Models and types of Peano's arithmetic (original) (raw)

Automorphisms of models of arithmetic: A unified view

Annals of Pure and Applied Logic, 2007

We develop the method of iterated ultrapower representation to provide a unified and perspicuous approach for building automorphisms of countable recursively saturated models of Peano arithmetic P A. In particular, we use this method to prove Theorem A below, which confirms a long standing conjecture of James Schmerl.

Automorphisms of models of bounded arithmetic

Fundamenta Mathematicae, 2006

We establish the following model theoretic characterization of the fragment I∆ 0 +Exp+BΣ 1 of Peano arithmetic in terms of fixed points of automorphisms of models of bounded arithmetic (the fragment I∆ 0 of Peano arithmetic with induction limited to ∆ 0 -formulae). Theorem A. The following two conditions are equivalent for a countable model M of the language of arithmetic: (a) M satisfies I∆ 0 + BΣ 1 + Exp. (b) M = I f ix (j) for some nontrivial automorphism j of an end extension N of M that satisfies I∆ 0 .

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).

Substructure lattices of models of arithmetic

Annals of Mathematical Logic, 1979

We completely characterize those distributive lattices which can be obtained as elementary substructure lattices of models of Peano arithmetic. Stated concisely: every plausible distributive Mice occurs abundantly. Our proof employs the notion of a strongly definable type in many variables. With slight modifications the method also yields a characterization of those distributive lattices which can be obtained uniformly hy Gaifman's methods oi definable and end extensional l-types. As :J special case this gives another proof of two conjectures involving finite distributive lattices and models of arithmetic posed by Gaifman and initially proved by Schmurl. We also show that every minimal type (in the sense of Gaifman) satisfies a strong partitton property which we will call being "uniformly Ramsey". (2) VMbPA 3N> M Lt (N/M)= D. (3) D is complete, compactly generated, and eacf~ compact element of D hots C X,, compact predecessors.

End extensions of models of fragments of PA

Archive for Mathematical Logic, 2020

In this paper, we prove results concerning the existence of proper end extensions of arbitrary models of fragments of Peano arithmetic (P A). In particular, we give alternative proofs that concern (a) a result of Clote (Fundam Math 127(2):163-170, 1986); (Fundam Math 158(3):301-302, 1998), on the end extendability of arbitrary models of n-induction, for n≥2, and (b) the fact that every model of 1-induction has a proper end extension satisfying 0-induction; although this fact was not explicitly stated before, it follows by earlier results of Enayat and Wong (

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.

Models with elementary end extensions I

Suppose L is any countable first order language including the symbol < in which < is always interpreted as a linear ordering and T is an L-theory such that T has a K-like model where K is a strongly inaccessible cardinal. In this paper which is the first of a series of papers, we study the model theory of T and initiate a new line of investigations towards the two open questions due to Schmerl and due to Enayat and Shelah on this topic.