Modèles saturés et modèles engendrés par des indiscernables

2001 ◽  
Vol 66 (1) ◽  
pp. 325-348
Author(s):  
Benoît Mariou
Keyword(s):  
Algebraic Closure ◽  
Saturated Model ◽  
Stable Theory ◽  
Independence Property ◽  
Complete Answer ◽  
Unstable Case ◽  
Complete Theories ◽  
Recursively Saturated ◽  
Saturated Structure

AbstractIn the early eighties, answering a question of A. Macintyre, J. H. Schmerl ([13]) proved that every countable recursively saturated structure, equipped with a function β encoding the finite functions, is the β-closure of an infinite indiscernible sequence. This result implies that every countably saturated structure, in a countable but not necessarily recursive language, is an Ehrenfeucht-Mostowski model, by which we mean that the structure expands, in a countable language, to the Skolem hull of an infinite indiscernible sequence (in the new language).More recently, D. Lascar ([5]) showed that the saturated model of cardinality ℵ1 of an ω-stable theory is also an Ehrenfeucht-Mostowski model.These results naturally raise the following problem: which (countable) complete theories have an uncountably saturated Ehrenfeucht-Mostowski model. We study a generalization of this question. Namely, we call ACI-model a structure which can be expanded, in a countable language L′, to the algebraic closure (in L′) of an infinite indiscernible sequence (in L′). And we try to characterize the λ-saturated structures which are ACI-models.The main results are the following. First it is enough to restrict ourselves to ℵ1-saturated structures: if T has an ℵ1-saturated ACI-model then, for every infinite λ, T has a λ-saturated ACI-model. We obtain a complete answer in the case of stable theories: if T is stable then the three following properties are equivalent: (a) T is ω-stable, (b) T has an ℵ1-saturated ACI-model, (c) every saturated model of T is an Ehrenfeucht-Mostowski model. The unstable case is more complicated, however we show that if T has an ℵ1-saturated ACI-model then T doesn't have the independence property.

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.

Automorphisms moving all non-algebraic points and an application to NF

1998 ◽  
Vol 63 (3) ◽  
pp. 815-830 ◽  
Author(s):  
Friederike Körner
Keyword(s):  
Set Theory ◽  
Sufficient Conditions ◽  
Saturated Model ◽  
Natural Numbers ◽  
Algebraic Point ◽  
Complete Theories ◽  
Necessary And Sufficient ◽  
Countable Theory ◽  
Recursively Saturated ◽  
Recursively Saturated Model

AbstractSection 1 is devoted to the study of countable recursively saturated models with an automorphism moving every non-algebraic point. We show that every countable theory has such a model and exhibit necessary and sufficient conditions for the existence of automorphisms moving all non-algebraic points. Furthermore we show that there are many complete theories with the property that every countable recursively saturated model has such an automorphism.In Section 2 we apply our main theorem from Section 1 to models of Quine's set theory New Foundations (NF) to answer an old consistency question. If NF is consistent, then it has a model in which the standard natural numbers are a definable subclass ℕ of the model's set of internal natural numbers Nn. In addition, in this model the class of wellfounded sets is exactly .

Models of arithmetic and closed ideals

1982 ◽  
Vol 47 (4) ◽  
pp. 833-840 ◽  
Author(s):  
Julia Knight ◽  
Mark Nadel
Keyword(s):  
Closed Ideal ◽  
Turing Degrees ◽  
Saturated Model ◽  
Recursive Saturation ◽  
New Information ◽  
Recursively Saturated ◽  
Recursively Saturated Model ◽  
Saturated Structure ◽  
Models Of Arithmetic ◽  
Natural Way

A set J of Turing degrees is called an ideal if (1) J ≠ ∅, (2) for any pair of degrees ã, , if ã, ϵ J, then ã ⋃ ϵJ, and (3) for any ⋃ ϵ J and any , if < ⋃, then ϵ J. A set J of degrees is said to be closed if for any theory T with a set of axioms of degree in J, T has a completion of degree in J.Closed ideals of degrees arise naturally in the following way. If is a recursively saturated structure, let I() = { for some ā ϵ }. Let D() = {: is recursive in d-saturated}. (Recursive in d-saturation is defined like recursive saturation except that the sets of formulas considered are recursive in d.) These two sets of degrees were investigated in [2]. It was shown that if is a recursively saturated model of P, Pr = Th(ω, +), or Pr′ = Th(Z, +, 1), then I() = D(), and this set is a closed ideal. Any closed ideal J can be represented as I() = D() for some recursively saturated model of Pr′. For sets J of power at most ℵ1, Pr′ can be replaced by P.Assuming CH, all closed ideals have power at most ℵ1, but if CH fails, there are closed ideals of power greater than ℵ1, and it is not known whether these can be represented as I() = D() for a recursively saturated model of P.In the present paper, it will first be shown that information about representation of closed ideals provides new information about an old problem of MacDowell and Specker [6] and extends an old result of Scott [8] in a natural way. It will also be shown that the representation results from [2] answer a problem of Friedman [1]. This part of the paper is aimed at convincing the reader that representation problems are worth investigating.

Expansions of models and turing degrees

1982 ◽  
Vol 47 (3) ◽  
pp. 587-604 ◽  
Author(s):  
Julia Knight ◽  
Mark Nadel
Keyword(s):  
Model Theory ◽  
Good Deal ◽  
Turing Degrees ◽  
Saturated Model ◽  
First Order ◽  
Recursive Saturation ◽  
Axiomatizable Theory ◽  
Recursively Saturated ◽  
Recursively Saturated Model ◽  
Saturated Structure

If is a countable recursively saturated structure and T is a recursively axiomatizable theory that is consistent with Th(), then it is well known that can be expanded to a recursively saturated model of T [7, p. 186]. This is what has made recursively saturated models useful in model theory. Recursive saturation is the weakest notion of saturation for which this expandability result holds. In fact, if is a countable model of Pr = Th(ω, +), then can be expanded to a model of first order Peano arithmetic P just in case is recursively saturated (see [3]).In this paper we investigate two natural sets of Turing degrees that tell a good deal about the expandability of a given structure. If is a recursively saturated structure, I() consists of the degrees of sets that are recursive in complete types realized in . The second set of degrees, D(), consists of the degrees of sets S such that is recursive in S-saturated. In general, I() ⊆ D(). Moreover, I() is obviously an “ideal” of degrees. For countable structures , D() is “closed” in the following sense: For any class C ⊆ 2ω, if C is co-r.e. in S for some set S such that , then there is some σ ∈ C such that . For uncountable structures , we do not know whether D() must be closed.

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.

scholarly journals Condensable models of set theory

2021 ◽  
Author(s):  
Ali Enayat
Keyword(s):  
Set Theory ◽  
Countable Model ◽  
Saturated Model ◽  
Infinitary Logic ◽  
First Order ◽  
Recursively Saturated ◽  
Recursively Saturated Model ◽  
Models Of Set Theory

AbstractA model $${\mathcal {M}}$$ M of ZF is said to be condensable if $$ {\mathcal {M}}\cong {\mathcal {M}}(\alpha )\prec _{\mathbb {L}_{{\mathcal {M}}}} {\mathcal {M}}$$ M ≅ M ( α ) ≺ L M M for some “ordinal” $$\alpha \in \mathrm {Ord}^{{\mathcal {M}}}$$ α ∈ Ord M , where $$\mathcal {M}(\alpha ):=(\mathrm {V}(\alpha ),\in )^{{\mathcal {M}}}$$ M ( α ) : = ( V ( α ) , ∈ ) M and $$\mathbb {L}_{{\mathcal {M}}}$$ L M is the set of formulae of the infinitary logic $$\mathbb {L}_{\infty ,\omega }$$ L ∞ , ω that appear in the well-founded part of $${\mathcal {M}}$$ M . The work of Barwise and Schlipf in the 1970s revealed the fact that every countable recursively saturated model of ZF is cofinally condensable (i.e., $${\mathcal {M}}\cong {\mathcal {M}}(\alpha ) \prec _{\mathbb {L}_{{\mathcal {M}}}}{\mathcal {M}}$$ M ≅ M ( α ) ≺ L M M for an unbounded collection of $$\alpha \in \mathrm {Ord}^{{\mathcal {M}}}$$ α ∈ Ord M ). Moreover, it can be readily shown that any $$\omega $$ ω -nonstandard condensable model of $$\mathrm {ZF}$$ ZF is recursively saturated. These considerations provide the context for the following result that answers a question posed to the author by Paul Kindvall Gorbow.Theorem A.Assuming a modest set-theoretic hypothesis, there is a countable model $${\mathcal {M}}$$ M of ZFC that is bothdefinably well-founded (i.e., every first order definable element of $${\mathcal {M}}$$ M is in the well-founded part of $$\mathcal {M)}$$ M ) andcofinally condensable. We also provide various equivalents of the notion of condensability, including the result below.Theorem B.The following are equivalent for a countable model$${\mathcal {M}}$$ M of $$\mathrm {ZF}$$ ZF : (a) $${\mathcal {M}}$$ M is condensable. (b) $${\mathcal {M}}$$ M is cofinally condensable. (c) $${\mathcal {M}}$$ M is nonstandard and $$\mathcal {M}(\alpha )\prec _{\mathbb {L}_{{\mathcal {M}}}}{\mathcal {M}}$$ M ( α ) ≺ L M M for an unbounded collection of $$ \alpha \in \mathrm {Ord}^{{\mathcal {M}}}$$ α ∈ Ord M .

scholarly journals SOME PROPERTIES OF ANALYTIC DIFFERENCE VALUED FIELDS

2015 ◽  
Vol 16 (3) ◽  
pp. 447-499 ◽  
Author(s):  
Silvain Rideau
Keyword(s):  
Algebraic Closure ◽  
Quantifier Elimination ◽  
Order Theory ◽  
Analytic Structure ◽  
Independence Property ◽  
Valued Fields ◽  
First Order ◽  
Witt Vectors ◽  
First Order Theory

We prove field quantifier elimination for valued fields endowed with both an analytic structure that is $\unicode[STIX]{x1D70E}$-Henselian and an automorphism that is $\unicode[STIX]{x1D70E}$-Henselian. From this result we can deduce various Ax–Kochen–Eršov type results with respect to completeness and the independence property. The main example we are interested in is the field of Witt vectors on the algebraic closure of $\mathbb{F}_{p}$ endowed with its natural analytic structure and the lifting of the Frobenius. It turns out we can give a (reasonable) axiomatization of its first-order theory and that this theory does not have the independence property.

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.

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