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.

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.

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 Interpreting the compositional truth predicate in models of arithmetic

2021 ◽  
Author(s):  
Cezary Cieśliński
Keyword(s):  
Proof Theory ◽  
Formal Proof ◽  
Truth Predicate ◽  
Saturated Model ◽  
First Order ◽  
Order Arithmetic ◽  
Recursively Saturated ◽  
Recursively Saturated Model ◽  
Models Of Arithmetic

AbstractWe present a construction of a truth class (an interpretation of a compositional truth predicate) in an arbitrary countable recursively saturated model of first-order arithmetic. The construction is fully classical in that it employs nothing more than the classical techniques of formal proof theory.

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.

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.

scholarly journals A note on standard systems and ultrafilters

2008 ◽  
Vol 73 (3) ◽  
pp. 824-830 ◽  
Author(s):  
Fredrik Engström
Keyword(s):  
Standard System ◽  
Saturated Model ◽  
Consistent Theory ◽  
First Order ◽  
Order Arithmetic ◽  
Recursively Saturated ◽  
Scott Set ◽  
Special Case ◽  
Recursively Saturated Model

AbstractLet (M, )⊨ ACA0 be such that , the collection of all unbounded sets in , admits a definable complete ultrafilter and let T be a theory extending first order arithmetic coded in such that M thinks T is consistent. We prove that there is an end-extension N ⊨ T of M such that the subsets of M coded in N are precisely those in . As a special case we get that any Scott set with a definable ultrafilter coding a consistent theory T extending first order arithmetic is the standard system of a recursively saturated model of T.

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.

Gödel, Kurt (1906–78)

2018 ◽  
Author(s):  
John W. Dawson
Keyword(s):  
Twentieth Century ◽  
Set Theory ◽  
Continuum Hypothesis ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Mathematical Platonism ◽  
The Axiom Of Choice ◽  
Fundamental Theorems ◽  
The Continuum

The greatest logician of the twentieth century, Gödel is renowned for his advocacy of mathematical Platonism and for three fundamental theorems in logic: the completeness of first-order logic; the incompleteness of formalized arithmetic; and the consistency of the axiom of choice and the continuum hypothesis with the axioms of Zermelo–Fraenkel set theory.

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