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.

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.

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.

MODELS OF PT– WITH INTERNAL INDUCTION FOR TOTAL FORMULAE

2016 ◽  
Vol 10 (1) ◽  
pp. 187-202 ◽  
Author(s):  
CEZARY CIEŚLIŃSKI ◽  
MATEUSZ ŁEŁYK ◽  
BARTOSZ WCISŁO
Keyword(s):  
Peano Arithmetic ◽  
Truth Predicate ◽  
Saturated Model ◽  
Internal Induction ◽  
Philosophical Project ◽  
Positive Truth ◽  
Recursively Saturated ◽  
Recursively Saturated Model

AbstractWe show that a typed compositional theory of positive truth with internal induction for total formulae (denoted by PTtot) is not semantically conservative over Peano arithmetic. In addition, we observe that the class of models of PA expandable to models of PTtot contains every recursively saturated model of arithmetic. Our results point to a gap in the philosophical project of describing the use of the truth predicate in model-theoretic contexts.

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.

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.

Two theorems on degrees of models of true arithmetic

1984 ◽  
Vol 49 (2) ◽  
pp. 425-436 ◽  
Author(s):  
Julia Knight ◽  
Alistair H. Lachlan ◽  
Robert I. Soare
Keyword(s):  
Standard Model ◽  
Binary Operation ◽  
Peano Arithmetic ◽  
Turing Degree ◽  
First Order ◽  
Nonstandard Model ◽  
Basis Theorem ◽  
The Standard Model ◽  
The Universe ◽  
Countable Models

Let PA be the theory of first order Peano arithmetic, in the language L with binary operation symbols + and ·. Let N be the theory of the standard model of PA. We consider countable models M of PA such that the universe ∣M∣ is ω. The degree of such a model M, denoted by deg(M), is the (Turing) degree of the atomic diagram of M. The results of this paper concern the degrees of models of N, but here in the Introduction, we shall give a brief survey of results about degrees of models of PA.Let D0 denote the set of degrees d such that there is a nonstandard model of M of PA with deg(M) = d. Here are some of the more easily stated results about D0.(1) There is no recursive nonstandard model of PA; i.e., 0 ∈ D0.This is a result of Tennenbaum [T].(2) There existsd ∈ D0such thatd ≤ 0′.This follows from the standard Henkin argument.(3) There existsd ∈ D0such thatd < 0′.Shoenfield [Sh1] proved this, using the Kreisel-Shoenfield basis theorem.(4) There existsd ∈ D0such thatd′ = 0′.Jockusch and Soare [JS] improved the Kreisel-Shoenfield basis theorem and obtained (4).(5) D0 = Dc = De, where Dc denotes the set of degrees of completions of PA and De the set of degrees d such that d separates a pair of effectively inseparable r.e. sets.Solovay noted (5) in a letter to Soare in which in answer to a question posed in [JS] he showed that Dc is upward closed.

scholarly journals On maximal subgroups of the automorphism group of a countable recursively saturated model of PA

1993 ◽  
Vol 65 (2) ◽  
pp. 125-148 ◽  
Author(s):  
Roman Kossak ◽  
Henryk Kotlarski ◽  
James H. Schmerl
Keyword(s):  
Automorphism Group ◽  
Maximal Subgroups ◽  
Saturated Model ◽  
Recursively Saturated ◽  
Recursively Saturated Model

The Recursively Saturated Part of Models of Peano Arithmetic

1986 ◽  
Vol 32 (19-24) ◽  
pp. 365-370
Author(s):  
Henryk Kotlarski
Keyword(s):  
Peano Arithmetic ◽  
Recursively Saturated

Nonstandard characterizations of recursive saturation and resplendency

1987 ◽  
Vol 52 (3) ◽  
pp. 842-863 ◽  
Author(s):  
Stuart T. Smith
Keyword(s):  
Peano Arithmetic ◽  
Recursive Saturation ◽  
Recursively Saturated ◽  
Satisfaction Class

AbstractWe prove results about nonstandard formulas in models of Peano arithmetic which complement those of Kotlarski, Krajewski, and Lachlan in [KKL] and [L]. This enables us to characterize both recursive saturation and resplendency in terms of statements about nonstandard sentences. Specifically, a model of PA is recursively saturated iff is nonstandard and -logic is consistent. is resplendent iff is nonstandard, -logic is consistent, and every sentence φ which is consistent in -logic is contained in a full satisfaction class for . Thus, for models of PA, recursive saturation can be expressed by a (standard) -sentence and resplendency by a -sentence.

Bad models in nice neighborhoods

1986 ◽  
Vol 51 (4) ◽  
pp. 1043-1055 ◽  
Author(s):  
Terry Millar
Keyword(s):  
Linear Order ◽  
Homogeneous Model ◽  
Order Type ◽  
Saturated Model ◽  
Countable Number ◽  
Decidable Theory ◽  
Type Spectrum ◽  
Image Position ◽  
Countable Models

This paper contains an example of a decidable theory which has1) only a countable number of countable models (up to isomorphism);2) a decidable saturated model; and3) a countable homogeneous model that is not decidable.By the results in [1] and [2], this can happen if and only if the set of types realized by the homogeneous model (the type spectrum of the model) is not .If Γ and Σ are types of a theory T, define Γ ◁ Σ to mean that any model of T realizing Γ must realize Σ. In [3] a decidable theory is constructed that has only countably many countable models, only recursive types, but whose countable saturated model is not decidable. This is easy to do if the restriction on the number of countable models is lifted; the difficulty arises because the set of types must be recursively complex, and yet sufficiently related to control the number of countable models. In [3] the desired theory T is such thatis a linear order with order type ω*. Also, the set of complete types of T is not . The last feature ensures that the countable saturated model is not decidable; the first feature allows the number of countable models to be controlled.

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