saharon shelah- beginning of stability theory for polish spaces

8/3/2019 Saharon Shelah- Beginning of Stability Theory for Polish Spaces

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BEGINNING OF STABILITY THEORY FOR POLISH SPACES

SH849

SAHARON SHELAH

Abstract. We consider stability theory for Polish spaces and more generallyfor definable structures (say with elements of a set of reals). We clarify byproving some equivalent conditions for 0-stability. We succeed to prove exis-tence of indiscernibles under reasonable conditions; this gives strong evidencethat such a theory exists.

0. Introduction

0(A). General Aims.

{0.4}Question 0.1. Is there a stability theory/classification theory of Polish spaces/algebras(more generally definable structures say on the continuum)?

Naturally we would like to develop a parallel to classification theory and inparticular stability theory (see [Sh:c]). A natural test problem is to generalizeMorley theorem = Los conjecture. But we only have one model so does it meananything?

Well, we may change the universe. If we deal with abelian groups (or any variety)

it is probably more natural to ask when is such (definable) algebra free. {0.5}Example 0.2. IfP is adding (20)+-Cohen subsets of then

(C)V and (C)V[P]

are both algebraically closed fields of characteristic 0 which are not isomorphic (asthey have different cardinalities).

So we restrict ourselves to forcing notions P1 P2 such that

(20)V[P1] = (20)V[P2]

and compare the Polish models in VP1 , VP2 . We may restrict our forcing notions

to c.c.c. or whatever. {0.6}Example 0.3. Under any such interpretation

(a) C = the field of complex numbers is categorical

(b) R = the field of the reals is not (by adding 20 many Cohen reals).

Date: March 14, 2011.The author would like to thank the Israel Science Foundation for partial support of this research

(Grant No. 242/03). This was part of [Sh:771] which was first done in June 2002; but separatedin Feb. 2009. First version - 09/Feb/13.

1


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2 SAHARON SHELAH

(Why? We say assume P1 P2, (20)V[P1] = (20)V[P2] but RV[P1] = RV[P2]. Triv-ially: RV[P2] is complete in V[P2] while RV[P1] in V[P2] is not complete, but there

are less trivial reasons).{0.7}

Conjecture 0.4. We have a dychotomy, i.e. either the model is similar to cate-gorical theories, or there are many complicated models under the present inter-pretation.

So in particular we expect the natural variants of central notions defined below(like categoricity) will be equivalent; in particular we expect that it will be enoughto consider the forcing notions of adding Cohen reals. Naturally those questionscall for the use of descriptive set theory on the one hand and model theory on theother hand; in particular to using definability in both senses and using L1,0(Q).

Presently, i.e. here there is no serious use of either; the questions are naturallyinspired by model theory. It would be natural to consider questions inspired by the

investigation of such specific structures; to some extent considering the freeness ofa definable Abelian group fall under this.

A priori, trying to connect different directions in mathematics is tempting, butit may well lack non-trivial results. We suggest that the result on the existence ofindiscernibility, 3.8, give serious evidence that this is not the case; note that eventhe weaker 1.7 gives more than set theory, i.e. Erdos-Rado theorem. That is, thequestion is: is there a non-trivial theory in this direction? While the present workdoes not achieve a real theory of this kind, we believe that it gives an existenceproof.

Let us elaborate suggestions for the definition of categorical and for defin-able. Recall that it is well known that: a Borel (set or structure) is a 11-one, a11-set/structure is a

12-one and also an 1-Suslin one, and a

12-set/structure is a

1-Suslin one, see e.g. [Jec03].{z7f}

Definition 0.5. 1) Let A denote a definition of a -structure, a countable vocab-ulary, the set of elements of the structure is the reals or a definable set of reals suchthat it is absolute enough, i.e. for any forcing notion PQ we have A[VP] A[VQ]but we may say the structure/model A.2) We say A is F/Borel/11/-Suslin, etc., if the definition mentioned above isF/Borel/11/-Suslin, etc.3) We say A is categorical (or categorical1, similarly below) when for any forcingnotions PQ such that (20)V[P] = (20)V[Q] the structures A[VP],A[VQ] areisomorphic in VQ.3A) We say A is categorical under (e.g. = (20 = 1) means that for anyforcing notions P Q satisfying P ,Q and Q (20)V[P] = (20)V[Q]the structures A[VP],A[VQ] are isomorphic in VQ.4) We say A is K-categorical when above P,Q K (or the pair (P,Q) K).5) If is a definition of a cardinal then A is categorical in , is defined as in (3A)but P ,Q is replaced by [VP] = [VQ] = (20)V[P] = (20)V[Q] and thisis a non-empty condition; similarly in (3A),(4).6) Let T be a set of (first order) equations in the countable vocabulary . Let Abe a model ofT. We say A is free1 for Kwhen for every P K the model A[VP] isa free algebra. Similarly, parallely to (3A),(4),(5).

{z7k}


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BEGINNING OF STABILITY THEORY FOR POLISH SPACES SH849 3

Conjecture 0.6. 1) If, e.g. a 11-structure A is categorical1 in some for K then it is categorical in every of cofinality > 0 for K where

K= {(P,Q) : PQ and (20

)[VP

] = (Z0

)[VQ

] [VP

], Card[VP

] = Card[VQ

]}.2) Or at least for every 1.3) Similarly for freeness.

{0.8}Thesis 0.7. 1) Classification theory for such models resemble more the case ofL1, than the first order.2) As there (see [Sh:87a], [Sh:87b], [Sh:600]) if the continuum is too small we mayget categoricity for incidental reasons.

See [Sh:h]; as support for this thesis, in [Sh:771, 5] we prove:{0.9}

Theorem 0.8. There is an F-abelian group (i.e. an F-definition, in fact a veryexplicit definition) such that V |= G is a free abelian group iff V |= 20 < 736.

Comments: In the context of the previous theorem we cannot do better than F,but we may hope for some other examples which is not a group or categoricity isnot because of freeness.The proof gives

{0.10}Conclusion 0.9. For any n < for some F-abelian group A,A is categorical in iff n.

A connection with the model theory is that by Hart-Shelah [HaSh:323] suchthings can also occur in L1, whereas (by [Sh:87a], [Sh:87b]) if

n

(2n < 2n+1)

and L1, is categorical in every n, then is categorical in every . See morein Shelah-Villaveces [ShVi:648].

The parallels here are still open.

Those questions may cast some light on the thesis that non-first order logics aremore distant from the so-called mainstream mathematics. This work originallywas a section in [Sh:771]; in it we try to look at stability theory in this context,proving the modest (in 3.8):

for 0-stable -Suslin models the theorem on the existence of indis-cernibles can be generalized.

We may consider another interpretation of categoricity. Of course, we can usemore liberal than L[A2, r] or restrict the As further (as in the forcing version), see(0C). A natural question is whether in the existence of indiscernibles (in Theorem

3.8) we can start with a set of cardinality rather than ++

. Another naturalquestion is whether categoricity implies that the L,0-theory of A is simple

clearly by (0A) we are sure this holds.Both questions (and maybe in the right parallel of stable) are addressed in a

work in progress, [Sh:F1134].

0(B). The Content of the Present Work.

Our context is a -candidate (A, ), so A is structure with a set of elements/aset of reals, which is reasonably definable: usually -Suslin; you can fix as theset of quantifier free formulas. In the present work forcing does not appear, so theuniverse is fixed.


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4 SAHARON SHELAH

In 1 we give some basic definitions in Definition 1.4 but note that the 0-stableand 0-unstable are proved to be complimentary only later. The main result is to

get the end-extension indiscernibility existence lemma. Using stability, we havethat for a : < we get a subsequence a : S, S stationary suchthat the -type of a1 , . . . , ak over {a : < 0} does not depend on ak. Wethen improve it to does not depend on k, k1, . . . , kn for a fix n, but thisdoes not give full indiscernibility. Note that the definition of stability and its usein 1.7 speak of definability of types but the type are model theoretic ones speakingonly on -formulas whereas the definable means in set theoretic sense, so thedefinition is arbitrary, just has to be in the submodel. This seems inherent in ourframework: for a predicate P (A), the set theoretic definition ofPA may involve,as approximations, relations on the reals which are very complicated.

In 2 we generalize the order property implies unstability, in 2.1 giving a criterionfor unstability in 2.3. Now the unstable in Definition 1.4(3) really speaks onhaving a perfect set of types; we here define apparently weaker version (, , )-unstable.

Lastly, in 3, we define ranks, but here the ranks are for subsets I of mA, notjust definable ones; as explained above this seems inherent in our framework.

We then prove (in 3.3,3.7) that A is (0, )-stable is really defined, i.e. thatwe have several equivalent definitions some from the structure side, some from thenon-structure side (generally on this see [Sh:E53, (1A),(2A),(2B)]).

Lastly, we prove a theorem on the existence of (fully) indiscernible sequence:

from a set of cardinality ++

to one of cardinality ; a parallel situation occurs forstrongly dependent T (by [Sh:863]). We end noting that, of course, being (0, )-unstable implies a failure of categoricity (strong one) as expected in our frame.

0(C). Further Comments.

{0.12}Definition 0.10. 1) For a definition A of a -model (usually with a set of elementsa definable set of reals) we say that A is categorical2 in 2

0 when : for somereal r: for every A1, A2 the models A

L[A1,r],AL[A2,r] are isomorphic (in V).2) For a class Kof forcing notions and cardinal we say A is categorical2 in (,K)when for every P K satisfying P 20 , we have in VP: the structure A iscategorical2 in , i.e. in the sense of part (1).

Comparing Definition 0.10(1) with the forcing version we lose when V = L, asit says nothing, we gain as (when 20 > 1) we do not have to go outside theuniverse. Maybe best is categorical in in VP for every c.c.c. forcing notion P

making 20

.Note also that it may be advisable in 0.10(1) to restrict ourselves to the case is regular as we certainly like to avoid the possibility (20)L[A1,r] = < (20)L[A2,r]

(see on this and for history in [Sh:g, Ch.VII]).

Of course, any reasonably absolute definition of unstability implies non-categoricity:if we have many types we should have a perfect set of them, hence adding Cohensubsets of adds more types realized. If we add i : i < 20 Cohen reals for everyA 20 , AV[i:iA] : A 20 are non-isomorphic over the countable set of pa-

rameters, if we get 220

non-isomorphic models, we can forget the parameters and


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retain our richness in models. The present work is to some extent a continuationof [Sh:202], [Sh:522], see history there.

{0.15

}Remark 0.11. For () = 1, 2 letting = ()1, we may replace everywhere -Suslin, , + by 1(), ()1, () respectively. Presently though, the first (i.e.

11) case is more restrictive than the second (1-Suslin) case, what we can prove isthe same. For () = 1 we have real equivalence.

Note that below the indiscernible sequence are indiscernible sets, justified by 2.1because of (see, e.g. [Sh:c]).

{z20}Claim 0.12. 1) Assume I is an infinite linear order as mA for s I, as : r I is (, n)-indiscernible sequence (see ?) but not a (, n)-indiscernible set (see1.9(6)). Then we can find = (x0, . . . , xn, y) and b

g(y)A and < n suchthat: if s0


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6 SAHARON SHELAH

1. Generalizing stability in 0

We may consider the dividing line for abelian groups from [Sh:402] and try togeneralize it for any simply defined (e.g. Polish algebra till -Suslin) model. Wedeal with having two possibilities, in the high, complicated side we get a parallelof non 0-stability (hence strong non-categoricity, see 3.10; in the low side we havea rank. But even for minimal formulas, the example in [Sh:771, 5] shows that weare far from being done, still we may be able to say something on the structure.

We may consider also ranks parallel to the ones for superstable theories. Notethat there are two kinds of definability we are considering: the model theoretic oneand the set theoretic one.

{7.0}

Context 1.1. 1) If not said otherwise, A will be a structure with countable vocab-ulary and its set of elements is a set of reals; usually a definition - see 0.5(1).2) L is a logic. We did not specify the logic; we may assume it is L1,0 or justL1,0(Q) where Q is the quantifier there are uncountably many.3) is a fix cardinality, let a -model mean a -Suslin model.

{6a.del}Definition 1.2. 1) For a structure A, an A-formula or (A,L)-formula is aformula in the languageL(A) so in the logic L, and the formula in the vocabularyof A with finitely many free variables, writing = (x) means that x is a finitesequence of variables with no repetitions including the free variables of .2) denotes a set of such formulas and denote a pair (0(x), 1(x)) of formulasso is a -pair when 0, 1 .3) We say (or or ) is -Suslin (or 11 or

12 or

10 (= Borel)) iff they are so

as set theoretic formulas.{7.0k}

Discussion 1.3. 1) So if the relation A, (0, . . . , n1) PA where P is

a predicate1 of A, are -Suslin, then they are upward and downward absolute (as

long as the relevant subtrees of T are in the universe).2) When we consider a formula (x) this is more complicated by we are assumingit, too, is -Suslin, meaning that this (set theoretic) definition defines (x)A alsoin the other universes we consider.

{7.1}

Definition 1.4. 1) We say (A, ) is a -candidate (or -Suslin candidate; as isconstant we may omit it) when :

(a) A is a -model

(b) is a countable set of (A,L)-formulas2 which, are in the set theory sense,-Suslin (we identify and )

(c) we consider changes of the universe (say by forcing) only which this ispreserved, anyhow we can use each (x) is quantifier free.

1A) We can replace being -Suslin by being 11 by 12, etc., (naturally we needenough absoluteness); if we replace it by we write -candidate. If does notappear we mean it is 11 or understood from the context.2) If (A, ) is a candidate we say A is (0, )-stable (or (A, ) is 0-stable), when is a countable set of (A,L)-formulas and for large enough and x H(),

1pedantically we should change a little in 0.13, or we can translate PA n() by a subsetof

2We may use is a set of pairs = (0(x), , (x) of such formulas, and is later demand . So far it does not matter.


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for every N (H(), , A and 0, 1 andA |= (x)(0(x, b) 1(x, b))},

() if = (0(x, b), 1(x, b)) m(A,) N and < 2 and

A |= [a, b] then = c().

3) We say that (A, ) is 0-unstable (or A is (0, )-unstable) when: there area mA for 2 and ,0(x, y) and ,1(x, y) and b

g(y)A for >2 such that:

(a) A |= (x)(,0(x, b) ,1(x, b))

(b) if 0, 1 and 0, 1 2, n = g() and 0(n) = 0, 1(n) = 1 then

A |= ,0[a0 , b ] ,1[a1 , b ].

4) Let (x, y) m(A,) means that (x, y) = (0(x, y), 1(x, y)) and 0(x, y), 1(x, y)

belongs to and A |= (x, y)(0(x, y) 1(x, y)).5) Let (x, y) means that (x, y) = (0(x, y), 1(x, y)) and 0(x, y), 1(x, y)belongs to .

Remark 1.5. 1) There are obvious absoluteness results (for m(A,), (A, ) is

0-unstable and 0-stable).2) On those notions being complimentary see Theorem 3.3.

{7.2}Observation 1.6. 1) If is closed under negation then in Definition 1.4(2) weequivalently can replace by:

for some c N we have: if (x, y) and b g(y)A and b N thenA |= (a, b) iff c((x, b)) = 1.

2) In Definition 1.4(2) we can fix (A, ) and omit


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8 SAHARON SHELAH

(i) if , S E then fn+1() = fn+1() fn() = fn()

(ii) if n < and < , then the sequence a : S E, fn() = is

(, n)-end extension indiscernible over A(ii)+ moreover, if n < and , < then a : S E\ and fn() = is

(, n)-end extension indiscernible over A {a : < }.{7.4y}

Remark 1.8. 1) This is a first round on indiscernibility.2) The assumption is closed under negation is quite strong.3) Really we get indiscernible sets by 0.12 and 2.1.4) The claim and proof are similar to [Sh:c, Ch.III,4.23,pg.120-1], but before provingwe define:

{7.4a}

Definition 1.9. 1) Let (A, ) be a candidate. We say A has (, )-order when :

() for some m() < and (x, y) m()A, with g(x) = g(y), this formula

linear orders some I m()A of cardinality , see part (2) for definition.

2) We say that the formula (x, y) linear orders I A iff for some at : t I wehave:

(a) I = {at : t I}

(b) I is a linear order

(c) = (0(x, y), 1(x, y)) m(A,)

(d) if s


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countable subsets of N, for some c the set N = {N : N N is countable andcN = c

} is a stationary subset of [N]0 , so c can serve for N.

2) Let N : < be an increasing continuous sequence of elementary submodelsof (H(), , 1 by the induction hypothesis we can find stationaryS1 S as required in

n. For each < we can choose , = (, ) for

n such that = ,0 < ,1 < . . . < ,n and 0 < n , S1. Leta = a,0 . . . a,n so a

m(n+1)A and apply the induction hypothesis or justthe case n = 1, i.e. part (2) to m (n + 1), S1, a

: < getting a stationary

S2 S1 as required in n or just in

1.

We claim that S2 is as required. So assume 0 < . . . <

n < and

0 < .. . < n < and

,

S2 for n.

Now, letting (, ) = , for n we have:

(i) a0

a1

. . . an and a0a(0,1) . . . a(0,n) realizes the same -typeover A {a : < } in A.

[Why? As {1, . . . , n} (1, ) S1, S2 S1 and the choice of S1).]

(ii) a0

a(0,1) . . . a(0,n) is equal to a0

.

[Why? By the choice of a0

.]

(iii) a0

, a0

realizes the same -type over A{a : < } hence over A{a :

< }.

[Why? By the choice of S2].Similarly

(iv) a0

is equal to a0

a(0 ,1) . . . a(0 ,n)

(v) a0

a(0 ,1) . . . a(0 ,n) and a0 a1 . . . an realizes the same -typeover A {a : < }.

By (i)-(v) the set S2 is as required in n+1 .

4) The proofs of parts (2), (3) actually give this. 1.7


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2. Order and unstability{7.7}

Claim 2.1. The order/unstability lemma:Assume that

1 (a) (A, ) is a -candidate, (e.g. () {1, 2}, = ()1 the relationare 1() hence a -Suslin)

(b) 0(x, y), 1(x, y) are contradictory inA

(c) J is a linear order of cardinality

(d) at mA for t J

(e) A |= 0[as, at] and 1[at, as]whenever s + For < + we let

P = {I : I mA is linearly ordered by getting an order of cardinality> + and density }.

This should be clear.By 2 of the assumption at least one of the cases holds. 2.1


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Similarly{b7}

Remark 2.4. Like 2.1, replacing clause (e) by

(e) for some < n and = (x0, . . . , xn, y), g(x) = m; b g(y)A and as

mA for s J and the conclusion of 0.12.

Proof. Proof of 2.3For each (x) as {a g(x)A : A |= [a]} is a -Suslin set and4

()(x) we can find C, : and < such that

(a) {a g(x)A : A |= [a]} = {a: for some < and we have(a, ) C,}

(b) if < then C, is closed subset of (g(x)+1)().

We can find functions F0, F1 such that if (x) and A |= (a) then F

0(a)

and F1(a) witnessing this and code a continuously. For notational simplicityand without loss of generality m = 1. Let W = {w : w >2 is a front5 hencefinite}.

For w W and n < let Qn,w be the family of objects x = (n, u, , , ) =(nx, . . .) such that:

()n,w,x for unboundedly many < + we can find a witness (or an -witness)y = (a : < n, B : w) which means:(a) u = (u0, u

1) : w and w u

0, u

1 n and =

i : i

() < >

(g) if w, b B, i {0, 1}, ui

theni

, F0i (b, a) and

i

, (F1

i

(b, a)).

Clearly

1 Q0,{} = .

4Pedantically note that we sometimes consider a g(x)A as a member of or better(H(0)) rather than g(x)() or see 0.13(4).

5i.e. for every 2 there is one and only one n < such that n w


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12 SAHARON SHELAH

[Why? Let x = (0,, (, ,,, (x)(x = x)) and if < + choose I Pwe let B = I; recall that by clause (c) of Claim 2.3 the family P is non-empty.]

2 if x Qn,w and for w then there is y such that:

y Qn,w uy = ux

ix,, iy,,

iy,, =ix,,

y = x.

[Why? As x Qn,w we know that for some unbounded Y + for each Ythere is an -witness y = a : < nB

: w as required in ()n,w. Let

< + and () = Min(Y\( + 1)). Now for each w we have B() P().

Let k > sup{g(ix,, : < 2, w and < n} + 1 and, of course, the set{F

i(b, a

() )k : b B

() } has cardinality for each i < 2, < 2, < n, w.

Hence by clause (e) of Claim 2.3 as < () we can find a subset B, of B()

from P and ,i,

w>w, ,i, > such that

b B, F1ix,

(b, a() )k =

i,

b B, F0ix,

(b, a() k =

,i, .

Similarly for some ,i, ,,i, : < 2, w, < n we have

Y = { Y : ,i, = ,i, and

,i, =

,i,} is unbounded in

+.

Now it is easy to choose y

3 if x Qn,w and w and u = (w\{}) { < 0 >, < 1 >} so u W,then there is y Qn+1,u such that:

() ny = nx + 1

() uy,

= ux,

for w\{}, = 0, 1

uy,

{0, . . . , n 1} = ux, for = 0, 1 and j = 0, 1

() iy,, =

ix,, for w\{}

iy,, = ix,,

() iy,, =

ix,, for w\{}

iy,,

= ix,,

() yn = x.

[Why? Similar to the proof of2 using clause (d) of the assumption of Claim 2.3this time.]

Together it is not hard to prove the non 0-unstability. 2.3{7.8}

Remark 2.5. 1) This claim can be generalized replacing 0 by , strong limit sin-gular of cofinality 0.


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{7.9}

Definition 2.6. 1) tp(a,A,A) = {(x, b) : (x, y) and b g(y)(A) and

A |= [a, b]}.2) pr,mA,,A = {(0(x, b), 1(x, b)) : 0(x, y), 1(x, y) belong to and b

g(y)A and

x = x : < m and A |= (x)[0(x, b) 1(x, b)]} where A A, a set ofA-formulas, and so pr,mA, = {(0(x, y), 1(x, y)) : 0, 1 ,A |= yx[0(x, y) 1(x, y)]}.3) Sm (A,A) = {tp(a,A,A) : a

mA} where A A and a set ofL(A)-formulas.{7.10}

Definition 2.7. 1) We say (A, ) is (, , )-unstable iff there are M A, m < and a : < such that:

(a) a mA

(b) if = are < then for some (0(x, b), 1(x, b)) pr,mA,,M (see Definition

2.6(2)) we have 0(x, b) tp(a, M, A) and 1(x, b) tp(a , M, A)

(c) M .

1A) Let A be (0, , per)-unstable mean that (A, ) is 0-unstable; here per standsfor perfect.2) In part (1) and (1A) we add weakly if we weaken clause (b) to

(b) tp(a, M, A) = tp(a , M, A) for = from X(so if is closed under negation there is no difference); in part (1),

X = and in part (1A), X = 2.

3) We use (0, , x,Q) where Q is a forcing notion iff the example is found in VQ

such that usually M is in V and we add an additional possibility if x = per thenM V and X = (2)V (here per stands for perfect).4) We may replace a forcing notion Q by a family K of forcing notions (e.g. thefamily of c.c.c. ones) meaning: for at least one of them.5) We replace unstable by stable for the negation.

{7.11}Observation 2.8. If is closed under negation, then A is weakly (0, , )-unstable iffA is (0, , )-unstable.


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14 SAHARON SHELAH

3. Rank and Indiscernibility{7.12}

Definition 3.1. Let (A, ) be a -candidate. Let 1 m N and assume D isa +-complete filter on mA; (or a filter on some I mA then we interpret it as{J mA : J I D}; writing rkm we meand rk

mD , D = {I

mA : mA\I hascardinality } similarly rkn


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Now let P = {{a : S} : S + is unbounded}. Now we can prove byinduction on that I P rkmD(I) .

(d) (a): Trivial.

(e) (c):

Let A A, |A| and let Sm (A,A) = {tp(a,A,A) : a mA for some m}.

Let be large enough, N (H(), ) be of cardinality such that A, A Nand + 1 N. By Definition 3.7(2) and the present assumption (e), for everym and p Sm (A,A) there is cp N such that p is disjoint to q

nocp

and if is

closed under negation is equal to pyescp

where for t {yes, no} we let qtc := {(x, b) :

(x, y) , b g(y)(A N) and c((x, b)) = t}. Hence if is closed undernegation, then Sm(A,A) has cardinality N = ; as this holds from everyA A of cardinality we get (c), as promised.

If is not closed under negation but (c) holds, let S = {tp(a, A,A) : < +}

be as in Definition 2.7(1), rest should be clear.(d) (e):

We have to prove (e), i.e. A is (, 0)-stable, see Definition 1.4(2).So assume M N, N (H(), ),A N, + 1 N and N = and a mA

and we should find a function c as there. Now let Z := {rkmk (I) : I B} whereB = {I : I N is a non-empty subset of mA to which a belongs}. Now this familyB is not empty (as mA B) hence Z = .

Of course, B,Z N and / Z by our present assumption, i.e. (d). Hence = min(Z) is a well defined ordinal, so by the definition ofZ there is I B suchthat rkm (I) = . Now why rk

m (I) := + 1? By Definition 3.1, case 3, clause

(a) or clause (b) there fail.If clause (a) fails, then there is a sequence I = Ii : i of subsets of I

with union I such that i < rkm (Ii) < , clearly I H() hence without

loss of generality I N. As N necessarily i < Ii N and obviouslyi < Ii I mA.

Lastly, as a I necessarily there is j < such that a Ij . So Ij witnessrkm (Ij) Z and as said above rk

m (Ij) < , so contradicting the choice of as

min(Z).So clause (b) of case 3 of Definition 3.1 fail (for our ,, I) so

1 for every (x, b) mA, we can choose < 2 such that rk

m ({a

I : A |=

[a, b]}) < .

So there is a function t N with domain mA, and range {0, 1} such that

2 if (x,b)

m

A, then t = t((x,b)) satisfies rk

m

(I(x,b),t) < where3 I(x,b),t = {a

I : A |= t[a, b]}.

However

4 if (x, b) mA, N and t(x, b) is satisfied by a, t < 2 and A |= t[a, b]

then

(a) I(x,b),t B

(b) rkm (I(x,b),t) Z


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16 SAHARON SHELAH

(c) rkm (I(x,b),t) =

(d) t = t((x, b)).

[Why? For clause (a): as (x, b) N clearly I(x,b),t N and it is I mA. Also,

a I(x,b),t by the assumption of 4. Hence by the definition ofZ also clause(b) holds. But by 3.2(3) we have rkm (I(x,b),t) rk

m (I) = so by the choice of

as min(Z) we get equality, i.e. clause (c) holds. Recalling 2 we get clause (d)contradicting the present assumption.]

Now by 4 (as t is a function from N), we are done proving (e). 3.3{7.14d}

Conclusion 3.4. 1) The property rkm (A) = is preserved by forcing.2) If rkm (

mA) = then for some U1 coding the definitions of A and of the-Suslin definitions of every (x) , andU1 Ord every forcing extension ofL[U1,U2] satisfies rk

m (

mA) = {7.15}

Definition 3.5. If p is a (1, m)-type over A in A (i.e. a set of formulas (x, a)with (x, y) 1, a A), we let (may write instead of < +)

rkm


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For m < and I mA let JI = J[I] be the family ofS S0 such that: we canfind Fx, cx : x H() (a witness) such that:

() cx : mA, {0, 1}

() Fx :>(H()) H()

() if M S0 is closed under Fx for x M then for every a I for somey M, cy is a witness for tp(aM, M A,A), see Definition 1.4(2).

Clearly JI is a normal ideal on S0. Also if m < S0 J[mA] then

increasing we get the desired result. Toward contradiction assume that m < and S0 / J[

mA] and let P (i.e. P = P for < +) be the family of I mAsuch that S0 / JI.

We now finish by 2.3 once we prove

if I P then for some (x, b) mA, for each < 2 the set I(x,b)

is

{a I : A |= (a,b)} belong to P.

If not, for every (x, b) mA, there is = c[(x, b)] < 2 and (F(x,b)x , c

(x,b)x ) :

x H() witnessing S0 J[I].

Define (Fy , cy) for y H() by: ify = x, (x, b) then Fy = F(x,b)x , cy =

c(x,b)x , otherwise c.

Clearly we can find M S0 such that

1 if (x, b) mA, M and x M then M is closed under F

(x,b)x

2 for some a mA, no cy, y M defines tp(a, M A,A), in the sense of

1.4(2).

But c does it! So we are done.

(h) (g).Obvious.

(g) (d).

Like (c) (d).

(c) (h).

Repeat the proof of (c) (d) in 3.3. 3.7{7.18}

Theorem 3.8. Assume that (A, ) is a -candidate and is 0-stable or just -stable.

For some < 1 we have: if ,m < , A A, |A| and a mA for

< + then for some S + of cardinality the sequence a : S is

-indiscernible over A inA

.Proof. Assume not. For < 1 let

P = {{a : < +} : for no S + of cardinality is a : S is -indiscernible over A in A}.

The central point is to note the obvious:

if + is regular, > 0, A A, |A| , a mA for < + and S +

is stationary then (a) or (b) where


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18 SAHARON SHELAH

(a) for some club E of , a : S E is -indiscernible over A in A

(b) for some m < and club Em of + we have

(b)m (i) a : S Em is (, m)-end extension indiscernible(ii) for no club E Em of

+ is a : S E1 a sequencewhich is (, m + 1)-end extension indiscernible.

Clearly clause (a) is impossible by our present assumptions so let E, m be as inclause (b). By claim 1.7(4) there is a club E of and fn : n < as there and letS = { S : fm+1() = }, so > ,Pm+1 = { : S

is stationary}. Without

loss of generality E E and / Pm+1 S = . Without loss of generality

fm+1 is as in claim 3.9 below.So by (b)m(ii) clearly Pm+1 is not a singleton (and it cannot be empty), so we

clearly have finished. 3.8{7.19}

Claim 3.9. LetA, a : < , E, fn : n < be as in 1.7(4). Then without loss

of generality (possibly shrinking E and changing the fns) we can add

(iii) if n < and 1 = 2 are in Rang(fn+1) but fn+1(1) = 1 fn+1(2) =2 fn(1) = fn(2) letting S = { : fn() = fn(1) = fn(2)} and = Min(S E\(1 + 1)\(2 + 1),

then for some formula (x0, . . . , xn) with parameters from A {a : < } suchthat:

() if i < 2, 0 < . . . < n+1 are from S E( n)()(fn() = fn(

)

fn+1() = i) and f(0) = i thenA |= [a0 , . . . , an ] i = 0.

Proof. Easy. 3.9{c19}

Claim 3.10. Assume (A, ) is a -candidate and it is (0, )-unstable, see Claim

3.3 and Definition 1.4(3). Then A is not categorical, even under 20 = 1, see0.5(3A).

Remark 3.11. Of course, we can get stronger versions: many models.

Proof. Let x = xm, = (,0(x, y), ,1(x, y)) m(A,) for

>2 and b :

>2, a : 2 be as in Definition 1.4(3).Without loss of generality this is absolutely, i.e. if P is a forcing extension, and

(2)V[P] then we can choose a.Let Q be the forcing of adding 2 Cohens,

=

: < 1, so P is trivial and

easily A = AV,AV[Q] are not isomorphic: if F is such that G Q be generic overV, =

[G] and toward contradiction p F is an isomorphism from AV[Q]

onto AV where p G. So for some () < 1, F

(b

) : >2 depend just

on

: < () so in V[()] we can compute it, so F (()) can have no

possible value contradiction. 3.10


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4. Private Appendix

Discussion 4.1. (2011.2.23)1) Can we characterize categoricity in 1. So assume 0-stability. We hope: theelmination in L(aa), has 0-homogeneity.2) Can we get elimination of quantifiers.3) Should we give details in 0.9 instead as in [Sh:771, 5].

Maybe see [Sh:F562].Moved from pg.5:

Metric ModelsConcerning [Sh:771, 1] we may think of a more general context.

{1.6}Definition 4.2. 1) We say a is a metric -model ( = a is a vocabulary, that is aset of function symbols and predicates; in the main case we say -algebra when has no predicates only functions) iff

(a) a is a metric space with metric da

(b) Ma = M(a) is a model or an algebra with universe |Ma|, (of course witha set of elements the same as the set of points of the metric space), withvocabulary = a

(c) if F a is (an n-place) function symbol, then FM(a) is (an n-place)continuous function from Ma to Ma (for da, i.e. by the topology which themetric dG induces, of course)

(d) if R a is an n-place predicate, then Ra = RM(a) is a closed subset ofMna =

n(|Ma|).

2) We say a is unitary if some e a is a unit ofa which means that e an individualconstant and {ea} is closed under FM(a) for F G and R G e , . . . Ra.3) We say a is complete if (|Ma|, da) is a complete metric space.4) We replace unitary by specially unitary above if we add:

() for every R+, the set {a Ma : da(a, ea) < } is a subalgebra ofA.

5) We replace unitary by specially-unitary if some witness it which means:

(a) is a decreasing sequence of positive reals with limit zero

(b) for every F a for some n = n(F, a) we have for every m [n, ) the set{a Ma : da(a, ea) < m} is closed under F

M(a).

6) We add the adjective partial if we allow FM(a) to be a partial function, so inclause (c) of part (1) means now:

(c) ifF a = (Ma) is an n-place function symbol then the set {(a1, . . . , an, FM(a)(a1, . . . , an) :a1, . . . , an Ma and F

M(a)(a1, . . . , an) is well defined} is a closed subsetof n+1(Ma).

7) We say a is specially-unitary iff some witnesses it which means (a),(b) frompart (5) and

(d) for any F a a k-place function for every m n(F, a) we also have:if da(x, y) < m+1 for = 1, . . . , k and da(x, ea) < m+1, da(y, ea)

saharon shelah- beginning of stability theory for polish spaces

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