Automorphisms of models of PA: The neglected cases. · 2012-05-29 · Automorphisms of models of...

Automorphisms of models of PA:The neglected cases.

Roman KossakCity University of New York

July 3, 2008

In memory of Henryk Kotlarski 1949-2008

OVERVIEW

I Models of PA in the 1980s

II Automorphisms of models of PA

III Automorphism groups of models of PA

IV The neglected cases. What is left to be done?

MODEL THEORY OF PA IN 1980s

I Hajek, Kirby, McAloon, Paris, Pudlak: Combinatorialproperties of cuts, indicators, independence results.

I Barwise, Kaufmann, Kaye, Lessan, Macintyre, Ressayre,Schlipf, Smorynski, Wilmers: Model theory of recursivelysaturated structures. Resplendence. Expandability.Automorphisms.

I Kotlarski, Krajewski, Lachlan, Murawski, Smith: Nonstandardsatisfaction classes.

I Abramson, Blass, Gaifman, Harrington, Mills, Schmerl, Wilkie:Extensions. Definable types. Iterated ultrapowers. Lattices ofelementary submodels. Uncountable models with interestingsecond-order properties.

I Harnik, Knight, Marker, Nadel, Pabion, Richard, Solovay:Turing degrees. Scott sets. Presburger Arithmetic. Reducts.Other...

Henryk Kotlarski, Automorphisms of Countable RecursivelySaturated Models of PA: a Survey, Notre Dame Journal of FormalLogic Volume 36, Number 4, 505-518 (1995)

Let me begin with the pre-history of the subject. Thequestion whether PA has a model with a nontrivialautomorphism (i.e., such that its elements cannot beindividualized) was due to Hasenjager. It was solvedpositively by Ehrenfeucht and Mostowski [3]. Their resultand the idea of indiscernibility is nowadays so well knownthat Hodges [5] writes “today model theorists use it atleast once a week,” so let me omit the statement of theEhrenfeucht-Mostowski Theorem.

Henryk Kotlarski, Automorphisms of Countable RecursivelySaturated Models of PA: a Survey, Notre Dame Journal of FormalLogic Volume 36, Number 4, 505-518 (1995)

Let me begin with the pre-history of the subject. Thequestion whether PA has a model with a nontrivialautomorphism (i.e., such that its elements cannot beindividualized) was due to Hasenjager. It was solvedpositively by Ehrenfeucht and Mostowski [3]. Their resultand the idea of indiscernibility is nowadays so well knownthat Hodges [5] writes “today model theorists use it atleast once a week,” so let me omit the statement of theEhrenfeucht-Mostowski Theorem.

THREE PAPERS ON EXTENDING AUTOMORPHISMS

1. Results on automorphisms of recursively saturated models ofPA, Fund. Math. 129 (1), 9-15 (1988).

2. On extending of automorphisms of models of Peanoarithmetic, Fund. Math. 149 (3), 245-263 (1996).

3. More on extending automorphisms of models of Peanoarithmetic, to appear in Fund. Math.

AUTOMORPHISMS OF COUNTABLE RECURSIVELYSATURATED STRUCTURES

Let M be countable and recursively saturated.

Proposition

For every infinite A ∈ Def(M) there are a, b ∈ A such that a 6= band tp(a) = tp(b).

Proposition

For all a, b ∈M, if tp(a) = tp(b), then there is f ∈ Aut(M) suchthat f (a) = b.

Proposition

|Aut(M)| = 2ℵ0 .

Proposition

|Aut(M)| = 2ℵ0 .

Proposition

|Aut(M)| = 2ℵ0 .

WHAT IS SPECIAL ABOUT PA?EHREFEUCHT-GAIFMAN LEMMA

TheoremIf M |= PA, f ∈ Aut(M), and f (a) 6= a, then f (a) /∈ Scl(a)1.

1Scl(a) is the definable closure of a in M.

AUTOMORPHIC IMAGES

DefinitionFor X ⊆M, a(X ) = |{f (X ) : f ∈ Aut(M)}|Basic facts:

X ∈ Def0(M)⇒ a(X ) = 1

X ∈ Def(M)⇒ a(X ) ≤ ℵ0

a(X ) > ℵ0 ⇒ a(X ) = 2ℵ0 (Kueker-Reyes)

AUTOMORPHIC IMAGES

X ∈ Def0(M)⇒ a(X ) = 1

X ∈ Def(M)⇒ a(X ) ≤ ℵ0

AUTOMORPHIC IMAGES

X ∈ Def0(M)⇒ a(X ) = 1

X ∈ Def(M)⇒ a(X ) ≤ ℵ0

AUTOMORPHIC IMAGES

X ∈ Def0(M)⇒ a(X ) = 1

X ∈ Def(M)⇒ a(X ) ≤ ℵ0

a(X ) IN MODELS OF PA

Assumption from now on: M |= PA is countable and recursivelysaturated

Theorem (Schmerl)

a. ∀X ⊆ M, a(X ) ∈ {1,ℵ0, 2ℵ0}b. ∀X ∈ Def(M) \ Def0(M), a(X ) = ℵ0.

Theorem (Schmerl)

a. ∀X ⊆ M, a(X ) ∈ {1,ℵ0, 2ℵ0}

b. ∀X ∈ Def(M) \ Def0(M), a(X ) = ℵ0.

Theorem (Schmerl)

a. ∀X ⊆ M, a(X ) ∈ {1,ℵ0, 2ℵ0}b. ∀X ∈ Def(M) \ Def0(M), a(X ) = ℵ0.

MANY AUTOMORPHIC IMAGES

Theorem (RK, Kotlarski)

(M, X ) |= PA∗ and X 6∈ Def(M)⇒ a(X ) = 2ℵ0

TheoremLet I ⊆end M. Then a(I ) < 2ℵ0 iff ∃a ∈ M

I = sup[Scl(a) ∩ I ] or I = inf[Scl(a) ∩ (M \ I )]

From now on M |= PA is countable and recursively saturated

Theorem (RK, Kotlarski, independently Tzouvaras)

f ∈ Aut(M) and f 6= id ⇒ |{f g : g ∈ Aut(M)}| = a(f ) = 2ℵ0

From now on M |= PA is countable and recursively saturated

Theorem (RK, Kotlarski, independently Tzouvaras)

f ∈ Aut(M) and f 6= id ⇒ |{f g : g ∈ Aut(M)}| = a(f ) = 2ℵ0

MOVING GAPS LEMMA

Theorem (Kotlarski)

If f ∈ Aut(M) \ {id}, then for arbitrarily large a ∈ M,

f (a) /∈ gap(a).

MOVING GAPS LEMMA

Theorem (Kotlarski)

f (a) /∈ gap(a).

Corollary

If p(v) is an unbounded type realized in M, f ∈ Aut(M) \ {id},then there is a ∈ pM such that f (a) 6= a.

Corollary

If K ≺ M and K ∩ gap(a) 6= ∅ for all a ∈ M, then K = M.

MOVING GAPS LEMMA

Theorem (Kotlarski)

f (a) /∈ gap(a).

Corollary

MOVING GAPS LEMMA

Theorem (Kotlarski)

f (a) /∈ gap(a).

Corollary

THE AUTOMORPHISM GROUP

M |= PA countable and recursively saturated, G = Aut(M).

Theorem (Kaye)

If H / G is closed then H = G{I} for some invariant I ⊆end M.

Theorem (Kaye)

Theorem (Kaye, RK, Kotlarski)

Aut(Q, <) ⊆ G ⊆dense Aut(Q, <), but G 6∼= Aut(M) for anyℵ0-categorical countable M, in particular G 6∼= Aut(Q, <).

Theorem (Schmerl)

Let A be a countable linerly ordered structure. There is K ≺end Msuch that G{K}/G(K)

∼= Aut(A).

Theorem (Kaye)

Theorem (Schmerl)

∼= Aut(A).

Theorem (Kaye)

Theorem (Schmerl)

∼= Aut(A).

Theorem (Schmerl)

∼= Aut(A).

For every K ≺end M

|{g ∈ Aut(K ) : ∀f ∈ G{K} g 6= f � K}| = 2ℵ0

i.e. many automorphisms of K do not extend to M.

Theorem (Schmerl)

∼= Aut(A).

For every K ≺end M

|{g ∈ Aut(K ) : ∀f ∈ G{K} g 6= f � K}| = 2ℵ0

i.e. many automorphisms of K do not extend to M.

ARITHMETIC SATURATION

DefinitionM is arithmetically saturated iff SSy(M) is arithmetically closed.

DefinitionM is arithmetically saturated iff M is recursively saturated SSy(M)is arithmetically closed.

Theorem (Scott)

If X ⊆ P(N) and |X| = ℵ0, then

∃M |= PA, X = SSy(M)⇔ (ω, X) |= WKL0.

Proposition

If M is recursively saturated, then M is arithmetically saturated iff(ω, SSy(M)) |= ACA0.

Theorem (Scott)

∃M |= PA, X = SSy(M)⇔ (ω, X) |= WKL0.

Proposition

Theorem (Scott)

∃M |= PA, X = SSy(M)⇔ (ω, X) |= WKL0.

Proposition

Theorem (Lascar)

If M is arithmetically saturated, then H < G is open iff[G : H] < 2ℵ0 .

M is arithmetically saturated iff M has a maximal automorphism.

Theorem (RK, Schmerl)

M arithmetically saturated iff cf(G ) > ℵ0.

Theorem (Enayat)

M is arithmetically saturated iff for each K ≺ M, then there isf ∈ G such that fix(f ) ∼= K .

Theorem (Lascar)

Theorem (Enayat)

Theorem (Lascar)

Theorem (Enayat)

Theorem (Lascar)

Theorem (Enayat)

THE SPECTRUM OF AUTOMORPHISM GROUPS.CODING SSy(M) IN G .

If M0 and M1 are arithmetically saturated and M0 ≡ M1, then

Aut(M0) ∼= Aut(M1) iff SSy(M0) = SSy(M1).

If at least one of M0 and M1 is arithmetically saturated, and one isa model of TA, and the other is not; then

Aut(M0) 6∼= Aut(M1).

THE SPECTRUM OF AUTOMORPHISM GROUPS.CODING (fragments of) Th(M) IN G .

Theorem (Nurkhaidarov)

There are countable arithmetically saturated M0, M1, M2, M3 suchthat for all i , j < 4, SSy(Mi ) = SSy(Mj) and for i 6= j

Aut(Mi ) 6∼= Aut(Mj).

If M0 and M1 are arithmetically saturated andAut(M0) ∼= Aut(M1); then for every n < ω

(ω, SSy0(M0)2) |= RTn2 iff (ω, SSy0(M1)) |= RTn

2SSy0(M) = Rep(Th(M))

THE FOUR MODELS OF NURKHAIDAROV

1. (ω, SSy0(M0)) |= ¬RT22 (Hirst).

2. (ω, SSy0(M1)) |= RT22 ∧ ¬RT3

2 (Seetapun, Slaman).

3. M2 6|= TA, (ω, SSy0(M2))) |= RT32 (folklore).

4. M3 |= TA.

1. (ω, SSy0(M0)) |= ¬RT22 (Hirst).

2. (ω, SSy0(M1)) |= RT22 ∧ ¬RT3

4. M3 |= TA.

1. (ω, SSy0(M0)) |= ¬RT22 (Hirst).

2. (ω, SSy0(M1)) |= RT22 ∧ ¬RT3

4. M3 |= TA.

1. (ω, SSy0(M0)) |= ¬RT22 (Hirst).

2. (ω, SSy0(M1)) |= RT22 ∧ ¬RT3

4. M3 |= TA.

THE FIRST NEGLECTED CASE: NOT ARITHMETICALLYSATURATED MODELS

QuestionDo recursively saturated but not arithmetically saturated modelshave the small index property?

QuestionIf M is recursively saturated but not arithmetically saturated, isthere f ∈ G such that [f ] is comeager?

QuestionLet T be a completion of PA. Are there recursively saturated butnot arithmetically saturated models M0 and M1 of T such thatAut(M0) 6∼= Aut(M1)?

Psychologists assure us that tall people command moreattention and respect than short ones. In asantropomorphic field as logic, it would follow that tallerconcepts excite more imagination than shorter ones.Thus, more is written about tall end extensions thanstubby cofinal ones, and asked for a preference betweentall and short models, most logicians would take the tallchoice. Such a high-minded strategy might work well inthe short run; but in the long run we must pay everythingits due.

C.SmorynskiCofinal Extensions of Nonstandard Models of Arithmetic, 1981

THE SECOND NEGLECTED CASE: SHORT MODELS

Notation: M(a) = sup(Scl(a)) =⋂{K ≺end M : a ∈ K}

G (a) = Aut(M(a))

G � M(a) = {f � M(a) : f ∈ G{M(a)}}

Theorem (Shochat)

G � M(a) is not normal in G (a), it is dense in G (a), and

[G (a) : G � M(a)] = 2ℵ0 .

QuestionIs G � M(a) maximal in G (a)?

G (a) = Aut(M(a))

G � M(a) = {f � M(a) : f ∈ G{M(a)}}

Theorem (Shochat)

[G (a) : G � M(a)] = 2ℵ0 .

G (a) = Aut(M(a))

G � M(a) = {f � M(a) : f ∈ G{M(a)}}

Theorem (Shochat)

[G (a) : G � M(a)] = 2ℵ0 .

DIVERSITY AMONG SHORT MODELS

Theorem (Kotlarski)

There are ai : i < ω such that for i 6= j , M(ai ) 6∼= M(aj).

Theorem (Shochat)

There are a0, a1 such that G (a0) 6∼= G (a1) as topological groups.

DIVERSITY AMONG SHORT MODELS

Theorem (Kotlarski)

There are ai : i < ω such that for i 6= j , M(ai ) 6∼= M(aj).

Theorem (Shochat)

TOPOLOGY ON G (a)

Theorem (Shochat)

I⋂{H / G (a0) : H is closed } = G{M[a0]}

I⋂{H / G (a1) : H is closed } = {id}

DIGRESSION: STRONG NON-RECONSTRUCTION RESULT

Theorem (Schmerl)

Let A be an infinite linearly ordered structure. There is M |= PA,|A| = |M| (and M is not recursively saturated) such thatAut(A) ∼= Aut(M).

THE THIRD NEGLECTED CASE: EXTENDINGAUTOMORPHISMS TO COFINAL EXTENSIONS

If K ≺end M, f ∈ Aut(K , Cod(M/K )), then3 f = g � K for someg ∈ Aut(M).

3With minor restrictions.

If K ≺cof M, f ∈ Aut(K , Cod(M/K )), and the extension K ≺cof Mhas the the description property, then f = g � K for someg ∈ Aut(M).

THE DESCRIPTION PROPERTY

DefinitionThe extension K ≺cof M has the description property if for everya ∈ M \ K there is a coded in M nested sequence 〈Ai : i < ω〉 ofK -finite sets such that

1. M |= a ∈ Ai for all i < ω;

2. For each K -finite B such that a ∈ B, there is an i < ω suchthat Ai ⊆ B.

For each M, there are K0, K1 such that K0≺cof M ≺cof K1 andboth extensions have the description property.

QuestionFor a given M is there an N such that M ≺cof N has thedescription property and SSy(M) = SSy(N)?

QuestionFor a given M, is there an N such that for each boundedA ∈ Cod(N/M) there is a b ∈ M such that b codes A?

ISOLATED GAPS

DefinitionIf M ≺cof N and b ∈ N \M, then gapN(b) is non-isolated if thereare d < gap(b) < e ∈ N such that [d , e] ∩M = ∅

ISOLATED GAPS

Every cofinal extension has non-isolated gaps.

No extension with the description property has isolated gaps.

QuestionAre there cofinal extensions with isolated gaps?

ISOLATED GAPS

Automorphisms of models of PA: The neglected cases. · 2012-05-29 · Automorphisms of models of...

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