moran cohen and saharon shelah- stable theories and representation over sets

8/3/2019 Moran Cohen and Saharon Shelah- Stable Theories and Representation Over Sets

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STABLE THEORIES AND REPRESENTATION OVER SETS

[COSH:919]

MORAN COHEN AND SAHARON SHELAH

Abstract

In this paper we give characterizations of the stable, superstable, and -stable

theories, in terms of an external property called representation. In the sense of

the representation property, the mentioned classes of first-order theories can be

regarded as not very complicated.

Introduction

Our motivation to investigate the properties under consideration in this paper comes

from the following

Thesis: It is very interesting to find dividing lines and it is a fruitful approach

in investigating quite general classes of models. A natural dividing prop-

erty should have equivalent internal, syntactical, and external properties.

( see [Sheb] for more)

The main results presented in this paper are:

Characterization of stable theories (Theorem 13):

For a complete first-order theory T, the following conditions are equiv-

alent

(1) T is stable(2) T is representable in Ex10,|T|(k

eq) (cf. definitions 2, 11, and 4).

(3) For some cardinals 1, 1, 2, 2, it holds that T is representable in

Ex11,1(Ex22,2

(keq)) (cf. definition 11).

(4) T is representable in Ex0,(keq) for some cardinals , (cf. definition

10).

(5) T is representable in Ex00,|T|(keq).

Characterization of -stable theories (Theorem 40):

For a complete first-order theory T, the following conditions are equiv-

alent

We thank Assaf Hasson for his constructive remarks.

1


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americanSTABLE THEORIES AND REPRESENTATION OVER SETS [COSH:919] 2

(1) T is -stable.

(2) T is representable in Ex2(keq).

(3) T is representable in Ex1,2(keq).

(4) T is representable in Ex0,lf,2(keq) (see definition 11)

Characterization of superstable theories (Theorem 42):

For a complete first-order theory T the following conditions are equiva-

lent

(1) T is superstable.

(2) T is representable in Ex

2

2|T|,0(keq

).(3) T is representable in Ex12|T|,2(k

eq).

(4) T is representable in Ex0,lf2|T|,2

(keq).

(5) T is representable in Ex2,0(keq) for some cardinal .

(6) T is representable in Ex0,lf,(keq) for some cardinals , .

Discussion 0.1. It would seem natural to conjecture that if keq is replaced by kor in

the above properties, we can find an analogous characterization for dependent the-

ories. However, such characterization would imply strong theorems on existence of

indiscernible sequences. Lately (see [KS]), some dependent theories were discovered

for which it is quite hard to find indiscernible subsequences, rendering the nave

conjecture false.

1. Preliminaries

The main model-theoretic definitions that we will use are

Definition 1. A theory T is called -stable iff for every model M of T and A

M, |A| it holds for every m < that |Sm(A, M)| .

T is called stable if T is -stable for some cardinal . T is called superstable if there

exists , such that T is -stable for every .

1.1. Structure classes and representation.

Convention 1. k denotes a class of structures with a given dictionary (signature)

k. A structure I = , |I| , |= is a triple of dictionary, universe(domain) and the

interpretation relation for formulas in the language (usually logical closure of atomic

formulas). The dictionary may contain constants, partial finitary and infinitary

functions and finitary and infinitary relations.

Discussion 1.1. We will always have a first-order complete theory T in the back-

ground. We separate the notions of structure and model to structures that

belong to a given class of structures and structures from EC(T), respectively.


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Now we reach the central definition

Definition 2. For structures M and I, and sets of formulas . The function

f : M I is called a -representation of M in I if

tpqf(f(a), , I) = tpqf(f(b), , I) tp(a, , M) = tp(b, , M)

for any two sequences a, b


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M0 := I

For limit : M(I) =


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Observation 1.4. Every free algebra has a minimal system of terms. This follows

from the axiom of choice.

1.3. Extensions of classes of structures.

Discussion 1.5. For a class of structures k, we define several classes of structures

that are based on k.

Definition 10. Ex0,(k) is the class of structures defined by extending each I k

to a structure whose dictionary is

I+ := I, P : < , F : <

By additional relations and functions P, F / I for all < , < , such that:

PI+

: <

is a partition on |I|.

FI+

: <

are partial unary functions.

Definition 11. Ex0,lf,(k) is the class of structures in Ex0,(k) for which the closure

of every element under the new functions if finite. (lf stands for locally finite).

Ex1,(k) is the class of structures in Ex0,(k) for which F(P) P


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Observation 1.12. Ex12,(keq) is qf-representable in Ex10,(k

eq).

Proof. Let I, P : < 2 , F : < be the dictionary ofI+. Without loss of

generalityPI+0

2 (every model such thatPI+0

= 1 can be represented in sucha model). We select two distinct t0, t1 P

I+

0 and let h : 2 P() a bijection.

Consider the structure I = I, F : < , G : < whose universe is |I+|,

FI

= FI+

and also define for all < , x PI+

,

GI

(x) =

t0 h()

t1 h()

it is easy to verify that the identity is a LqfI+

-representation of I+ in I.

Observation 1.13. Ex2(keq) is qf-representable in Ex1,2(k

eq) for 0.

Proof. Let (x) : < be a minimal system of terms ofM(S) (see definition

9, is the upper bound on function symbol arities, is the number of functions).

W.l.o.g lg (x0) = 1, 0(x0) = x0. Consider the structure I+ whose dictionary is

I, flast, fhead, P : < and has universe

{,i,s0 . . . si : < , i < lg (x) , s0 . . . si1 S}

Let , i : i < enumerate the pairs {, i : < , i < lg (x)} in increasing

lexical order. Let PI+

the set of sequences in |I+| whose head is , i, and let

fI+

last (,i,s0 . . . si) := 0, 0, si

fI+

head (,i,s0 . . . si) := , i 1, s0 . . . si1 (i > 0)

we define a map h : M(S) |I+| as follows:

h((v)) =

, lg (x) 1, v0, . . . , vlg(x)1

That h is a qf-representation of M(S) in I+ is easy to verify.

Definition 12. We say that a function f with domain and range contained in a

structure I is a partial automorphism when for every sequence a |I| it holds that

tpqf(a, , I) = tpqf(f(a), , I).

2. Stable theories

The central result for this section is


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Theorem 13. for a complete first-order theory T, the following conditions are

equivalent

(1) T is stable

(2) T is representable in Ex10,|T|(keq) (cf. definitions 2, 11, and 4).

(3) T is representable in Ex1|T|+,|T|(keq).

(4) T is representable in Ex12|T|,|T|(keq).

(5) For some cardinals 1, 1, 2, 2, it holds thatT is representable inEx11,1

(Ex22,2(keq))

(cf. definition 11).

(6) T is representable in Ex0,(keq) for some cardinals , (cf. definition 10).(7) T is representable in Ex00,|T|(k

eq).

Proof. 1 3 is by theorem 30. 2 3 4 5 are immediate by 1.10. 4 5, 7

6 are immediate. 5 1 by theorem 15. 3 2 by observations 1.6,1.12 giving us

equivalence of conditions 1 5. 2 7 by 1.7. We leave 6 1 without a complete

proof, since it is very similar to 5 1.

Stability of representable theories.

Discussion 2.1. We shall first prove the first direction of the main theorem.

Namely, that a theory which is representable in Ex22,2

Ex

11,1(keq)

is stable.

The method relies on the combinatorial properties of models of stable theories, par-

ticularly that all order indiscernibles are indiscernible sets.

For this subsection we assume that T is representable in Ex22,2

Ex11,1(keq)

, for

some cardinals $1, 2, 1, 2$.

Theorem 14. For cardinals , , if the following holds:

a. 2 + +1

b. > + 2,2 + 1 a regular cardinal (see definition 9)

g. >


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Each a is closed under the partial functions F : To find the closure of a

under the functions we need to add at most 1 elements, so the closure of

a is < .

lg a = =|| for all < : since =

is regular, and by reordering.

The rest of the proof is by taking subsequences of the original sequence, while

preserving the length , as follows (in brackets we note the common property of

the sought subsequence):

First subsequence (sequences constructed by the same terms) : By 2, for each i . Let T1 a skolemization of T. We choose

M1 |= T1 and sequence a =

a :

such that

(1) a M1, lg(a) = nlg()

(2) (x, y) LT

(3)

a : <

is an indiscernible sequence over

a : ,

for

all n, {n : n < } = and = n : n < and it is easy to get

a contradiction to 4 in both cases.

Theorem 44. Every superstable T is representable in Ex22|T|,0(keq).

Proof. Let T superstable. Let M |= T. We choose Bn, as, us : s Sn by induction

on n < such that:

(1) Sn Sk = (k < n)

(2) as : s Sn M

(3) Bn = acl({as : s {Sk : k < n}}) M

(4) as : s Sn is without repetitions and independent over Bn.

(5) for all s S, us S


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(2) v v Mv Mv(3) tp(Mv , M {M : < }) does not fork over

Mv : v v, <

{as : s v}

The induction is //clearly// possible.

Now Mv : v I is a stable system of models (except for I being closed to finite

subsets). (see [She90, XII, p.598]). For all v I let bv enumerate Mv such that if

vs = us{s} then b0v = as (we use superscript indexes in bv). Therefore,

bv : v I

is without repetitions.

For all < we define I as follows:

I0 = {},

Ik = {v I : v S


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Then the sequences bv0 . . . bvn1 and bu0

. . . bun1 realize the same complete type

over . (This follows from the definitions of the equivalence relations E and

above)

W.l.o.g I S = . We define a structure A with universe |A| = I S as follows:

PA := I, PA,i := I,i, S

An := Sn.

Define partial functions FA for all < |T| , n < as FA (s) = b

us{s}

. Define

partial functions Gn such that {Gn(u) : n} = {v : v u}. Now, the function f well

defined as f(a) = s a = as is a representation of M in A. ( proof should beclear from the definitions )

5. Characterization of (T)

Theorem 45. For a complete theory T, and regular the following are equivalent:

(1) (T) .

(2) T is representable in Ex22|T|,(keq).

(3) T is representable in Ex2,(keq), for some .

(4) T is representable in Ex12|T|,(keq).

(5) T is representable in Ex12|T|,(keq), for some .

6. Miscellaneous

This final section completes the proofs of several claims that were used in the main

theorems above.

Theorem 46. (Fodor) Let a regular cardinal, f : such that f() < for

all 0 < < , Then { < : f() = } is a stationary set of for some < .

(see [Jec78])

Corollary 47. If f : , > regular, then f1({}) is a stationary set of for some < .

Theorem 48. Let a regular cardinal, |W| = ,|St| < (t W) such that

<


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(c) For some equivalence relation E on 0 it holds that t W zt =

zt (, ) E.

Proof.

(1) See [Jec78, She90].

(2) The map t (t) fulfills the assumptions of Fodors theorem ((t)

moran cohen and saharon shelah- stable theories and representation over sets

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