Saharon Shelah- On λ Strong Homogeneity Existence for Cofinality Logic

8/3/2019 Saharon Shelah- On Strong Homogeneity Existence for Cofinality Logic

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ON STRONG HOMOGENEITY EXISTENCE FOR COFINALITY

LOGIC

SH750

SAHARON SHELAH

Abstract. Let C= Reg be a non-empty class (of regular cardinals). Then

the logic L(QcfC

) has additional nice properties: it has the homogeneous modelexistence property.

Date: December 23, 2009.The author thanks Alice Leonhardt for the beautiful typing. The author would like to thank

the Israel Science Foundation for partial support of this research (Grants No. 451/99 and 710/07).First version 1996. First Typed - 08/June/17.

1


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2 SAHARON SHELAH

0. Introduction

We deal with logics gotten by strengthening of first order logic by generalizedquantifiers, in particular compact ones. We continue [Sh:199] (and [Sh:43])

A natural quantifier is the cofinality quantifier, Qcf (or QcfC), introduced in

[Sh:43] as the first example of compact logic (stronger than first order logic, ofcourse). Recall that the uncountably many xs quantifier Qcard1 , is 0-compact

but not compact. Also note that L(Qcf) is a very nice logic, e.g. with a niceaxiomatization (in particular finitely many schemes) like the one of L(Qcard1) ofKeisler. By [Sh:199], e.g. for = 20, its Beth closure is compact, giving thefirst compact logic with the Beth property (i.e. implicit definition implies explicitdefinition).

Earlier there were indications that having the Beth property is rare for such logic,see e.g. in Makowsky [Mak85]. A weaker version of the Beth property is the weakBeth property dealing with implicit definition which always works; H. Friedmanclaim that historically this was the question. Mekler-Shelah [MkSh:166] prove thatat least consistently, L(Qcard1) satisfies the weak Beth property. Vaananen in themid nineties motivated by the result of Mekler-Shelah [MkSh:166] asked whetherwe can find a parallel proof for L(Qcf) in ZFC.

A natural property for a logic L is{0z.5}

Definition 0.1. A logic L has the (strong) homogeneous model existence propertywhen every theory T L(), (so has a model) has a strongly (L, 0)-homogeneousmodel M, so M = and M is a model of T and M satisfies: if a, b

>M realizethe same L()-type in M then there is an automorphism of M mapping a to b.

This property was introduced in [Sh:199] being natural and also as it helps toinvestigate the weak Beth property.

In 1 we prove that L(QcfC) has the strongly 0-saturated model existence prop-erty. The situation concerning the weak Beth property is not clear.

{0z.9}Question 0.2. 1) Does the logic L(QcfC) have the weak Beth property?2) Does the logic L(Qcf1 , Q

cf2

) has the homogeneous model existence property?The first version of this work was done in 1996.

{0z.21}

Notation 0.3. 1) denotes a vocabulary, L a logic, L() the language for thelogic L and the vocabulary .2) Let L be first order logic, L(Q) be first order logic when we add the quantifierQ.3) For a model M and ultrafilter D on a cardinal , let M/D be the ultrapowerand jM,D = j

M,D be the canonical embedding of M into M

/D; of course, we can

replace by any set.4) Let LST (theorem/argument) stand for Lowenheim-Skolem-Tarski (on existenceof elementary submodels).

Concerning 0.1, more generally{0z.23}

Definition 0.4. 1) M is strongly (L, )-saturated (in L = L we may write just )when

(a) it is -saturated (i.e. every set ofL(M)-formulas with < parameters fromM and < free variables which is finitely satisfiable in M is realized in M


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ON STRONG HOMOGENEITY EXISTENCE FOR COFINALITY LOGIC SH750 3

(b) if < and a, b M realize the same L(M)-type in M, then someautomorphism of M maps a to b.

2) M is a strongly sequence (L, )-homogeneous when clause (b) above holds.3) M is sequence (, )-homogeneous when: L(M) and if < , a M, b M and tp(a, , M) = tp(b, , M) then for every c M for some d M we havetp(bd, , M) = tp(ac, M).3A) 1() is the set of formulas of the form (x) = (y)(x, y) where (x, y) isquantifier free first order formula in the vocabulary .4) We may omit sequence.

{0z.26}Definition 0.5. 1) The logic L has the strong -homogeneous existence propertywhen every theory T L(1) has a strongly (L, )-homogeneous model.2) Similarly the strong -saturated existence property, etc.


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4 SAHARON SHELAH

1. On strongly saturated models

We prove that any theory in L(QcfC) has strongly (L(QcfC), )-saturated model

when C\ / {, Reg\} of course.{2b.1}

Definition 1.1. Let {1, 2} and C be a class of regular cardinals such thatC = , Reg.

1) The quantifier Qcf()C is defined as follows:

syntactically: it bounds two variables, i.e. we can form (Qcf()

C x, y), with its set offree variables being defined as FVar()\{x, y}.

syntactically: M |= (Qcf()

C x, y)(x,y, a) iff (a) + (b) holds where

(a) relevancy demand:the case = 1: the formula (, ; a)M define in M a linear order with no

last element called

M,a on the non-empty set Dom(

M,a) = {b M :M |= (y)((b, y; a)}The case = 2: similarly but M,a is a quasi linear order on its domain

(b) the actual demand: M,a has cofinality cf(M,a), (necessarily an infinite

regular cardinal) which belongs to C.{2b.1d}

Convention 1.2. 1) Writing QcfC we mean that this holds for Qcf()C for = 1 and

for = 2.2) Let -order mean order when = 1 and quasi order when = 2; but when we

are using Qcf()

C then order means -order.{2b.2}

Definition 1.3. 1) As { L(QcfC) : has a model} does not depend on C(and is compact, see [Sh:43]) we may use the formal quantifier Qcf, so the syntex

is determined but not the semantics, i.e. the satisfaction relation |=. We shallwrite M |=C or M |=C T for the interpretation of Qcf as QcfC, but also can sayT L(Qcf)() has model/is consistent.2) If C is clear from the context, then Qcf stands for Q

cfC if = 1 and Q

cfReg\C if

= 0.{2b.4}

Convention 1.4. 1) T is a complete (consistent has models) theory in L(Qcf)which is closed under definitions i.e. every formula = (x) is equivalent to apredicate P(x) so P (T), as in [?], i.e. T (z)[(z) P(z)].2) Let T = T (first order logic), i.e. T = T L(T), it is a complete first ordertheory.3) C Reg, we let C1 = C and C0 = Reg\C, both non-empty.

{2b.6}

Theorem 1.5. Assume = cf(), =


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(b) if (z) = (Qcf )(x, y; z) then: M |= P(x)[a] iff (y, z; a) define in M alinear order with no last element and cofinality

(c) M is strongly1 -saturated model of T. {2b.7}Remark 1.6. 1) We can now change ,, and M by LST. Almost till the endinstead 2|T| + + just |T| + + suffice. The proof is broken to a seriesof definitions and claims. The 2|T| is necessary for 0-saturativity.3) We can assume V satisfies GCH high enough and then use LST. So + = 2

below is not a real burden.{2b.8}

Definition 1.7. 0) ModT is the class of models of T.1)

(a) K = {(M, N) : M N are from ModT}

(b) K = {M : M = Mi : i < satisfies Mi ModT and i < j Mi Mj}(so K = K2)

(c) K = {M K : Mi for i < }, but then we (naturally) assume < +

(d) let = T {P : < } {R(x,y,z), : (x,y, z) L(T), < }, eachP a unary predicate and each R(x,y,z), is an (g(z) + 1)-place predicateand no incidental identification (so P / , etc.)

(e) for M K let m(M) be the -model M with universe {M : < }

MT = {M : < }

PM = M

RM(x,y,z), = {ca : (x,y, a) a linear order, a

g(z)(PM ) such that

M |= P(Q

cf

0 x,y)(x,y,z)

[a] and c Dom(

M,a

) and [b Dom(

M,a) PM b

M,a c}

(f) let m0(M) be the -model {M : < } so m0(M) = m(M).

2) Assume (M, N) K for = 1, 2 let (M1, N1) (M2, N2) mean that clauses(a),(b),(c) below hold and let (M1, N1) K (M2, N2) mean that in addition clause(d) below holds, where:

(a) M1 M2

(b) M2 N1 = M1

(c) N1 N2

(d) ifM1 |= P(Qcf0 x,y)(x,y,z)[a], c N1, c Dom(N1,a) and in N

1 the element

c is N1,a

-above all d Dom(M1,a

), then in N2 the element c is N2,a

-

above all d Dom(M2,a).

3) For M1, M2 K let M1 M2 means < < (M1 , M1) (M

2 , M

2);

similarly M1 K M2 means M1, M2 K and < < (M1 , M

1) K

(M2 , M2).

4) For M K, D an ultrafilter on we define N = M/D,jM,D = jM,D naturally:

N = M /D for < and jM,D = {jM ,D : < }, recalling 0.3.

1as T has elimination of quantifiers, doing it for L(QcfC

) or for L is the same


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6 SAHARON SHELAH

{2b.10}Fact 1.8. 0) For M1, M2 K we have

(a) M1 K M2 iff m(M1) m(M2)

(b) (m(M)PM )T = M

(c) M1 K M1 implies M1 M2.

1) (K, ) and (K, K) are partial orders.2a) If M1 K M

2 in K and 0 < < then (


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(b) jM ,D(M) : < = M K M

: < , so

(c) for many Y [{M : < }] we have j

M,D

(M) K MY : <

K ; see 1.8(5), 1.12(3).

Proof. 1) See [Sh:199, 4]; just let D be a regular ultrafilter on M1 + ||, letg an elementary embedding of M1 into (M0)

/D extending j = jM0,D, necessarilyexists.

Lastly, let M2 (M1)/D include jM1,D(M

1) g(M1) be of cardinality . Iden-

tifying M1 with jM1,D

(M1) (M1)/D we are done.

2) Similarly.3) Trivial.4) Should be clear. 1.9

{2b.14}Definition 1.10. Kec is the class of M K such that: if M K N K, then

m(M) 1 m(N), i.e. () below and Kec,

= Kec K where

() if a1, . . . , an m(M), b1, . . . , bk m(N), L(+ ) is quantifier free andm(N) |= [a1, . . . , an, b1, . . . , bk] then for some b

1, . . . , b

k


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8 SAHARON SHELAH

Now let D be a regular ultrafilter on = m(M2) and use 1.9(2),(3). This isfine to get (f2, N) with N K and by 1.8(5) without loss of generality N K

and by 1.11(1) without loss of generality N Kec, . 1.12{2b.18}Claim 1.13. 1) (Kec, , K ) has the JEP.

2) Suppose M1, M2 K , , f is an elementary embedding of


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Proof. First assume = 1, without loss of generality = 0 as we can let N =M.

By 1.13(2) without loss of generality = 2. Now this is obvious by [Sh:43],[Sh:199]; in details by [Sh:43] there is a +-saturated model M of T such thatM1 M and M |=C T

whenever, e.g. ++ C + / C. Let {i(x,y, a

i ) :

i < } list {(x,y, a) : L(T), M0 |= P(Qcfx,y0 )(x,y;z)[a]}, and for each i < let

ci,, < + be iM,a

i-increasing and cofinal. For < + let f be an elementary

embedding of M1 into M over M0 such that:

() ifc Dom(iM,ai) is a iM,ai

-upper bound of Dom(iM0,ai) then ci,

iM,a

i

c.

Let c M be a M,a

-upper bound of Dom(M0,a). Choose N0 M of cardi-

nality be such that M0 {c} N0 and choose < + large enough such that:

() ifi < and d N0 is a i

M,ai-upper bound of Dom(iMi,ai) then d

iM,ai

ci,.

Let N1 M be of cardinality be such that N0 f(M1) N1. Renaming, f isthe identity and (N0, N1) is as required.

Second, assume = 0 is even easier (again without loss of generality first, = + 2 and second = 0, = 2 and use N0 = M0, N1 satisfies M1 N1 andN1 = and N1 realizes the relevant upper). 1.16

{2b.24}Conclusion 1.17. In 1.15 the model M = m(M) =


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10 SAHARON SHELAH

1-type off(a) in m(N), and the 1-type of b in m(M) is equal to the 1-type of

f(a) in m(N). But f(a) = f0(a) = g0(b) = g(b), so we have gotten the promised

equality of 1-types. 1.18{2b.27}Observation 1.19. 1) If M Kec and < then M

:= M[, ) = M+ : < belongs to Kec.

2) If M K, < and M[, ) K, N then for some N K we have

M K N and N[, ) = N.

Proof. 1) If not, then there is N K such that M K N

but m(M) 1m(N). Define N = N : < by: N is M if < and is N if [, ).

Easily M K N K but m(M) 1 m(N), contradiction to the assumptionM Kec .2) The proof is included in the proof of part (1). 1.19

{2b.28}Claim 1.20. In 1.15 for each < we have

(a) M+ : < is homogeneous universal for K

(b) if = , i.e. + = then there is an isomorphism from M ontoM+ : < , in fact, we can determine f(a) = b if a

(M0 ), b (M ) and tp(a, , M

) = tp(b, , M

).

Proof. Chase arrows as usual recalling 1.19. 1.20

Proof. Proof of Theorem 1.5:Without loss of generality there is such that 2 = + (why? let >

be regular, work in VLevy(+,2) and use absoluteness argument, or choose set A of

ordinals such that P() L[A] hence T, T L[A] and regular large enough suchthat L[A] |= 2 = +, work in L[A] a little more; and for the desired conclusion

(there is a model of cardinality such that ...) it makes no difference). Let = and let M Kec, be as in 1.15 for :=

+ and let M = {M : < }.Now

()1 M is a model of T by the {+}-interpretation.

[Why? By 1.17.]

()2 M is -saturated.

[Why? Clearly M is -saturated for each < . As is regular and M : <

is increasing with union M, also M is -saturated.]

()3 M is strongly 0-saturated and even strongly -saturated, see Definition

0.4(1).[Why? Let < and a, b (M) realize the same q.f.-type (equivalently the firstorder type) in M. As < for some < we have a, b (M). Now by 1.20we know that M+ : <

= M : < , and by 1.18 the sequences a, brealize the same 1-type in m(M

+ : < ) hence by clause (c) of 1.15 there

is an automorphism of it mapping a to b. So is also an automorphism of Mmapping a to b as required.]

Lastly, we have to go back to models of cardinality =


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More fully, first let M : < be K -increasing continuous sequence with

union M. For < and a, b (M) let fa,b be an automorphism of M mapping

a to b. Now the set of < satisfying below is a club of hence if cf() = then M = {M : < } is as required except of being of cardinality , where

(a) if < , < and a, b ({M : < }) realize the same 1-type

in M then {M : < } is closed under fa,b and under f1a,b

(b) the witnesses for the cofinality work, i.e.1 if < , a

>(M), M |= P(Qcf0 y,z)(y,z,x)[a] then for some <

we have a M and for every (, ) there is c = c,a, M+1

which is a M+1,a-upper bound of Dom(M ,a), hence this holds for

any [, )

2 if < , a >(M ) and M

|= P(Qcf1 y,z)(y,z,x)[a] then for arbitrarily

large < we have a M

and there is c = c,a M+1

which is a(

M+1 ,a

)-upper bound of Dom(M ,a) for every [, ).

By a similar use of the LST argument we get a model of T of cardinal . 1.5

Remark 1.21. If you do not like the use of (set theoretic absoluteness) you may dothe following. Use 1.22 below, which is legitimate as

(a) the class (Kec , K) is an a.e.c. with LST number |T| + 0 and amalga-mation, so 1.22(1) apply

(b) using 1-types, it falls under [Sh:3] more exactly [Sh:54], so 1.22(3) apply

(c) we can define Kec()

by induction on = 0: K

= 1: Kec

= n + 1: Kec(n+1)

= {M Kec(n) : ifM N K

ec(n) then m(M) n+1

m(N)}

= : Kec() = {K

ec(n) : n < }.

On Kec() apply 1.22(2).

{2b.35}Remark 1.22. 1) Assume k = (Kk, k) is a a.e.c. with > LST(k) and = M tp(a, , M) D andM D N iff M N are from KD and a >M tp(a, , M) = tp(a, , N).Assume further D is good, i.e. for every M KD and there is a sequence


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(D, )-homogeneous model N KD which D-extends M. Then for every = |T| + 0 and M KD of cardinality there is a strongly sequence (, )-

homogeneous.{2b.33}Conclusion 1.23. 1) The logic L(QcfC) has the strong 0-saturated model existenceproperty (hence the strong 0-homogeneous model existence property).2) If = cf() Min(C) and Min(Reg\C) then in part (1) we can replace0 by .

Proof. Choose C, Reg\C and apply 1.5. 1.23


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2. Private Appendix

Moved from pgs.7,8:{2b.17d}

Claim 2.1. IfM K andM0 K M

1 andM0 M2. IfM0 Kec, , M0 K

M2 or M0 1 M2, then we can find(N , f2) such that:

M1 K N K , moreover N K

ec, and f2 is a K-embedding of M

2 into

N over M0.

Proof. By the definition of M0 Kec, in both cases we can assume M0 1 M

2.

For a = a : < list the elements of m(M2) and let = tpqf(a, , m(M2))

{(x0 , . . . , xn1 , b) : L(T) is quantifier free, b m(M0) and m(M2) |=

[a0 , . . . , an1 , b]} {P(x))t(,) : < , < and t(, ) is the truth value of

a M2}.

Now let D be a regular ultrafilter on = m(M2) and use 1.9(2),(3). This is

fine to get (f2, N) with N K and by 1.8(5) without loss of generality N K

and by 1.11(1) without loss of generality N Kec, . 2.1

References

[Mak85] Johann A. Makowsky, Compactnes, embeddings and definability, Model-Theoretic Logics(J. Barwise and S. Feferman, eds.), Springer-Verlag, 1985, pp. 645716.

[Sh:h] Saharon Shelah, Classification Theory for Abstract Elementary Classes, Studies in Logic:Mathematical logic and foundations, vol. 18, College Publications, 2009.

[Sh:3] , Finite diagrams stable in power, Annals of Mathematical Logic 2 (1970), 69118.[Sh:43] , Generalized quantifier s and compact logic, Transactions of the American Mathe-

matical Society 204 (1975), 342364.[Sh:54] , The lazy model-theoreticians guide to stability, Logique et Analyse 18 (1975),

241308.

[Sh:88r] , Abstract elementary classes near 1, Chapter I. 0705.4137. 0705.4137.[MkSh:166] Alan H. Mekler and Saharon Shelah, Stationary logic and its friends. I, Notre Dame

Journal of Formal Logic 26 (1985), 129138, Proceedings of the 1980/1 Jerusalem Model Theoryyear.

[Sh:199] Saharon Shelah, Remarks in abstract model theory, Annals of Pure and Applied Logic29 (1985), 255288.

Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram, The He-

brew University of Jerusalem, Jerusalem, 91904, Israel, and, Department of Mathe-

matics, Hill Center - Busch Campus, Rutgers, The State University of New Jersey, 110

Frelinghuysen Road, Piscataway, NJ 08854-8019 USA

E-mail address: [email protected]: http://shelah.logic.at

Saharon Shelah- On λ Strong Homogeneity Existence for Cofinality Logic

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