M |= [a]}

Then (L 1 , L 3) has the weak Local Automorphism Property (which means, in 2.15(1)we add to the hypothesis |P M | > h ( | |), h a function depending on L ).

Of course,

2.18 Observation. If the pair ( L 1 , L 2) has the Local Homogeneity Property thenthis pair has the Local Automorphism Property.

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3 -Closure does not help Beth

Our main Theorem is 3.5, but most of the work is done in 3.13, and the interestis exposed in the conclusions 3.6 (specic logics), 3.7 (counterexamples in abstractmodel theory).

Essentially the theorem says that ( L , ) is far from having Beth property:even for implicit denitions in quite weak logics (like L( 1 ), L 2 , ).

The proof uses the idea of Morley omitting type theorem (in Stage I of the proof of 3.13), Hutchingtons idea of using a model with large conality so that it hasmany L , -elementary submodels with conality 0 and with conality 1 , andthe proof of not Beth( L ) (see the exposition [Ma]).

However, the following theorem isolates a sufficient condition for the failure of the Beth (weak Beth) Property for -logics.

Recall:

3.1 Denition. For a logic L , let 0(L ) be the minimal cardinal 0 such thatfor every vocabulary and L [ ] for some of cardinality < : if M 1 , M 2are models and M 1 = M 2 then M 1 |= M 2 |= .

3.2 Denition. Let L be a logic with dependence number 0( L ) . Let (P ) =1(P ) 2(P ) be a L ( { P })-formula and 1(P ), 2(P ) are contradictory.1) M is a (, )-counter example for (strictly speaking for 1(P ), 2(P ), P ) if M has vocabulary , | | < and: for every expansion of M to a -structure M ,and T T hL (M ), where |T | < , (note ), P / , and card( ) < , thereare -structures N 1 , N 2 such that:

(i) N |= T (ii ) N |= ( P ) [(N , P ) |= (P )] for = 1 , 2.

2) If = 0 , = 0( L ) we say M is an L -counterexample for .

3.3 Remark. We can use for (P ), (P ) (P ), (P ) (P ).

3.4 Fact: Let L be a logic with dependence number 0( L ) . Let (P ) be aL ( { P })-formula which is an implicit (weak implicit 1) denition of P.1) Assume that M is an L -counter example for . Then ( L ) does not have the

1 i.e. for every -model there is a unique P

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14 SAHARON SHELAH

(Weak) Beth Property.2) If M is a (, )-counter example for then ( L ) fails the FWROB (see [Ma]).

Proof. 1) Let (P ) be a L ( { P })-formula which is an implicit (weak implicit)denition of P . Assume for contradiction that there is an explicit denition for P given by a formula (x) ( L ). Since ( L ) has the same dependence numberas L , there are a vocabulary # with # and card( # ) < and 1(x),2(x) L ( # ) which forms the -denition of . This means that for every -model M and a M there is a unique { 1, 2} such that for some expansion M #of M to a # -model M |= [a]. Let for simplicity # = {R i : i < }, and letR i a predicate with ( n(R i ) + 1)-place. Now let M a L -counterexample to (P ),for every a M let (a) { 1, 2}, R ai n (R i ) |M | for i < be such that:

(M, Rai )i b : b R ai where a M }, M =: ( M, R i )i

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3.6 Conclusion. 1) If > 0 is regular, any cardinal then Beth( L, ( ), ( L , ) )and FWROB( L, ( ), ( L, )) fail.2) In particular this holds for the logic with the quantier: there are uncountably

many.3) Beth( L2 , , ( L , ), FWROB( L 2 , , ( L, )) fail.4) For > 1 Beth(( L , )) fail and Beth(( L, )), FWROB(( L, )) fail.

3.7 Corollary. Let L 1 , L 2 be two logics. Then

(i) -Int (L 1 , L 2) (see denition below) does not imply BETH (L 1 , L 2)(ii ) -Int (L 1 , L 2) does not imply WFROB (L 1 , L 2).

The subsequent denitions and lemmas lead to a proof of 3.5.3.8 Denition. -Int( L 1 , L 2) means: if L 1[ ] for = 1 , 2, = 1 2 andthe class of -models is the disjoint union of

K 1 = {M : M a 1-model satisfying 1} andK 2 = {M : M a 2-model satisfying 2}then K 1 is the class of -models of for some L 2[ ].

Explanation: What is the point of the following game? We, on the one hand,want to build a type of indiscernibles to be able to control the conality. On theother hand, if the type say it is bounded, we are lost. If the conality is weaklycompact, we can use partition theorems, but maybe the class of weakly compactis bounded. But restricting ourselves to rapid sequences solve the dilemma itensures unboundedness, and gives one type.

3.9 Denition. Suppose M is a model, A a subset of |M |, < a two place relationwhich linearly orders A with no last element, and a set of formulas (x, d) in thevocabulary of M with parameters from |M | .1) We say a i : i < is rapid for ( A, < ) over inside M if in the following gameplayer I has no winning strategy:

a play last moves:in the -th move player I choose bi A and player II choose ci A,such that bi < c iin the end player II wins if for every (x1 , x2 , . . . ,x n , d) , and i1 appearH () } D .

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3.13 Main Lemma. : Suppose

(a) L + , [ ], | | (where is a vocabulary);

(b) M |= , ( = vocabulary of ), |= T hL , (M )(c) T is a non-empty set of nite sequences of ordinals, closed under initial

segments, T , and for every T for some cardinal > :

 T i <

(d) T = P M , < M = { , < j > ) : T , i < j < }, F M () = ,P M n = { T : g() = n}, R a coding function on { : i < }

(e) = = for , T ; k < for k < g(), T and for simplicity [, T , g() < g ( ) |= < ]

(f ) for T , is regular and (2 )+ ( ) where =: + { : < })(g) 1 = 2 are regular cardinals < Min

T

(h) 0 , the relation belongs to 0 , K 0 a class of 0-models, (closed under isomorphism), such that for every T : (M 0) { : i < }belongs to K 0

(i) if M K 0 then < M linearly ordered |M |( j ) if M K 0 , { 1, 2}, M = and cf (|M | , < M ) = then for some Q,

(M, Q ) K (k) K is a class of ( 0 { Q})-models closed under isomorphism and K = {N

0 : N K } and K 0 , K 1 are disjoint

then there is a -model N of and Qn

P N n such that:

(1) P N 0 Q (necessarily |P N 0 | = 1 )(2) for every x P N n letting N x = ( N 0) {y P N n +1 : F n (y) = x}, the

following are equivalent:(i) cf (N x , < ) = 1

(ii ) cf (N x , < ) = 2(iii ) (N x , Q N x ) K 1(iv) (N x , Q N x ) / K 2 (hence N x / K 1)


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( ) (M 2 , H 2) is an approximation, ( M 1 , H 1) (M 2 , H 2).( ) there is a predicate R (M 2) (M ), R = {(, a) : a H 1()}.

Proof. First nd an expansion M 3 of M 1 by suitable R, so | (M 3) | . Second ndan expansion M 2 of M 3 which has Skolem functions and | (M 2)| . Lastly wend H 2 : just apply 3.11 for each Dom H 1 and apply 3.11 with H 1(), (B , < ), = { : an L, [ (M 1)]-formula with parameters from A }, = { : anL, [ (M 2)]-formula with parameters from A } here standing for a i : i < ,(A, < ), , there. What we get ( bi : i < ( )+ ) will be called H 2().

Stage F - Claim: If ( M 1 , H 1) is an -approximation + k m , n ,m,k < then there is a -approximation ( M 2 , H 2) which is (n,m,k )-indiscernible and

(M 1

, H 1

) (M 2

, H 2

).

Proof. By polarized partition relation of Erdos Hajnal Rado [EHR]. See represen-tation (in our terminology) [Sh 18], (or [ \Sha , AP]) (or Erdos Hajnal Mate Rado[\EHRM ]).

Stage G: Rest of the proof of Theorem 3.13.We use the style of [ \Sh18 , 5] see [\Sha , VII,5].

Let be regular, large enough, such that M H ( ), < a well ordering of H ( ),and let A = {i : i } (M ).

LetA

= ( H ( ), , < , M , a)a A . There is a modelA

of T hL , (A

) such thatA omit every type which A omits and {t : A |= t is an ordinal < (2 ) + ()} isnot well ordered. So there are tn A such that:

|= tn is an ordinal < (2 )+ ()

|= tn +1 + ( n + 1) 2 < t n .

In A we dene by induction on n, Q n , (M n , H n ) such that:

( ) A |= ( M n , H n ) is a tn -approximation( ) A |= ( M n , H n ) (M n +1 , H n +1 )( ) If n = 2( m2 + k), k < m then ( M n +1 , H n +1 ) is (m, k )-indiscernible and

(M n +1 , H n +1 ) relates to ( M n , H n ) as in Stage E () R n (M n +1 ) (M n ), for n = 2 m + 1

R n = Rn +1 = {(, a) : a H n ()}.


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20 SAHARON SHELAH

For n = 0 use stage D.For n = + 1, odd use stage E to dene ( M n , H n ).For n = + 1, even use stage F to dene.

Now let n = (M n ), N n be the Skolem Hall of in M n

(more exactly M n

asinterpreted in A). So (N n ) = n , n n +1 , | n | , N n N n +1 (i.e. N n(N n +1 n )) and let N =

n

N n , = n . So N n N .

We now dene by induction on n model N n and Qn such that:

(A) N 0 = N , (N n ) =

(B ) N n N n +1 , N =n

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Proof. Immediate by Gitik [G], but to clarify we deal with it lengthily. We use thefollowing theorem of Gitik [G]:

( ) if is a measurable of order , a regular cardinal < and < then forsome forcing notions P ,, ,Q

,, ( V P ,, ) the following holds.

P ,, is -pseudo complete (see [Sh:b, Ch.X]) does not collapse cardinals, retainthe regularity of .Q

,, is -pseudo complete, does not collapse cardinals, change the conality of

to and add no new sequences of ordinals of length < .We also have (really follows generally, see [Sh 250]) minimal elements and order 0 witnessing to -pseudo completeness.

Suppose W is a set of triples ( ,, ), of regular cardinals < < such that

=1 ,2

(, , ) W 1 < 2 1 < 2 .

Let W [ ] = {(,, ) W : < }. We dene P W as

p : p = p(,, ) : (,, ) W ,

p(,, ) is a P W [ ]-name of a member of

P ,, such that

{(,, ) : 0 p(,, ) } is nite

with obvious order (both and 0 ).Next QW V P W is dened as:

q :q = q(,, ) : (,, ) W

q(,, ) (Q( ,, ) )V P

W [ ] and

{(,, ) W : 0 0 q(,, ) } is nite

order the obvious one ( and 0 ).

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22 SAHARON SHELAH

Now P W , QW has the expected properties. We force with the iteration R =R i : i < (support as in P W ) where for each i for some i , R i H (i ),

R i { P W i , P W i , QW i }, 2 i < Min{ : (,, ) W i }.

The rest is left to the reader.References

[DSh 65] K. Devlin and S. Shelah, A weak form of the diamond which follows from2 0 < 2 1 , Israel J. of Math. 29(1978), 239-247.

[EHR] P. Erdos, A. Hajnal and R. Rado, Partition relations for cardinal numbers,Acta Math. Sci Hung. 16(1965), 193-196.

[EHMR] P. Erdos, A. Hajnal, A Mate and R. Rado, Combinatorial set theory:partition relations for cardinals, North Holland Publ. Co., Amsterdam (1984).

[G] M. Gitik, Changing conalities and the non-stationary ideal, Israel J. Math(1986,7).

[HoSh 109] W. Hodges and S. Shelah, Innite reduced products, Annals of MathLogic 20(1981), 77-108.

[HoSh 271] , There are reasonable nice logics, J. of Symb. Logic.

[Ma] J.A. Makowsky, Compactness embeddings and denability, Model TheoreticLogics, ed. J. Barwise and S. Feferman, Springer-Verlag group, pp. 645-716.

[MaSh 62] J.A. Makowsky and S. Shelah, The theorems of Beth and Craig inabstract logic I, Trans. A.M.S. 256(1979), 215-239.

[MaSh 101] , The theorems of Beth and Craig in abstract modeltheory, II, Compact Logics, Proc. of a Workshop, Berlin, July 1977, Archive furMath. Logik, 21(1981), 13-36.

[MaSh 116] , Positive results in abstract model theory; a theoryof compact logics, Annals of Pure and Applied Logic, 25(1983), 263-299.

[MShS 47] J.A. Makowsky, S. Shelah and J. Stavi, -logics and generalized quan-tiers, Annals of Math. Logic 10(1976), 155-192.

[MkSh 166] A.H. Mekler and S. Shelah, Stationary logic and its friends, I, Proc. of

the 1980/ 1 Jerusalem Model Theory year, Notre Dame J. of Formal Logic, 26(1985),129-138.

[MkSh 187] , Stationary logic and its friends II, Notre DameJ. of Formal Logic, 27(1986), 39-50.

[PiSh 130] A. Pillay and S. Shelah, Classication over a predicate I, Notre DameJ. of Formal Logic, 26 (1985), 361-376.


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