Axiomatizable (original) (raw)

Abstract elementary classes and infinitary logics

Annals of Pure and Applied Logic, 2008

In this paper we study abstract elementary classes using infinitary logics and prove a number of results relating them. For example, if (K, ≺ K) is an a.e.c. with Löwenheim-Skolem number κ then K is closed under L ∞,κ +-elementary equivalence. If κ = ω and (K, ≺ K) has finite character then K is closed under L ∞,ω-elementary equivalence. Analogous results are established for ≺ K. Galois types, saturation, and categoricity are also studied. We prove, for example, that if (K, ≺ K) is finitary and λ-categorical for some infinite λ then there is some σ ∈ L ω 1 ,ω such that K and Mod(σ) contain precisely the same models of cardinality at least λ.

Infinitary logics and abstract elementary classes

Proceedings of the American Mathematical Society

We prove that every abstract elementary class (a.e.c.) with LST number κ and vocabulary τ of cardinality κ can be axiomatized in the logic L 2 (κ) +++ ,κ + (τ). In this logic an a.e.c. is therefore an EC class rather than merely a PC class. This constitutes a major improvement on the level of definability previously given by the Presentation Theorem. As part of our proof, we define the canonical tree S = S K of an a.e.c. K. This turns out to be an interesting combinatorial object of the class, beyond the aim of our theorem. Furthermore, we study a connection between the sentences defining an a.e.c. and the relatively new infinitary logic L 1 λ .

An Arithmetical-like Theory of Hereditarily Finite Sets

Anais do II Workshop Brasileiro de Lógica (WBL 2021), 2021

This paper presents the (second-order) theory of hereditarily finite sets according to the usual pattern adopted in the presentation of the (second-order) theory of natural numbers. To this purpose, we consider three primitive concepts, together with four axioms, which are analogous to the usual Peano axioms. From them, we prove a homomorphism theorem, its converse, categoricity, and a kind of (semantical) completeness.

Internal End-Extensions of Peano Arithmetic and a Problem of Gaifman

Journal of the London Mathematical Society, 1976

A well known result of M. Rabin states that the only existentially complete model of full arithmetic is the standard one. H. Gaifman [1], raised the parallel question for end-extensions of full arithmetic, i.e. does every non-standard model of full arithmetic have an end-extension in which a diophantine equation unsolvable in the original model has a solution. A. Wilkie provided a partial answer [4] when he proved that every countable model of P, Peano Arithmetic, has such an end-extension (which is in fact isomorphic to the original model).

Classical mereology is not elementarily axiomatizable

Logic and Logical Philosophy, 2015

By the classical mereology I mean a theory of mereological structures in the sense of [10]. In [7] I proved that the class of these structures is not elementarily axiomatizable. In this paper a new version of this result is presented, which according to my knowledge is the first such presentation in English. A relation of this result to a certain Hsing-chien Tsai's theorem from [13] is emphasized.

Degrees of insolubility of extensions of arithmetic by true propositions

Russian Mathematical Surveys, 1988

By the degree of insolubility (or simply degree) d (T) of a theory Τwe mean the Turing degree of the set of Godel numbers of the theorems of Τ [1]. In this paper we solve the problem of the degrees of extensions of the arithmetic PA by true propositions. It follows from the classical theorems on insolubility that all these degrees are greater than or equal to O'• In this paper we prove that the set of these degrees coincides with the ideal/(0')={a| O'C"}· The main theorem also implies that every degree inKO) contains a theory of the class under ...

A standard model of Peano arithmetic with no conservative elementary extension

Annals of Pure and Applied Logic, 2008

The principal result of this paper answers a long-standing question in the model theory of arithmetic Question 7] by showing that there exists an uncountable arithmetically closed family A of subsets of the set ω of natural numbers such that the expansion Ω A := (ω, +, ·, X) X∈A of the standard model of Peano arithmetic has no conservative elementary extension, i.e., for any elementary extension Ω * A = (ω * , · · ·) of Ω A , there is a subset of ω * that is parametrically definable in Ω * A but whose intersection with ω is not a member of A.

A Note on Induction, Abstraction, and Dedekind-Finiteness

2011

The purpose of this note is to present a simplification of the system of arithmetical axioms given in previous work; specifically, it is shown how the induction principle can in fact be obtained from the remaining axioms, without the need of explicit postulation. The argument might be of more general interest, beyond the specifics of the proposed axiomatization, as it highlights the interaction of the notion of Dedekind-finiteness and the induction principle.