saharon shelah- maximal failures of sequence locality in a.e.c

8/3/2019 Saharon Shelah- Maximal Failures of Sequence Locality in A.E.C.

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MAXIMAL FAILURES OF SEQUENCE LOCALITY IN A.E.C.

SH932

SAHARON SHELAH

Abstract. We are interested in examples of a.e.c. with amalgamation hav-ing some (extreme) behaviour concerning types. Note we deal with k beingsequence-local, i.e. local for increasing chains of length a regular cardinal.For any cardinal 0 we construct an a.e.c. with amalgamation k withL.S.T.(k) = , |K | = such that { : is a regular cardinal and K is not(2 , )-sequence-local} is maximal. In fact we have a direct characterizationof this class of cardinals: the regular such that there is no uniform +-

complete ultrafilter. We also prove a similar result to (2

, )-compact fortypes.

0. Introduction

Recall a.e.c. (abstract elementary classes); were introduced in [Sh:88]; and their(orbital) types defined in [Sh:300], see on them [Sh:h], [Bal0x]. It has seemed tome obvious that even with k having amalgamation, those types in general lack thegood properties of the classical types in model theory. E.g. (, )-sequence-localitywhere

{z1}

Definition 0.1. 1) We say that an a.e.c. k is a (, )-sequence-local (for types)when is regular and for every k-increasing continuous sequence Mi : i ofmodels of cardinality and p, q S(M) we have (i < )(p Mi = q Mi) p = q. We omit when we omit Mi = .2) We say an a.e.c. k is (, )-local when: LST(k) and if M k and p1, p2 S(M) and N k M N p1N = p2N then p1 = p2.3) We may replace by ,< , [, ] with the obvious meaning (and allow tobe infinity).

Of course, being sure is not a substitute for a proof, some examples were providedby Baldwin-Shelah [BlSh:862, 2]. There we give an example of the failure of (, )-sequence-locality for k-types in ZFC for some , , actually = 0. This was doneby translating our problems to abelian group problems. While those problems seemreasonable by themselves they may hide our real problem.

Here in 1 we get k, an a.e.c. with amalgamation with the class { : (< , )-sequence-localness fail for k} being maximal; what seems to me a major missingpoint up to it, see Theorem 1.3. Also we deal with compactness of types gettingunsatisfactory results - classes without amalgamation; in [BlSh:862] this was doneonly in some universes of set theory but with amalgamation; see 2.

We relay on [BlSh:862] to get that k has the JEP and amalgamation.

Date: October 19, 2009.The author thanks Alice Leonhardt for the beautiful typing.

1


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2 SAHARON SHELAH

{z2}Question 0.2. Can { : k is (< , )-local}, e.g. can it be all odd regular alephs?

etc?Note that for this the present translation theorem of [BlSh:862] is not suitable.


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MAXIMAL FAILURES OF SEQUENCE LOCALITY IN A. E. C. SH932 3

1. An a.e.c. with maximal failure of being local{e1}

Claim 1.1. Assume

1 (a) = cf() > 0 or just = cf() 0, 0

(b) there is no uniform +-complete ultra-filter D on

(c) is a vocabulary of cardinality consisting of n-place predicateswith each n.

Then

there are I, M,, , (for = 1, 2 and ), g (for < ) satisfying(a) I, a set of cardinality

, is -increasing continuous with

(b) M,, a -model of cardinality , is increasing continuous with

(c) , is a function from M, onto I, increasing continuous with

(d) |1,{t}| for t I, and = 1, 2

(e) if t I+1\I then 1,{t} M,+1\M,

(f) for < , g is an isomorphism from M1, onto M2, respecting(1,2, 2,) which means a M1, 1,(a) = 2,(g(a))

(g) for = there is no isomorphism from M1, onto M2, respecting(1,, 2,).

Proof. Follows from 1.2 which is just a fuller version adding to unary functionFc for c G.

{e2}Claim 1.2. Assuming1 of 1.1 we have

there are I, M,, , (for = 1, 2, ) and g (for < ) and G suchthat(a) G is an additive (so abelian) group of cardinality 0

(b) I is a set, increasing continuous with , |I| =

(c) M, is a + -model, increasing continuous with , of cardinality

where + = {Fc : c G}, Fc a unary function symbol

(d) , is a function from M, onto I increasing continuous with

(e) FM,c (c G) is a permutation of M,, increasing continuous with

(f) ,(a) = ,(FM,c (a))

(g) FM,c1 (F

M,c2 (a)) = F

M,c1+c1(a)

(h) ,(a) = ,(b)

cG F

M,

c (a) = b

(i) for < , f is an isomorphism from M1, onto M2, which respects(1,, 2,) which means a M1, 1,(a) = 2,(f(a))

(j) there is no isomorphism from M1, onto M2, respecting(1,, 2,).

Proof. Let

()0 = 0 so = 0


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4 SAHARON SHELAH

()1 (a) let G = ([]


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(f) if a n(A) where 0 < n < then a/En, hasexactly members (we shall use only its having 2 members).

Now we choose a vocabulary of cardinality 2 (but see ()12) and for we

choose a -model M1, with universe A such that:

()6 (a) M1, increasing with

(b) assume that a, b are En,-equivalent (so a, b n(A) and

(a) = (b); then a, b realize the same quantifier free typein M1, iff aEn,b.

(c) for every function e there is a two-place predicate Re

such thatR

M1,e = {(af1,1,u1 , af2,2,u2) A A : f1 = e f2 and i u1

iff (|{j u2 : e(j) = i}| is odd)}.

[Why is this possible? First as, for each < , g maps RM1,e onto itself.Why? Assume we are given a pair (af1,1,u1 , af2,2,u2) from A A so 1, 2 1, > 2 so af1,1,, af2,1, M,.

Recall that h maps RM1,e onto R

M2,e by and R

M2,e = R

M1,e because g maps

RM1,e onto itself (see in ()6 the why...). Now see ()6(c), i.e. the definition of

RM1,e , i.e. obviously (af1,1,, af2,2,) RM1,e so as h is an isomorphism we have

(h(af1,1,), h(af2,2,)) RM2,e so by the previous sentence and the definitions of

uf,( = 1, 2) in 2 we have (af1,1 , uf1,1, (af2,2,uf2,2 ) RM1,e which by the

definitions of RM1,e in 6(c) implies uf1,1 {e(i) : i uf2,2} as promised.]

7 (a) |uf,1 | = |uf,2| for 1, 2 < , f

(b) n(f) = |uf,| is well defined

(c) if f1 f2 then n(f1) n(f2).


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[Why? For clause (a) use 6 for e = id and f1 = f2 = f. Clause (b) follows.Clause (c) holds by 6(c).]

8 there are f and < such that:(i) if f f and < then |uf, | = |uf,|

(ii) moreover if f = e f where e and f , < then e uf,is one-to-one from uf, onto uf, so n(f) = n(f)

(iii) if < ,f = e1, f1, f1 = e2, f2 then e2uf2, is one-to-one onto uf1,.

[Why? First note that clause (ii), (iii) follows from clause (i). Second, if not thenwe can find a sequence (fn, n, en) : n < such that

() n < , fn for n <

() fn fn+1 say fn = en fn+1 and en

() (en, fn+1, n+1) witness that (fn, n) does not satisfy the demands (i)+ (ii)

on (f, ) hence n(fn) < n(fn+1).

Let un = ufn,n for n < . For n < and i < let An,i = < : fn() = i,so An,i : i < is a partition of and An+1,i An,en(i). So lettingA = {An,(n) : n < } for

clearly A : is a partition of .

As we have = 0 by ()0, there is a sequence en : n < satisfying en and f such that fn = en f for each n < . So n < fn f whichby 7(c) implies n(fn) n(f). As n(fn) : n < is increasing, easily we get acontradiction.]

9 n(f) > 0, i.e. < uf, = .

[Why? If (f )( < )(uf, = ) then (by 3) we deduce h is the identitycontradiction. Otherwise assume uf, = hence as in the proof of8 there is f

such that f f f f so by 5 and 8 we have 0 < |uf,| |uf,| = |uf, |.]

10 if f , < and i uf, then = sup{ < : < and f() = i}.

[Why? If not, let () < be > sup{ < : < ,f() = i} and > .Let Y = {(af,,u, af,(),u) : u G,i / u}. Now for every ((), ) the functiong maps the set Y onto itself hence by the definition of E2,()+1 it follows thata Y a/E2,()+1 Y and as h respects (1,()+1, 1,()+1) it follows thath(a) a/E2,()+i and so > () + 1 h

+1 (h(a)) a/E

2,()+1.

Now for a Y, the pairs a, h(a) realizes the same quantifier free type inM1,()+1, M2,()+1 respectively, hence by the choice ofM2,()+1 the pairs a, g

1()+2, h(a))

realize the same quantifier free type in M1,. By ()6(b) recalling g

1

()+1(h(a)) a/E2,()+1 this implies that a, g

1()+1(h(a)) are E2,()+1-equivalent. By the def-

inition of E2,(), g1()+1(h(a)) belongs to the closure of {a} under {g

1 :

((); )} hence h(a) belongs. But by an earlier sentence Y is closed under thosefunctions so h(a) Y. Similarly h1(a) Y, h maps Y onto itself, recalling ()2this implies i / uf,.]

Now fix f, for the rest of the proof, without loss of generality f is onto and let uf, = {i

: i < ()} with i

: < () increasing for simplicity. Now

for every f such that f f and < by 8(ii), (iii) we know that if


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8 SAHARON SHELAH

e f = e f then e is a one-to-one mapping from uf, onto uf, ; but soe uf, is uniquely determined by (f, , f , ) so let if,, uf, be the unique

i uf, such that e(i) = i .

Now if f f and 1, 2 < and e and we choose e = id sonecessarily f uf,1 = e f uf,2, then e Rang(f uf,2) map uf,2 ontouf,1 but e is the identity so we can write uf instead of uf, let if, = if,, for < (), < .

Let

A = {A : for some f, f f and < we have f1{if,0}\ A}

1 A P()\[]


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So by the assumptions of 1.2 we are done. 1.2{e3}

Claim 1.3. For every there is an k = k

such that

(a) k is an a.e.c. with LST(k) = , |k| =

(b) k has the amalgamation property

(c) k admits closure (see below)

(d) if is a regular cardinal and there is no uniform +-completeultrafilter on , then : k is not ( 2, )-sequence-local for types,i.e., we can find ank-increasing continuous sequence Mi : i of models and p = q Sk(M) such that i < p Mi = q Miand M is of cardinality 2.

We shall prove 1.3 below. As in [BlSh:862] the aim of the definition of admitclosures is to ensure types behave reasonably.

{e4}

Definition 1.4. We say an a.e.c. k admits closure when for every M Kk andnon-empty A M there is B = ck(A, M) such that: MB Kk, MB k M andA M1 k N M k N B M1; we may use clk(A, M) for Mck(A, M).

{e5}Claim 1.5. Assume k is an a.e.c. admitting closure. Then tpk(a1, M , N 1) =tp

k(a2, M , N 2) iff letting M = Nck(M {a}), there is an isomorphism from

M1 onto M2 over M mapping a1 to a2.

Remark 1.6. In Theorem 1.3 we can many times demand M = , e.g., if()( = 2).

Note{e6}

Claim 1.7. 1) If k satisfies clause (a) of 1.3, i.e. is an a.e.c. with LST-number

and fails the assumption of clause (d) of 1.3, that is there is a uniform +

-complete ultrafilter on , then the conclusion of clause (d) of 1.3 fails, that is k is-sequence local for types.2) If D is a +-complete ultrafilter on and k is an a.e.c. with LST(k) thenultraproducts by D preserve M k, M k N, i.e.

ifMi, Ni(i < ) are (K)-models andM =i


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10 SAHARON SHELAH

By (e) we have

(e)+ for each i there is ni < and Ni,m : n ni such that() Ni0 = Ni

() Ni,mi = Ni

() a Ni,() if n < ni then Ni,2m+1 k Ni,2m, Ni,2m+2.

As = cf() > 0 without loss of generality i < ni = n. Let be such that

Mi : i , Ni,n : n n : i < and kLST(k) all belongs to H(). Let B be theultrapower (H(), )/D and j the canonical embedding of (H(), ) into B and

j1 be the Moskolski-Collapse of B to a transitive set Hi and let j = j1 j0. So jis an elementary embedding of (H(), ) into (H, ) and even L+,+-elementaryone. Without loss of generality k H() hence j(k) = k( hence by part (2), jpreserves N Kk, N

2k N

2.

So j(Mi : i ) has the form Mi : i j() but j() > =i


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MAXIMAL FAILURES OF SEQUENCE LOCALITY IN A.E.C. SH932 11

() for a HM, FM2 (a, b) : b GM list a/EM2 = {a

HM:M(a) = M(a)} with no repetitions

() if a HM and b, c G then FM2 (a, b +M c) = FM2 (FM2 (a, b), c),on the + see clause (b)

() F(a, 0GM ) = a for a HM

() for n < and < the relation RMn, is an n-place relation

{n(a/EM1 ) : a HM}.

We define k as being a submodel. Easily

3 k = (K, k) is an a.e.c.

For A M K let

(a) c0M(A) = the subgroup of (GM, +M) generated by (A GM) {b :

for some a1 = a2 A we have a1EM2 a2 and F

M2 (a1, b) = a2}

(b) c1M(A) = (A IM) {M(a) : a A HM}

(c) c2M(A) = (A JM) {FM1 (a) : a A H

M}

(d) c3M(A) = {a HM : for some b c0M(A) and a

1 A H

M, a =FM2 (a1, b)}

(e) c(A, M) = cM(A) = M ({cM(A) : = 1, 2, 3}).

Now this function c(A, M) shows that k admits closure (see 1.4) so

4 k admits closure and LST(k) + |k| = .

Assume is as in clause (d) of 1.3, we use the M,( = 1, 2, )constructed in 1.1 (and more of their actual construction as stated in 1.2).They are not in the right vocabulary so let M

,

be the following -model:

5 (a) elements GM, = G

IM

, = IJM

, = {t}, t just a new element

HM

, = A,(we assume disjointness)

(b) (GM

, , +M

,) is G = ([]


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12 SAHARON SHELAH

Let M0, = M, (G

M, IM

,) for = 1, 2 and (we get the sameresult).

Note easily6 M0, k M,, M, : is k-increasing (check)

7 tpk(t1, M

0,, M

1,) = tpk(t

2, M

0,, M

2,) for < .

[Why? By the isomorphism from M1, onto M2, respecting (1,, 2,) in1.1.]

8 tpk(t1, M

0,, M

1,) = tpk(t

2, M

0,, M

2,).

[Why? By the non-isomorphism in 1.1; extension will not help.]

Now by thetranslation theorem of [BlSh:862, 4] we can find k which has all theneeded properties, i.e. also the amalgamation and JEP. 1.3


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2. Compactness of types in a.e.c.

Baldwin [Bal0x] ask can we in ZFC prove that some a.e.c. has amalgamation,JEP but fail compactness of types. The background is that in [BlSh:862] weconstruct one using diamonds.

To me the question is to show this class can be very large (in ZFC).Here we accomplish both by direct translations of problems of existence of models

for theories in L+,+, first in the propositional logic. So whereas in [BlSh:862] we

have an original group GM, here instead we have a set PM of propositional vari-ables and PM, set of such sentences (and relations and functions explicating this;so really we use coding but are a little sloppy in stating this obvious translation).

In [BlSh:862] we have IM, set of indexes, 0 and H, set of Whitehead cases, Htfor t IM, here we have IM, each t IN representing a theory PMt P

M and inJM we give each t IM some models MMs : P

M {true,false}. This is set up sothat amalgamation holds.

Notation 2.1. In this section types are denoted by p, q as p, q are used for propo-sitional variables.

{b2.0}

Definition 2.2. 1) We say that an a.e.c. k has ( , )-compactness (for types)when: ifMi : i is k-increasing continuous and i < Mi andpi S


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() if = (q

j


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(a) |Mi| = Pi i

(b) PM = Pi and Mi = i

(c) the natural relations and functions.

Let Mi : Pi {true false} be a model of i.We define a model Ni Kk for i < (but not for i = !)

(a) Mi k Ni

(b) PNi = PMi

(c) Ni = Mi

(d) IM = {ti}

(e) JM = {si}

(f) FNi

(si) = t

i

(g) RNi1 = i {ti}

(h) RNi2 is chosen such that MNisi

is Mi

(i) FNii (i < ) are defined naturally.

Now

()1 pi = tpk(ti, Mi, Ni) S1(Mi).

[Why? Trivial.]

()2 i < j < pi = pjMj.

[Why? Let Ni,j = Nj(Mj {sj, tj}).]Easily tp(tj , Mi, Ni,j) pj and tp(tj , Mi, Ni,j) = pi by the claim 2.7 above.]

()3 there is no p S1(M) such that i < pMi = pi.

Why? We prove more:

()4 there is no (N, t) such that(a) M k N

(b) t IN

(c) ( M)[RN1 t].

[Why? As then = M has a model contradiction to an assumption.] 2.8

So e.g.{b2.13}

Conclusion 2.10. If > is regular with no +-complete uniform ultrafilter on and = 2, then k is not (, )-compact.

Remark 2.11. Recall ifD is an ultrafilter on then min{ : D is not -complete}is 0 or a measurable cardinality.


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Proof. (Well known).Let M be the model with universe 2, PM0 = and R

M be such that

{{ < : RM

} : < } = P(), 2 andk is not (, )-stable then for some + satisfying

> we have < (k).2) If > , > then < (k) then k not (, )-stable.

{a1.21}Conclusion 3.7. (, )-stability spectrum - behave as in [Sh:3].

{a1.23}Discussion 3.8. We can look at [, 2) using splitting rather than stronglysplitting.

It seems to me the main question is{a1.26}

Question 3.9. Assume ( LS(k)((k) > 0).What can you say on Min{ : (h) > 0, LST(k)}?

{a1.31}Question 3.10. Assume GCH can we find an a.e.c. k such that: ( LST(k))((k) =0) but unstable in every regular > LST(k)?

References

[Bal0x] John Baldwin, Categoricity, vol. to appear, 200x.[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:88] , Classification of nonelementary classes. II. Abstract elementary classes, Classi-

fication theory (Chicago, IL, 1985), Lecture Notes in Mathematics, vol. 1292, Springer, Berlin,1987, Proceedings of the USAIsrael Conference on Classification Theory, Chicago, December1985; ed. Baldwin, J.T., pp. 419497.

[Sh:88r] , Abstract elementary classes near 1, Chapter I. 0705.4137. 0705.4137.[Sh:300] , Universal classes, Classification theory (Chicago, IL, 1985), Lecture Notes in

Mathematics, vol. 1292, Springer, Berlin, 1987, Proceedings of the USAIsrael Conference onClassification Theory, Chicago, December 1985; ed. Baldwin, J.T., pp. 264418.

[Sh:702] , On what I do not understand (and have something to say), model theory, Math-

ematica Japonica 51 (2000), 329377, math.LO/9910158.[Sh:734] , Categoricity and solvability of A.E.C., quite highly, 0808.3023.[BlSh:862] John Baldwin and Saharon Shelah, Examples of non-locality, Journal of Symbolic

Logic 73 (2008), 765782.

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- maximal failures of sequence locality in a.e.c

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