AUTOMORPHISM GROUPS OF SATURATED MODELS OF PEANO ARITHMETIC

2014 ◽  
Vol 79 (2) ◽  
pp. 561-584
Author(s):  
ERMEK S. NURKHAIDAROV ◽  
JAMES H. SCHMERL
Keyword(s):  
Automorphism Groups ◽  
Peano Arithmetic ◽  
Saturated Model ◽  
Saturated Models ◽  
Image Position

AbstractLet κ be the cardinality of some saturated model of Peano Arithmetic. There is a set of ${2^{{\aleph _0}}}$ saturated models of PA, each having cardinality κ, such that whenever M and N are two distinct models from this set, then Aut(${\cal M}$) ≇ Aut ($${\cal N}$$).

Automorphism groups of arithmetically saturated models

2006 ◽  
Vol 71 (1) ◽  
pp. 203-216 ◽  
Author(s):  
Ermek S. Nurkhaidarov
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Open Subgroup ◽  
Peano Arithmetic ◽  
Standard System ◽  
Permutation Groups ◽  
Saturated Model ◽  
Saturated Models ◽  
Homogeneous Set ◽  
Image Position

In this paper we study the automorphism groups of countable arithmetically saturated models of Peano Arithmetic. The automorphism groups of such structures form a rich class of permutation groups. When studying the automorphism group of a model, one is interested to what extent a model is recoverable from its automorphism group. Kossak-Schmerl [12] show that if M is a countable, arithmetically saturated model of Peano Arithmetic, then Aut(M) codes SSy(M). Using that result they prove:Let M1. M2 be countable arithmetically saturated models of Peano Arithmetic such that Aut(M1) ≅ Aut(M2). Then SSy(M1) = SSy(M2).We show that if M is a countable arithmetically saturated of Peano Arithmetic, then Aut(M) can recognize if some maximal open subgroup is a stabilizer of a nonstandard element, which is smaller than any nonstandard definable element. That fact is used to show the main theorem:Let M1, M2be countable arithmetically saturated models of Peano Arithmetic such that Aut(M1) ≅ Aut(M2). Then for every n < ωHere RT2n is Infinite Ramsey's Theorem stating that every 2-coloring of [ω]n has an infinite homogeneous set. Theorem 0.2 shows that for models of a false arithmetic the converse of Kossak-Schmerl Theorem 0.1 is not true. Using the results of Reverse Mathematics we obtain the following corollary:There exist four countable arithmetically saturated models of Peano Arithmetic such that they have the same standard system but their automorphism groups are pairwise non-isomorphic.

scholarly journals AUTOMORPHISM GROUPS OF COUNTABLE ARITHMETICALLY SATURATED MODELS OF PEANO ARITHMETIC

2015 ◽  
Vol 80 (4) ◽  
pp. 1411-1434
Author(s):  
JAMES H. SCHMERL
Keyword(s):  
Automorphism Groups ◽  
Peano Arithmetic ◽  
Saturated Models ◽  
Image Position

AbstractIf ${\cal M},{\cal N}$ are countable, arithmetically saturated models of Peano Arithmetic and ${\rm{Aut}}\left( {\cal M} \right) \cong {\rm{Aut}}\left( {\cal N} \right)$, then the Turing-jumps of ${\rm{Th}}\left( {\cal M} \right)$ and ${\rm{Th}}\left( {\cal N} \right)$ are recursively equivalent.

Some highly saturated models of Peano arithmetic

2002 ◽  
Vol 67 (4) ◽  
pp. 1265-1273
Author(s):  
James H. Schmerl
Keyword(s):  
Regular Cardinal ◽  
Peano Arithmetic ◽  
Saturated Model ◽  
Elementary Extension ◽  
Saturated Models ◽  
The One ◽  
Recursively Saturated

Some highly saturated models of Peano Arithmetic are constructed in this paper, which consists of two independent sections. In § 1 we answer a question raised in [10] by constructing some highly saturated, rather classless models of PA. A question raised in [7], [3], ]4] is answered in §2, where highly saturated, nonstandard universes having no bad cuts are constructed.Highly saturated, rather classless models of Peano Arithmetic were constructed in [10]. The main result proved there is the following theorem. If λ is a regular cardinal and is a λ-saturated model of PA such that ∣M∣ > λ, then has an elementary extension of the same cardinality which is also λ-saturated and which, in addition, is rather classless. The construction in [10] produced a model for which cf() = λ+. We asked in Question 5.1 of [10] what other cofinalities could such a model have. This question is answered here in Theorem 1.1 of §1 by showing that any cofinality not immediately excluded is possible. Its proof does not depend on the theorem from [10]; in fact, the proof presented here gives a proof of that theorem which is much simpler and shorter than the one in [10].Recursively saturated, rather classless κ-like models of PA were constructed in [9]. In the case of singular κ such models were constructed whenever cf(κ) > ℵ0; no additional set-theoretic hypothesis was needed.

Large resplendent models generated by indiscernibles

1989 ◽  
Vol 54 (4) ◽  
pp. 1382-1388 ◽  
Author(s):  
James H. Schmerl
Keyword(s):  
Peano Arithmetic ◽  
Saturated Model ◽  
Infinite Cardinal ◽  
Common Generalization ◽  
Saturated Models ◽  
Uncountable Cardinal ◽  
Finite Language ◽  
Recursively Saturated ◽  
Recursively Saturated Model

The motivation for the results presented here comes from the following two known theorems which concern countable, recursively saturated models of Peano arithmetic.(1) if is a countable, recursively saturated model of PA, then for each infinite cardinal κ there is a resplendent which has cardinality κ. (See Theorem 10 of [1].)(2) if is a countable, recursively saturated model of PA, then is generated by a set of indiscernibles. (See [4].)It will be shown here that (1) and (2) can be amalgamated into a common generalization.(3) if is a countable, recursively saturated model of PA, then for each infinite cardinal κ there is a resplendent which has cardinality κ and which is generated by a set of indiscernibles.By way of contrast we will also get recursively saturated models of PA which fail to be resplendent and yet are generated by indiscernibles.(4) if is a countable, recursively saturated model of PA, then for each uncountable cardinal κ there is a κ-like recursively saturated generated by a set of indiscernibles.None of (1), (2) or (3) is stated in its most general form. We will make some comments concerning their generalizations. From now on let us fix a finite language L; all structures considered are infinite L-structures unless otherwise indicated.

Remarks on weak notions of saturation in models of Peano arithmetic

1987 ◽  
Vol 52 (1) ◽  
pp. 129-148 ◽  
Author(s):  
Matt Kaufmann ◽  
James H. Schmerl
Keyword(s):  
Peano Arithmetic ◽  
Saturated Model ◽  
Simple Extension ◽  
Minimal Models ◽  
Recursively Saturated ◽  
Countable Models

This paper is a sequel to our earlier paper [2] entitled Saturation and simple extensions of models of Peano arithmetic. Among other things, we will answer some of the questions that were left open there. In §1 we consider the question of whether there are lofty models of PA which have no recursively saturated, simple extensions. We are still unable to answer this question; but we do show in that section that these models are precisely the lofty models which are not recursively saturated and which are κ-like for some regular κ. In §2 we use diagonal methods to produce minimal models of PA in which the standard cut is recursively definable, and other minimal models in which the standard cut is not recursively definable. In §3 we answer two questions from [2] by exhibiting countable models of PA which, in the terminology of this paper, are uniformly ω-lofty but not continuously ω-lofty and others which are continuously ω-lofty but not recursively saturated. We also construct a model (assuming ◇) which is not recursively saturated but every proper, simple cofinal extension of which is ℵ1-saturated. Finally, in §4 we answer another question from [2] by proving that for regular κ ≥ ℵ1; every κ-saturated model of PA has a κ-saturated proper, simple extension which is not κ+-saturated.Our notation and terminology are quite standard. Anything unfamiliar to the reader and not adequately denned here is probably defined in §1 of [2]. All models considered are models of Peano arithmetic.

PROVABILITY LOGICS RELATIVE TO A FIXED EXTENSION OF PEANO ARITHMETIC

2018 ◽  
Vol 83 (3) ◽  
pp. 1229-1246
Author(s):  
TAISHI KURAHASHI
Keyword(s):  
Peano Arithmetic ◽  
Provability Logic ◽  
Consistent Extension ◽  
Image Position ◽  
Computably Enumerable

AbstractLet T and U be any consistent theories of arithmetic. If T is computably enumerable, then the provability predicate $P{r_\tau }\left( x \right)$ of T is naturally obtained from each ${{\rm{\Sigma }}_1}$ definition $\tau \left( v \right)$ of T. The provability logic $P{L_\tau }\left( U \right)$ of τ relative to U is the set of all modal formulas which are provable in U under all arithmetical interpretations where □ is interpreted by $P{r_\tau }\left( x \right)$. It was proved by Beklemishev based on the previous studies by Artemov, Visser, and Japaridze that every $P{L_\tau }\left( U \right)$ coincides with one of the logics $G{L_\alpha }$, ${D_\beta }$, ${S_\beta }$, and $GL_\beta ^ -$, where α and β are subsets of ω and β is cofinite.We prove that if U is a computably enumerable consistent extension of Peano Arithmetic and L is one of $G{L_\alpha }$, ${D_\beta }$, ${S_\beta }$, and $GL_\beta ^ -$, where α is computably enumerable and β is cofinite, then there exists a ${{\rm{\Sigma }}_1}$ definition $\tau \left( v \right)$ of some extension of $I{{\rm{\Sigma }}_1}$ such that $P{L_\tau }\left( U \right)$ is exactly L.

Rigid homogeneous chains

1981 ◽  
Vol 89 (1) ◽  
pp. 07-17 ◽  
Author(s):  
A. M. W. Glass ◽  
Yuri Gurevich ◽  
W. Charles Holland ◽  
Saharon Shelah
Keyword(s):  
Automorphism Group ◽  
General Problem ◽  
Automorphism Groups ◽  
Ordered Sets ◽  
First Order ◽  
Image Position

Classifying (unordered) sets by the elementary (first order) properties of their automorphism groups was undertaken in (7), (9) and (11). For example, if Ω is a set whose automorphism group, S(Ω), satisfiesthen Ω has cardinality at most ℵ0 and conversely (see (7)). We are interested in classifying homogeneous totally ordered sets (homogeneous chains, for short) by the elementary properties of their automorphism groups. (Note that we use ‘homogeneous’ here to mean that the automorphism group is transitive.) This study was begun in (4) and (5). For any set Ω, S(Ω) is primitive (i.e. has no congruences). However, the automorphism group of a homogeneous chain need not be o-primitive (i.e. it may have convex congruences). Fortunately, ‘o-primitive’ is a property that can be captured by a first order sentence for automorphisms of homogeneous chains. Hence our general problem falls naturally into two parts. The first is to classify (first order) the homogeneous chains whose automorphism groups are o-primitive; the second is to determine how the o-primitive components are related for arbitrary homogeneous chains whose automorphism groups are elementarily equivalent.

Saturation of homogeneous resplendent models

1986 ◽  
Vol 51 (1) ◽  
pp. 222-224 ◽  
Author(s):  
Julia F. Knight
Keyword(s):  
Continuum Hypothesis ◽  
Homogeneous Model ◽  
Peano Arithmetic ◽  
Saturated Model ◽  
Relation Symbol ◽  
First Order ◽  
Recursively Saturated ◽  
Recursively Saturated Model ◽  
Saturated Structure ◽  
The Continuum

The complete diagram of a structure , denoted by Dc(), is the set of all sentences true in the structure (, a)a∈. A structure is said to be resplendent if for every sentence θ involving a new relation symbol R in addition to symbols occurring in Dc(), if θ is consistent with Dc(), then there is a relation P on such that (see[1]).Baldwin asked whether a homogeneous recursively saturated structure is necessarily resplendent. Here it is shown that this need not be the case. It is shown that if is an uncountable homogeneous resplendent model of an unstable theory, then must be saturated. The proof is related to the proof in [5] that an uncountable homogeneous recursively saturated model of first order Peano arithmetic must be saturated. The example for Baldwin's question is an uncountable homogeneous model for a particular unstable theory, such that is recursively saturated and omits some type. (The continuum hypothesis is needed to show the existence of such a model in power ℵ1.)The proof of the main result requires two lemmas.

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