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 .

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 .

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.

scholarly journals Strong Tolerance and Strong Universality of Interval Eigenvectors in a Max-Łukasiewicz Algebra

Mathematics ◽  
2020 ◽  
Vol 8 (9) ◽  
pp. 1504
Author(s):  
Martin Gavalec ◽  
Zuzana Němcová ◽  
Ján Plavka
Keyword(s):  
Set Theory ◽  
Binary Operation ◽  
Sufficient Conditions ◽  
Discrete Events ◽  
Linear Combinations ◽  
Main Method ◽  
Max Algebra ◽  
Discrete Events Systems ◽  
Necessary And Sufficient ◽  
Theoretical Results

The Łukasiewicz conjunction (sometimes also considered to be a logic of absolute comparison), which is used in multivalued logic and in fuzzy set theory, is one of the most important t-norms. In combination with the binary operation ‘maximum’, the Łukasiewicz t-norm forms the basis for the so-called max-Łuk algebra, with applications to the investigation of systems working in discrete steps (discrete events systems; DES, in short). Similar algebras describing the work of DES’s are based on other pairs of operations, such as max-min algebra, max-plus algebra, or max-T algebra (with a given t-norm, T). The investigation of the steady states in a DES leads to the study of the eigenvectors of the transition matrix in the corresponding max-algebra. In real systems, the input values are usually taken to be in some interval. Various types of interval eigenvectors of interval matrices in max-min and max-plus algebras have been described. This paper is oriented to the investigation of strong, strongly tolerable, and strongly universal interval eigenvectors in a max-Łuk algebra. The main method used in this paper is based on max-Ł linear combinations of matrices and vectors. Necessary and sufficient conditions for the recognition of strong, strongly tolerable, and strongly universal eigenvectors have been found. The theoretical results are illustrated by numerical examples.

Mathematical Prelude: A Selective Introduction to Logic and Set Theory for Social Scientists

2012 ◽  
Author(s):  
Gary Goertz ◽  
James Mahoney
Keyword(s):  
Qualitative Research ◽  
Set Theory ◽  
Quantitative Research ◽  
Sufficient Conditions ◽  
Social Scientists ◽  
Qualitative And Quantitative ◽  
Statistics And Probability ◽  
Aggregation Techniques ◽  
The Social ◽  
Necessary And Sufficient

This chapter considers some key ideas from logic and set theory as they relate to qualitative research in the social sciences, including ideas concerning necessary and sufficient conditions. It also highlights a major contrast between qualitative and quantitative research: whereas quantitative research draws on mathematical tools associated with statistics and probability theory, qualitative research is often based on set theory and logic. The chapter first compares the natural language of logic in the qualitative culture with the language of probability and statistics in the quantitative culture. It then considers the necessary conditions and sufficient conditions as basis for qualitative methods, focusing on set theory and Venn diagrams, two-by-two tables, and truth tables. It also discusses the use of qualitative and quantitative aggregation techniques and concludes by explaining the criteria for assessing the “fit” of the model or the “importance” of a given causal factor.

scholarly journals The Complexity of Classification Problems for Models of Arithmetic

2010 ◽  
Vol 16 (3) ◽  
pp. 345-358 ◽  
Author(s):  
Samuel Coskey ◽  
Roman Kossak
Keyword(s):  
Classification Problem ◽  
The Other ◽  
Saturated Model ◽  
Classification Problems ◽  
Finitely Generated ◽  
Saturated Models ◽  
Other Hand ◽  
Recursively Saturated ◽  
Recursively Saturated Model ◽  
Models Of Arithmetic

AbstractWe observe that the classification problem for countable models of arithmetic is Borel complete. On the other hand, the classification problems for finitely generated models of arithmetic and for recursively saturated models of arithmetic are Borel; we investigate the precise complexity of each of these. Finally, we show that the classification problem for pairs of recursively saturated models and for automorphisms of a fixed recursively saturated model are Borel complete.

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.

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.

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