Saharon Shelah- Few Non Minimal Types and Non-Structure

8/3/2019 Saharon Shelah- Few Non Minimal Types and Non-Structure

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FEW NON MINIMAL TYPES

AND NON-STRUCTURE

SH603

Saharon Shelah

The Hebrew University of JerusalemEinstein Institute of Mathematics

Edmond J. Safra Campus, Givat Ram Jerusalem 91904, Israel

Department of MathematicsHill Center-Busch Campus

Rutgers, The State University of New Jersey

110 Frelinghuysen RoadPiscataway, NJ 08854-8019 USA

Abstract. We deal with abstract elementary classes K which has amalgamation in

, categorical in and in +. Our main result is that if ( 2 < 2+

and) theminimal types on members of K are not dense (among non algebraic (complete)types over models in K, extending our given model), then the number of models inK of cardinality ++ is maximal. For this we deal with some claims in pcf. Thisimproves a result in [Sh 576], it is essentially self-contained.

Saharon:(a) in 3.4 proof: |S(M)| lim(T), reference (but 3.4 contains a proof, 1.15contains a statement)(b) also in 1.12; there is fact in end of 1(c) weaker coding: 3.4 refer to Definition ?

2000 Mathematics Subject Classification. - 03C45, 03C75, 03C95, 03E04.Key words and phrases. model theory, abstract elementary classes, classification theory, cate-

goricity, nonstructure theory, pcf theory.

I thank Alice Leonhardt for the beautiful typingSplit from Sh600; work done Spring of 1995Revised after proof reading for the Proc! and moreLatest version - 04/May/28

Typeset by AMS-TEX

1


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Annotated Content

0 Introduction

[We explain our aim and define our framework.]

1 Non minimal types and nonstructure

[We define unique amalgamation, UQ, and try to use it for building many

models in + when 2 < 2+

(so the weak diamond holds). If this approachfails we still get the many models in ++ by the easy criterion of [Sh 576,3] or [Sh:F603] but it works only if the weak diamond ideal on + is not++-saturated.]

2 Remarks on pcf

[We prove some pcf observations needed here.]

3 Finishing the many models

[We prove the result of 1 without the extra assumption on the saturationof the weak diamond ideal.]

4 A minor debt

[There was one point in [Sh 576] where we use > 0

, though our aim therewas to generalize theorem known for = 0. We eliminate this use.]


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FEW NON MINIMAL TYPES AND NON-STRUCTURE SH603 3

0 Introductions

In [Sh 576] there was an important point where we used as assumption I(+3, K) =0. This was fine for the purpose there, but is unsuitable in other frameworks, like[Sh 600]: we would like to analyze what occurs in higher cardinals, so our main aimhere is to eliminate its use and add to our knowledge on non-structure.

The point was the minimal triples in K3 are dense ([Sh 576, 3.30,pg.88,3.17t]).For this we assume we have a counterexample, and try to build many non-isomorphicmodels. Hence we get cases of amalgamation which are necessarily unique. Thoseunique amalgamations are normally too strong (even for first order superstabletheories), but here they help us to prove positive theorems, controlling omittingtypes. So we try to build many models in + by omitting types over models ofsize , in a specific way where unique amalgamation holds. If this argument fails,

we prove C1K, has weak +-coding (see [Sh 838]) and by it get 2++

non-isomorphicmodels except when the weak diamond ideal on + is ++-saturated; this is donein 1. In 3 we work harder and by partition to cases relying on pcf theory wesucceed to get the full result. We work also to get large IE (many models no oneK-embedding to another). The pcf lemmas (which are pure infinite combinatorics)are dealt with in 2.

Compared to its original [Sh 603] we:

(a) there was also another point left in [Sh 576, 4.2t], for the case = 0 only,this was filled in 4 but hence omitted as it was moved to citeSh:EF6

(b) we rely on [Sh 838] which correct, improves and simplify [Sh 576, 3](c) the main changes in this version is improving 1.8 to 1.11 and simplification

of Definition 1.6.

The result of 3.3 is not meaningful outisde the general aim here; move to [Sh 838,1]?An aim of the present revision is to weaken the assumption K is categorical in+ to having an intermediate number of models in +.

0.1 Definition. We say K= (K, K) is an abstract elementary class, aec or a.e.c.in short, if ( = K is a fixed vocabulary, K a class of -models (and Ax0 holdsand) AxI-VI hold where:Ax0: The holding of M K, N K M depends on N, M only up isomorphism i.e.[M K, M = N N K], and [if N K M and f is an isomorphism from Monto the -model M mapping N onto N then N K M].


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4 SAHARON SHELAH

AxI: If M K N then M N (i.e. M is a submodel of N).

AxII: M0 K M1 K M2 implies M0 K M2 and M K M for M K.

AxIII: If is a regular cardinal, Mi (for i < ) is K-increasing (i.e. i < j < implies Mi K Mj) and continuous (i.e. for limit ordinal < we have

M =i


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M K N2 K N and a2 N

2 and (M, N1, a1) EatM (M, N2, a2).

(Note: ifM0 is an amalgamation base, see below, then the difference between real-ize and strongly realize disappears).

4) We say M0 K is an amalgamation base if: for every M1, M2 K andK-embeddings f : M0 M (for = 1, 2) there is M3 K andK-embeddings g : M M3 (for = 1, 2) such that g1 f1 = g2 f2.5) We say K is stable in if LS(K) and M K |S(M)| .6) We say N is -universal over M if for every M, M K M

K, there is aK-embedding of M

into N over M. If we omit we mean N.7) We say N is (, )-brimmed over M if there is as K-increasing continuoussequence Mi : i , such that M = M0, M = N, Mi K, Mi+1 is an amal-gamation base, Mi+1 is universal over Mi. Replacing by means for some = cf() . We omit over M to mean for some M K.8) K3

= {(M , N , a) : M K N, a N\M and M, N K}, with the partial order defined by (M , N , a) (M, N, a) iff a = a, M K M and N K N.9) We say (M , N , a) K3 is minimal if (M , N , a) (M

, N, a) K3 for = 1, 2implies tp(a, M, N1) = tp(a, M

, N2). We say p S(M) is minimal if p =tp(a,M,N) for some minimal triple (M , N , a) from K3, so = M.

10) We say K has amalgamation in if every M K is an amalgamation base.


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6 SAHARON SHELAH

1 Non-minimal triples and non-structure

We shall quote here [Sh 838] but in a black box nature.

1.1 Context.

(a) K abstract elementary class with LS(K) and K =

(b) K has amalgamation in .

Remark. Alternatively we can use a weaker context.

1.2 Definition. 1) For x {a, d} we say UQx(M0, M1, M2, M3) if:

(a) M K for 3

(b) M0 K M for = 1, 2

(c) if for i {1, 2} we have Mi K, for < 4 and Mi0 K M

i K M

i3 for

i = 1, 2, = 1, 2 and [x = d Mi1 Mi2 = M

i0] and f

i is an isomorphism

from M onto Mi for < 3 and f

i0 f

i1, f

i0 f

i2 then there are M

3, f3 such

that M23 K M3 and f3 is a K-embedding of M

13 into M

3 extending

(f21 (f11 )

1) (f22 (f12 )

1) i.e. f3 f11 = f21 & f3 f

12 = f

22

(d) M0 K M K M3 K for = 1, 2

(e) x = d M1 M2 = M0.

2) We say UQx(M0, M1, M2) if UQx(M0, M

1, M

2, M3) for some M3 and M

1, M

2

isomorphic to M1, M2 over M0 respectively.3) If we omit x, we mean x = a.

4) K3, is the family of triples (M , N , a) K3 such that there is no minimal triple

above it.5) K2, is the family {(M, N) : for some a, (M , N , a) K

3, }.

6) For M K let S(M) = {p S(M): for some (M , N , a) K3, we have

p = tp(a,M,N)}.7) For M K+ let S(M) = {tp(a,M,N) : (M , N , a) K+ and for some

M0 K N0 K N we have M0, N0 K, M0 K M and (M0, N0, a) K3, }.

The reader may find it helpful to look at the following example.

1.3 Example. Let K be the class of M = (|M|, EM), M and EM is anequivalence relation on |M| and K be being a submodel. Then UQ(M0, M1, M2)iff


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(a) M K for = 0, 1, 2

(b) M0 M for = 1, 2

(c) if aEM1c,bEM2d, b M1\M0, d M2\M0 then (aEM0b)

(d) for some {1, 2} for every c M we have (c/EM) M0 = 0.

1.4 Claim. 1) Symmetry: assuming x {a, d} we have UQx(M0, M1, M2, M3) UQx(M0, M2, M1, M3); we can also omit M3.

2) UQa(M0, M1, M2) UQd(M0, M1, M2) recalling M0 is an amalgamation base

(inK) by clause (b) of 1.1.3) UQa(M0, M1, M2, M3) iff clauses (a), (b), (d), (e) of Definition 1.2(1),(2) holdsand also (c), i.e., clause (c) restricted to the case M1 = M for 3.4) If UQx(M0, M1, M2, M3), M3 K M

3 K then UQ

x(M0, M1, M2, M

3); and

also the inverse: if UQx

(M0

, M1

, M2

, M3

) and M1

M2

M3

K M

3then

UQx(M0, M1, M2, M3).5) Assume (M , N , a) K3 and it is not minimal (even less) then UQ(M , N , N ).

Proof. 1),2) Trivial.3) Chasing arrows, we should prove clause (c) of Definition 1.2(1). Assume we aregiven M1 : < 4, M

2 : < 4, f

i : < 3 as there for i = 1, 2. First for

i = 1, 2 apply clause (c) to Mi : < 4, fi : < 3. So there are N

i3, f

i3 such

that: Mi3 K Ni3 K, and f

i3 a K-embedding of M3 into N

i3 extending f

i1 f

i2.

As K has amalgamation in (by 1.1(b)) there are N K and K-embeddings

gi : Ni N such that g1 f13 = g2 f23 , so we are done.4) Again by the amalgamation i.e., 1.1(b).5) Let N0 = N, a0 = a.We can find N1, a1 such that

N K N1 K, a N1\{a} and a = a

(this follows from non-minimality, and is all that we need).

Hence we can find N2, f1 such that: N1 K N2, f1 is a K-embedding of N intoN2 and fa(a) = a1. Clearly we have gotten two contradictory amalgamations, sowe are done. 1.4

1.5 Claim. 1) transitivity: If UQ(M, N, M+1, N+1) for = 0, 1 thenUQ(M0, N0, M2, N2).2) If = cf() < +, andMi : i is K-increasing continuous and Ni : i is K-increasing and UQ(Mi, Ni, Mi+1, Ni+1) for each i < thenUQ(M0, N0, M, N).3) Assume:


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8 SAHARON SHELAH

(a) , < +

(b) Mi,j K for i , j

(c) i1 i2 & j1 j2 Mi1,j1 K Mi2,j2(d) Mi,j : i is K- increasing continuous for each j

(e) Mi,j : j is K-increasing continuous for each i

(f) UQ(Mi,j , Mi+1,j, Mi,j+1, Mi+1,j+1) for every i < , j < .

Then UQ(M0,0, M,0, M0,, M,).4) If UQ(M0, M1, M2) and M0 K M

1 K M1 and M0 K M

2 K M2 then

UQ(M0, M1, M

2).

5) If M K N for = 1, 2 and N1 can be K-embedded into N2 over M, thenUQ(M, N2, M

) implies UQ(M, N1, M).

Proof. Chasing arrows (note: UQ = UQa is easier than UQd, for UQd the parallelclaim is not clear at this point, e.g. the straightforward proof of transitivity failsand we can construct a counterexample). 1.5

1.6 Definition. 1) Kdis [K] = {(M, ) : M K, K, || satisfies N M


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1.8 Claim. Assume 2 < 2+

or at least the definitional weak diamond; i.e.,DfWD+(+), see [Sh:E35, 1.7]. If () or at least () below holds (hence above

some triple from K3 there is no minimal one), then I(+, K) wd(+) where

() for every (M, ) Kdis [K] for some M, we have (M, )


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1.9 Claim. 1) An equivalent condition for () from 1.8 is (respectively):

() for every M K N from K for some M we have M


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Proof. By induction on < , we choose (M, , + ) : 2 such that:

(a) M K has universe < +

(b) (M, ) Kdis [K](c) N (N\M) + =

(d) (M , ) 2, { < 0 >, < 1 >} and (M , ) (M, ) Kdis [K]for = 1, 2 then we can find (M+2, +2) for = 1, 2 such that (M, ) (M+2, +2) and M3, M4 are isomorphic over M.

There is no problem to carry the induction, to guarantee (i) use assumption 3.Clearly, by clauses (g) + (h), as in the proof of 1.8

1 if +>2 and (M, ) (M, ) Kdis [K] for = 0, 1 then

we cannot find (M, ) Tdis [K] and functions f0, f1 such that (M, ) f

(M, ) for = 0, 1 and f0 M = f1 M1.

Now we apply [Sh 838, 1.4K], we have to check it assumption.The main point is proving () there, a stronger version of 1 above which says

2 the following is impossible

() < < +

() 1, 2 2

() 1 < 0 > 1 2 and 1 < 1 > 1 2

() 2 < 0 > 2 2 and 2 < 0 > 2

2

() f is an isomorphism from M1 onto M2 mapping M1 onto M2

() g is an isomorphism from M1 onto M2 mapping M1 onto M1

() g M1 = f M1 .

Proof of2. We can find 2

, 2 such that


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12 SAHARON SHELAH

(i) (M2 , 2) (M2 ,

2

) Kdis [K]

(ii) (M2 , 2) (M2 ,

2

) Kdis [K]

(iii) if N 1

then f can be extended to an isomorphism from N onto someN 2

(iv) if N 1 then g can be extended to an isomorphism from N onto someN 2 .

So (M2, 2) (M2 , 2)

(M2 , 2

) and (M2, 294)

(M2 , 2) (M2 ,

2

).By applying clause (i) with 2, 2 < 0 >, (M2, 2), (M2,

2

), (M2, 2

)

here standing for ,, (M1, 1), (M2, 2) there we can find (N, ) for = 1, 2 andh such that

(v) (M2 , 2) (N1, 1)

(vi) (M2 , 2

) (N2, 2)

(vii) h is an isomorphism from N1 onto N2 over M2 .

But now the mapping hf and g contradict the choice of (M1, 1), (M1, 1)that is 1 above.

Having proved 2 we are done. 1.10

1.11 Conclusion. If 2 < 2+

(or at least the definitional weak diamond for +)

and () below then I(+, K) = 2+ , where

() for some M0 K if M0 K M K N K then for some M we have

M


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So we can carry the definition as there because we assume that 3 of 1.8 fail.

M : +

2 are pairwise non-isomorphic over M (hence over M) and thissuffices. 1.11

1.12 Claim. Assume

(a) () of 1.11 fails (equivalently 1.9)

(b) M K |S(M)| > + (follows from above (M , N , a) K3 there

is no minimal triple +2 > + see [Sh:E46, xx?])

(c) K is categorical in

(d) K is categorical in +

(e) 2 < 2+

< 2++

.

Then I(++, K) wd(++, 2+

), (which is = 2++

when ) except possiblywhen

WDmId(+) is ++-saturated (normal ideal on +).

Remark. (See [Sh 838, 4]) 1) We may omit the model theoretic assumption (d) in

1.12 if we strengthen the set theoretic assumptions, e.g.

()1 for some stationary S S++

+we have S I[] but S / WDmId(++).

2) Note that: if =


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14 SAHARON SHELAH

Then we can find (1), a1, N(1), f such that

() (0) (1) < +, M(1) K N1,

() f is a K-embedding of N0 into N1() f(a0) = a1

() tp(a1, M(1), N1) is not realized in M.

Question: Can we find (1), a1,, N(1),, f for < 2 such that each (a1,, N(1),, f)

is as above and tp(a1,, M(1), N(1),) : < 2 are pairwise distinct? [??]

1.14 Remark. 1) We can get more abstract results.2) Note () of 1.12 is a light assumption, in fact, e.g. its negation has high

consistency strength.

1.15 Fact. Assume K is an abstract elementary class with amalgamation in , andabove (M , N , a) K3 there is no minimal pair and K

3 has the weak extension

property.1) Assume T is a tree with < + levels and nodes2. Then we can find(M, N, a) K3 above (M , N , a) for lim(T) such that tp(a, M

, N) for lim(T) are pairwise distinct so |S(M)| | lim(T)|. We can add (M, N, a)

is reduced, (see [Sh 576]).

2) If M K is universal thenS(M) sup{lim(

T) :T

a tree with nodesand levels}.

Proof. 1) Straight (or see the proof of 3.4(1)).2) As for any N K there is a model N K M isomorphic to N, now p p N isa function from S(M) onto S(N) by [Sh 576, 2] hence |S(M)|+ |S(N)| =|S(N)|. Now use part (1). 1.15

2If 2 > + then for some such T and , + < lim(T); as in 3.4


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2 Remarks in pcf

The following will provide us a useful division into cases (it is from pcf theory; onwd() see [Sh 576, 1.1t]), we can replace + by regular such that 2 = 2 2

() ()(cf() + < < pp() < ) hencecf() + < < pp+(

) <

() for every regular cardinal in the interval (, ] there is an increasingsequence i : i < + of regular cardinals > + with limit such

that = tcf

i; we can get, ofcourse, a tree T with cf() levels, nodes and |limcf()(T)|

).

(B) for some , we have: clauses () () from above (so 2 < ) and

() there is T : < such that: T +>2 a tree, of cardinality

+ and +

2 =


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16 SAHARON SHELAH

(C) letting = 2 we have (), (), (), (), () of clause (B) and

() for no (+, 2] do we have cf() +,pp() > 2 equivalently

2 > + cf([2]+ , ) = 2 hence (see the proof) wd(+, 2) =

2+

except (maybe) when < and there is A [wd(+, 2)]

such that A = B A |A B| = 0.

Remark. Remember that

cov(,,,) = + Min|P| :P [] 2 = and

on the other hand ()0 = (2)0 = 2 = necessarily < wd(+, 2) so

clause () follows; now clause () follows from clause () as WDmTId(+) is (2)+-

complete by [Sh 576, 1.2t](5) and we have chosen = 2. Now if 2+

> ++, (so

2+

+3), then for some < ,T is (a tree with + nodes, + levels and) atleast +3 +-branches which is well known (see e.g. [J]) to imply no normal idealon + is ++-saturated; so we got clause (). As for () the definition of and

the assumption = 2 we have the first two phrases, as for wd(+, 2) = 2

+

by [Sh:f, AP1.14,pg.956 + 1.16,pg.958] there is A as mentioned in () and by [Sh460] we get < . The equivalently holds as (2

)0 = 2.

Possibility 2: =: cov(2, ++, ++, 1) > 2.

Let = Min{ : cf() +, + < 2 and pp() = }; clearly we mayreplace pp() = by pp() =+ . We know by [Sh:g, II,5.4] that exists and(by [Sh:g, II,2.3](2)) clause () holds, also 2 < pp() cf() hence cf() = +.


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So clauses (), (), () hold (of course, for clause () use [Sh:g, Ch.II,5.4](2)), andby () + [Sh:g, VIII,1] also clause () holds.

For clause () let = Min{ : 2 }, clearly < 2|| < and

(as 2

) hence cf() < +

= cf() hence 2


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18 SAHARON SHELAH

We shall prove (B), so we are left with proving clauses ()() when < 2+

.By the choice of , easily clause () (in Case B of 2.1) holds. In clause (), 2 < holds as we are in possibility 2.

Also as pp() = and cf() = + by the choice of necessarily (by transitivityof pcf, i.e., [Sh:g, Ch.II,2.3](2)) cf() > + but > +. Easily cf() + pp() hence cov(, ++, ++, 1) = by [Sh:g, Ch.II,5.4], which

gives clause (). Now let A []+

exemplify cov(, ++, ++, 1) = and

let A = {B : B is an infinite countable subset of some A A}. So A []0

and easily A []+

(B A)(B A) and |A| as (+)0 2 <

certainly there is no family of > subsets of each of cardinality + withpairwise finite intersections, hence (by [Sh:b, Ch.XIV,1] or see [Sh 576, 1.2](1) or[Sh:f, AP,1.16]) we have < wd(

+, 2) thus completing the proof of ().

Now clause () follows by () + () by [Sh 576, 1.2t](5). Also if 2+

= ++ then

2+ +3 so by clause () (as < 2+), we have |lim+(T)| +3 for some which is well known to imply no normal ideal on + is ++-saturated; i.e., clause(). So we have proved that case (B) holds.

Subpossibility 2b: = 2+

.We have proved that case (A) holds, as we already defined and

and proved(), (), (), (), () we are done.

Still we depend on 2.3 below but first we prove

2.2 Claim. Assume

(a) < = cf() 0 1 < 1(b) ()[0 < < & cf() pp() < ]

(c) 2 = 1 + cov(1, +0 ,

+, +) (by clause (b) and [Sh:g, Ch.II,5.4] we knowthat it is < )

(d) pp() = cf() > 2 ( 1 ).

Then 0 .

Remark. In fact

2 cov(1, +

0 , +

, 2).

Proof. Assume toward contradiction 0 < . By [Sh:g, Ch.II,2.3](2) and clause (b)of the assumption we have sup{pp() : +0

0 and cf() } < hence

by [Sh:g, Ch.II,5.4] it follows that

=: cov(0 , +0 ,

+, +) < .


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We can by assumptions (b) + (d) and [Sh:g, Ch.II,3.5] + [Sh:g, Ch.VIII,1] findT , a tree with nodes, |lim(T)| , (if = pp(), the supremum in thedefinition of pp() is obtained by [Sh:g, II,5.4](2)). Moreover, by the construction

there is lim(T), || = such that

& |

| |{ : < , }| = . By renaming (and also by the construction), without loss of generality

if 00 = 11 belongs to T then 0 = 1.

So let i lim(T) for i < be pairwise distinct, listing .As 1 there is a sequence F = F : < satisfying: F a function from to1 such that < < ( < )F() = F().

Let wi, = {F(i()) : < }, so wi, [1]. By assumption (c) we have

2 = 1 + cov(1, +0 ,

+, +) so there is P [1]0 , 2 = |P| such that: any

w [1] is included in a union of members ofP. So we can find Xi,, P

for < such that wi,


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20 SAHARON SHELAH

TW is (essentially) a tree with 0 nodes and contradict the choice of = {i :i < }.(We could have instead using ,QY to fix Yi = Y as |P| < = cf().) 2.2

2.3 Claim. Assume

(a) < = cf() 0 1 <

(b) ()[0 < < & cf() pp() < ]

(c) T is a tree with 1 nodes, levels and -branches

(d) pp() = cf() > 2 where

(e) 2 = 1 + cov(1, +0 ,

+, +)

(f) = cov(0 , +0 ,

+, +) < .

Then for some subtree Y T, |Y| 0 and |lim(Y)| (for 2.1 it is enoughto prove for any given < ).Saharon: use of F??

Proof. Let T, = {i : i < } be as in the proof of the previous claim. Let{ : < } list distinct -branches of T

(see clause (c)). Without lossof generality the set of nodes of T is 1. Choose for each < the functionF : 1 by F() = (). Define wi,,P, Xi,,, Yi,P as in the proof of 2.2.But for Y P we change the choice of ZY, first

Y = { < 1 : for some Y, we have


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So let i lim(T) for i < be pairwise distinct, listing .Let { : < } list distinct -branches ofT (see clause (c)). Without loss ofgenerality the set of nodes ofT is 1 and let () be the unique < 1 which

belong to the branch and is of level . Define for each < the functionF : 1 by F() = () hence < < ( < )F() = F().]

For < and i < let wi, = {F(i()) : < }, so wi, [1]. By

assumption (e) we have2 = 1 + cov(1,

+0 ,

+, +) so there is P [1]0 , 2 |P| such that: any

w [1] is included in a union of members ofP. So we can find Xi,, P

for < such that wi,


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22 SAHARON SHELAH

Now TW is (essentially) a tree with 0 nodes and contradict the choice of ={i : i < }.(We could have instead using ,QY to fix Yi = Y as |P| < = cf().)

2.3, 2.1

2.4 Remark. 1) We could have used in 2.2, 2.3,2 = cov(

0 ,

+0 ,

+, +) instead cov(0 , +0 ,

+, +) and similarly in the proof of2.1.2) We can also play with assumption (b) as 2.2, 2.3.

It may be useful to note (actually tr = suffice)

2.5 Fact. IfT +>2 is a tree, |T| + and then for every regular

< large enough, we can find Y : < +

, cf() = , |Y| such that:for every lim+(T) for a club of < + we have

cf() = Y.

The following is needed when we like to get in the model theory not just manymodels but many models no one K-embeddable into another.2.6 Fact: Assume:

(a) cf() < , pp() =+ , moreover pp() =+ and + < <

(b) F is a function, with domain [], such that: for a [], F(a) is a familyof < members of []

(c) F is a function with domain [] such that

a [] a F(a) F(a).

Then we can find pairwise distinct ai [] for i < such that I = {ai : i < }

is (F, F)-independent which means

()F,F,I if a = b & a I & b I & c F(a) (F(b) c).

2.7 Remark. 1) Clearly this is similar to Hajnals free subset theorem [Ha61].2) Note that we can let F(a) = a.3) Note that if = cf([], ) then for some F, F as in the Fact

() if ai [] for i < + are pairwise distinct then not every pair {ai, aj} is(F, F)-independent[why? let P [] be cofinal (under ) of cardinality , and let F be such


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thatF(a) {b [] : a b and b P} has a -maximal member F(a);clearly there is such F. Now clearly

()1 if a = b are from [] and F(a) F(b) = then {a, b} is not (F, F)-independent.Also if 1 , cf(1) + < 1 and pp(1) then by[Sh:g, Ch.II,2.3] the Fact for 1 implies the one for .]

Proof. We can define g : [] [] and F, F functions with domain [] asfollows:

g(a) = { + : a} { : < }

F(a) = {{ : + b} : b F(g(a))}

F(a) = { : + F(g(a))}.

Now F, F are as above only replacing everywhere [] by [], and ifI = {ai :i < } [] with no repetitions satisfying ()F,F,I then I

= {g(ai) : i < }

is with no repetitions and ()F,F,I .So we conclude that we can replace [] by []. In fact we shall find the ai in[]a where a chosen below.Without loss of generality ++ < .

Assume < = cf(a/J) where a Reg\+, |a| , sup(a) = , Jbda Jand for simplicity = max pcf(a) and let f = f : < be a sequenceof members of a,


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24 SAHARON SHELAH

As + < 1, there is a stationary S { < +1 : cf() =

+} which is in I[+1 ],by [Sh 420, 1] and let di : i <

+1 witness it, so otp(di)

+, di i, [j di dj = di i] and i S i = sup(di), and for simplicity: for every club E of

+1 for

stationarily many S we have ( d)[( E)(sup( d) < < )], existsby [Sh 420, 1]. Now try to choose by induction on i < +1 , a triple (gi, i, wi) suchthat:

(a) gi a

(b) j < i gj sup(j


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[why use the ideal? In order to show that b = .] Now cf(+1 ) =

+1 > 1,

for some () < +1 we have A = {i : i = ()} is unbounded in +1 . Hence

E = { < +1 : a limit ordinal and A is unbounded in } is a club of +1 . So

for some S we have = sup(A ), moreover ifd = { : < +

} (increasing)then ()[E (sup

.


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26 SAHARON SHELAH

Define: X = {(, ) : < cf(), < }

F

(, )

=

(

,

) :(

,

) X\{(, )} and for somed F(Rang(f)) we have { a : f

() d} / J

so F

(, )

is a subset of X of cardinality < cf()+ + < .So by Hajnals free subset theorem [Ha61] we finish (we could alternatively, for

singular, have imitated his proof). 2.6


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3 Finishing the many models

Recall from [Sh 576, 3.13t](3) (see [Sh 838]??).

3.1 Claim. 1) Assume

(a) 2 < 2+

and Case A or B of Fact 2.1 holds for , (or just the conclusionthere)

(b) K is an abstract elementary class with LS(K)

(c) K+ = 0

(d) K has amalgamation in

(e) in K3, the minimal triples are not dense.

Then

()1 for any regular < we have:

()1 there is M K, |S(M)| > .

2) If in part (1) we strengthen clause (d) to (d)+, then we get ()+1 where:

(d)+ K has amalgamation in and has a universal member in

()+1 for some M K we have |S(M)| .

3) Assume (a), (b), (c), (e) of part (1) and (d)+ of part (2) then:

()2 I(+, K) and if (2)+ < then IE(+, K) .

4) If in clause (a) of part (1) we restrict ourselves to Case A of 2.1, then

= 2+

so in part (3) we get

()+2 I(+, K) = 2

+

and (2)+ < 2+

IE(+, K) 2+

.

3.2 Remark. 1) We can restrict clause (b) to K, interpreting in (c) + (e), K+ as{i


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28 SAHARON SHELAH

4) We can state the part of (A) of 2.1 used (and can replace 2+

by smallercardinals).5) We can replace + by a weakly inaccessible cardinal with suitable changes.

Proof. 1) Note that is singular (as by clause () of (A) of 2.1 (so also (B)),cf() = + < ). By 1.15(1) it suffices for each < to have < + and a treewith nodes and -branches. They exist by clause () of (A) of 2.1 (so alsoof (B)).2) Similarly using 1.15(2).3) By part (2) we can find M K satisfying S(M) has cardinality , andapply 3.3 elow.4) Should be clear from the proof of part (3). 3.1

3.3 Fact. Assume

(a) = cf([]+

, ) > 2

(b) K is an a.e.c. with LS(K)

(c) M K is an amalgamation base

(d) M K satisfies |S(M)| .

Then I(+, K) and if (2)+ < then IE(+, K) .

Proof. Let p S

(M

) for Z be pairwise distinct, |Z| and let M

KN K, p = tp(a, M, N).

Now for every X [Z]+

, as Khas amalgamation in there is MX K+ such thatM K MX and X N is embeddable into MX over M

(hence p is realized

in MX ). Let Y[X] = { Z : p is realized in MX}. So X Y[X] [Z]+

, so

{Y[X] : X [Z]+

} is a cofinal subset of [Z]+

, hence (see clause () of case (A)of Fact 2.1)

|{(MX , c)cM/ =: X [Z]+}|

|{Y[X] : X [Z]+

}| cf([Z]+

, )

cf([]+

, ) pp() = .

As 2 < also |{MX/ =: X [Z]+}| (clear or see [Sh:a, Ch.VIII,1.2]

because MX = +, M = and (+) < ) but IE(+, K) is than theformer.


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Lastly we shall prove (2)+ < 2+

IE(+, K) (so the reader may skipthis, sufficing himself with the estimate on IE(+, K)).

For each X []+

, let FX = {f : f a K -embedding of M into MX}, and for

f FX let

ZX,f =

X1 [Z]+ : there is a K -embedding of MX1 into MX extending f

,

and let SX,f = {p S(M) : f(p) is realized in MX}, so ZX,f {X1 : X1

SX,f} and |SX,f| +.Now the result follows from the the fact 2.6 above. 3.3

3.4 Claim. 1) Assume

(a) 2 < 2+

< 2+2

and case B or C of Fact 2.1 for occurs(so ,T are determined)

(b) K is an abstract elementary class LS(K)

(c) K++ = ,

(d) K has amalgamation in [was: and in +, seem irrelevant]

(e) in K3, the minimal triples are not dense.

Then

() for each < for some M K+ we have |S(M)| | lim+(T)|(see 1.2(7) the tree from clause () of 2.1); onS see 1.2(7).

2) If K satisfies (a)-(e) and is categorical in + or just has a universal member in

+ and amalgamation in +, then for some M K+ we have |S(M)| = 2+ .

3) If clauses (a)(e) from above and clause (f)+, then I(++, K) wd(++, 2+)

where

(d)+ K has amalgamation also in +

(f)+ K is categorical in and +.

Remark. 1) Assume (a)-(e) of part (1) then C0K, has weaker -coding if we have

restricted to (M , N , a) above which there is not minimal triple in Definition of C).But not used.2) Note that for 3.6 below we do not use 3.4.3) We would like to weaken in [Sh:E46] the assumption K categorical in + to


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30 SAHARON SHELAH

no maximal model in K. So by 3.3 for amalgamation bases M K+ ,S(M)

cannot be too large (used in the proof of 3.6 and as I(++, K) < wd(++, 2

+

),there are many amalgamation basis and by 3.4(1)-pf there are many M K+ with

S(M) large. But we have to put them together (20045/4).

Proof. 1) Let < . Recall the T is a subtree of+>2 of cardinality +,

hence let T =


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3) Assume that the conclusion fails. Now if (A) (B) of 2.1 let be as therethen by 3.1(3) we get I(+, K) > 1 contradicting assumption (f)+. Hencein 2.1, case (C) holds, so let T : < be as there, so by part (2) for some

M K+ we have |S(M)| = 2+

. By 3.3 with + here standing for there (!)by our assumption toward contradiction we deduce cf([2

+

]++

, ) < 2++

. By 2.1with + here standing for there; case (A)+ cannot hold (recall that in this case

= 2++

), so we can assume (C)+ (B)+ occurs.Now if (2 > +)+ (WDmId(+) is not ++-saturated) we get the desired result

as follows.

Case 1: () of 1.9 holds.The result follows by 1.12.

Case 2: () of 1.11 holds.By 1.11 we get a contradiction to I(+, K) = 1.

So one of the assumptions of the previous paragraph fails. If the second fails (i.e.WdmId(+) is ++-saturated as we are in case (B) or (C) (see (a) of 3.4(1))

so by clause () of 2.1 we have 2 = +, 2+

= ++. So in both cases 2 = +.However, once we know 2 = + we deduce that there is a model in + saturatedover and we apply the claim below. 3.4

3.5 Claim. Assume (a)(e), (f) of 3.4 and(g) below thenI(K, ++) wd(++, 2)

where

(g) there is M K+ saturated over .

Proof. Claim [Sh 576, 3.16](3) possibility ()2 applies.Alternatively see [Sh 838]. 3.4

3.6 Claim. Assume

(a) 2 < 2+

< 2++

(b) K an a.e.c. is categorical in , + and LS(K)

(c) 1 I(++, K) < wd(++, 2

+

)

(d) K3 has the weak extension property above (M, N, a).

Then above (M, N, a) K3 the minimal triples are dense.


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32 SAHARON SHELAH

Proof. Assume toward contradiction that above (M, N, a) K3 there is nominimal type. If 2 = +, then there is a M K+ saturated over hence wefinish by 3.5 above. So we can assume 2 > +, hence [Sh 430, 6.3] (with +

here standing for there so there is so < +

hence |T| || andlet = cf()) there are and4 tree T with nodes and levels with|lim(T)| > + hence for some M K, |S(M)| >

+ (e.g. by the proof of3.4(1)). If WDmId(+) is not ++-saturated then in 1.12 assumption (b) holds,and assumptions (c) + (d) + (e) holds by the assumptions of the present claim butnot the conclusion, so (a) fails, that is () of 1.9 holds hence by 1.9, () of 1.8holds. But now 1.8 contradicts clause (b) of the assumption, so we have to assumethat WDmId(+) is ++-saturated. Hence clause () of 2.1, Case B does not occur,hence Cases B,C of 2.1 do not occur and hence Case A occurs. So by 3.1(3) we geta contradiction to categoricity in +.

3.6

4let = Min{ : 2 > +}, s o i f 22, ) is okay, otherwise >2 =[

i


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4 A minor debt

4.1 Claim. We can prove [Sh 576, 4.2t] also for = 0.

Proof. We ask:Question 1: are there M


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34 SAHARON SHELAH

REFERENCES.

[Ha61] Andras Hajnal. Proof of a conjecture of S.Ruziewicz. FundamentaMathematicae, 50:123128, 1961/1962.

[KjSh 409] Menachem Kojman and Saharon Shelah. Non-existence of UniversalOrders in Many Cardinals. Journal of Symbolic Logic, 57:875891, 1992.math.LO/9209201.

[Sh:E45] Saharon Shelah. Basic non-structure for a.e.c.

[Sh 600] Saharon Shelah. Categoricity in abstract elementary classes: going upinductively. math.LO/0011215.

[Sh:E46] Saharon Shelah. Categoricity of an abstract elementary class in twosuccessive cardinals, revisited.

[Sh:F603] Saharon Shelah. Non-structure in ++. moved to 838.[Sh 838] Saharon Shelah. Non-structure in ++ using instances of WGCH.

Chapter VII, in series Studies in Logic, volume 20, College Publica-tions. 0808.3020.

[Sh:a] Saharon Shelah. Classification theory and the number of nonisomorphicmodels, volume 92 of Studies in Logic and the Foundations of Mathe-matics. North-Holland Publishing Co., Amsterdam-New York, xvi+544pp, $62.25, 1978.

[Sh:b] Saharon Shelah. Proper forcing, volume 940 of Lecture Notes in Math-

ematics. Springer-Verlag, Berlin-New York, xxix+496 pp, 1982.[Sh 420] Saharon Shelah. Advances in Cardinal Arithmetic. In Finite and Infi-

nite Combinatorics in Sets and Logic, pages 355383. Kluwer AcademicPublishers, 1993. N.W. Sauer et al (eds.). 0708.1979.

[Sh:g] Saharon Shelah. Cardinal Arithmetic, volume 29 of Oxford LogicGuides. Oxford University Press, 1994.

[Sh 430] Saharon Shelah. Further cardinal arithmetic. Israel Journal of Mathe-matics, 95:61114, 1996. math.LO/9610226.

[Sh:f] Saharon Shelah. Proper and improper forcing. Perspectives in Mathe-

matical Logic. Springer, 1998.

[Sh 460] Saharon Shelah. The Generalized Continuum Hypothesis revisited. Is-rael Journal of Mathematics, 116:285321, 2000. math.LO/9809200.

[Sh 576] Saharon Shelah. Categoricity of an abstract elementary class in twosuccessive cardinals. Israel Journal of Mathematics, 126:29128, 2001.math.LO/9805146.


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[Sh 603] Saharon Shelah. Few non minimal types and non-structure. In Proceed-ings of the 11 International Congress of Logic, Methodology and Philos-ophy of Science, Krakow August99; In the Scope of Logic, Methodology

and Philosophy of Science, volume 1, pages 2953. Kluwer AcademicPublishers, 2002. math.LO/9906023.

[Sh:E35] Saharon Shelah. A bothersome question. Internat. Math. Nachrichten,208:2730, 2008.

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Saharon Shelah- Few Non Minimal Types and Non-Structure