Near coherence of filters. I. Cofinal equivalence of models of arithmetic (original) (raw)

Existentially Closed Models in the Framework of Arithmetic

Journal of Symbolic Logic, 2016

We prove that the standard cut is definable in each existentially closed model of I Δ 0 +exp by a (parameter free) Π 1-formula. This definition is optimal with respect to quantifier complexity and allows us to improve some previously known results on existentially closed models of fragments of arithmetic. 1. Introduction. This work was initially motivated by a gap in the proof of Corollary 1.3 of [2] providing a parameter free Π 1-definition of the standard cut, N, in each existentially closed (e.c.) model of I Δ 0 + exp. Our aim is to provide a correct proof of the above result and, use it to obtain an updated view of the theory of e.c. models of I Δ 0 +exp. Existentially closed models of arithmetic were investigated in the 1970's as a part of the efforts to get a full understanding of the model theory of existentially closed structures (existence of model completions and companion theories, finite and infinite forcing, etc.). The results obtained in the early 1970's by A. Robinson, J. Hirschfeld, D. C. Goldrei, A. Macintyre, and H. Simmons pointed out the most important property of e.c. models of sufficiently strong arithmetic theories: there exist formulas defining N in each such model. These results were not stated in their full generality. In the 1970's a systematic study of fragments of Peano arithmetic PA was still to come and the authors focused essentially on e.c. models of Π 2 (N) (thesetoftrueΠ 2-sentences) or of Π 2 (PA)(thesetofΠ 2 consequences of PA), and more generally on e.c. models of Π 2 (T B), where T B is any extension of Π 2 (PA). Regarding Π 2 (N), Robinson (see [14]) proved N to be Σ 3-definable in every e.c. model of Π 2 (N) and Hirschfeld (see [7]) improved Robinson's result obtaining a Σ 2-definition of N,or even aΠ 1-definition, if parameters are allowed. Hirschfeld also showed that these definitions are optimal (in terms of quantifier complexity) for e.c. models of Π 2 (N). As to Π 2 (T B), in [11] Macintyre and Simmons (see also [5]) extended Hirschfeld's Σ 2-definition of N to all e.c. models of Π 2 (T B) and showed that the parametric Π 1definition can be extended to those e.c. models of Π 2 (T B)inwhichtheΣ 1-definable elements are not cofinal. However, these definitions are not best possible, since there is no general result ruling out the possibility of a parameter free Π 1-definition of N valid in all e.c. models. As a matter of fact, such an optimal definition was Key words and phrases. fragments of Peano arithmetic, existentially closed models, turing degrees of arithmetic theories.

ON THE SYMBIOSIS BETWEEN MODEL-THEORETIC AND SET-THEORETIC PROPERTIES OF LARGE CARDINALS

We study some large cardinals in terms of reflection, establishing new connections between the model-theoretic and the set-theoretic approaches. §1. Introduction. First-order logic alone cannot express important properties such as finiteness or uncountability of the model, well-foundedness of a binary predicate, completeness of a linear order, etc. This led to Mostowski [10] and later Lindström [6] to introduce the concept of a generalised quantifier. This made it possible to compare model-theoretic and set-theoretic definability of various mathematical concepts. It turned out that there is a close connection between the two. Following [14], we call this connection symbiosis.

Computable Quotient Presentations of Models of Arithmetic and Set Theory

2017

We prove various extensions of the Tennenbaum phenomenon to the case of computable quotient presentations of models of arithmetic and set theory. Specifically, no nonstandard model of arithmetic has a computable quotient presentation by a c.e. equivalence relation. No \(\Sigma _1\)-sound nonstandard model of arithmetic has a computable quotient presentation by a co-c.e. equivalence relation. No nonstandard model of arithmetic in the language \(\{+,\cdot ,\le \}\) has a computably enumerable quotient presentation by any equivalence relation of any complexity. No model of ZFC or even much weaker set theories has a computable quotient presentation by any equivalence relation of any complexity. And similarly no nonstandard model of finite set theory has a computable quotient presentation.

Inconsistent Models (and Infinite Models) for Arithmetics with Constructible Falsity

Logic and Logical Philosophy, 2019

An earlier paper on formulating arithmetic in a connexive logic ended with a conjecture concerning C♯ , the closure of the Peano axioms in Wansing's connexive logic C. Namely, the paper conjectured that C ♯ is Post consistent relative to Heyting arithmetic, i.e., is nontrivial if Heyting arithmetic is nontrivial. The present paper borrows techniques from relevant logic to demonstrate that C♯ is Post consistent simpliciter, rendering the earlier conjecture redundant. Given the close relationship between C and Nelson's paraconsistent N4, this also supplements Nelson's own proof of the Post consistency of N4♯. Insofar as the present technique allows infinite models, this resolves Nelson's concern that N4 ♯ is of interest only to those accepting that there are finitely many natural numbers.

Existentially Closed Models and Conservation Results in Bounded Arithmetic

Journal of Logic and Computation, 2009

We develop model-theoretic techniques to obtain conservation results for first order Bounded Arithmetic theories, based on a hierarchical version of the well-known notion of an existentially closed model. We focus on the classical Buss' theories S i 2 and T 2 i and prove that they are ∀ i b conservative over their inference rule counterparts, and ∃∀ i b conservative over their parameter-free versions. A similar analysis of the i b-replacement scheme is also developed. The proof method is essentially the same for all the schemes we deal with and shows that these conservation results between schemes and inference rules do not depend on the specific combinatorial or arithmetical content of those schemes. We show that similar conservation results can be derived, in a very general setting, for every scheme enjoying some syntactical (or logical) properties common to both the induction and replacement schemes. Hence, previous conservation results for induction and replacement can be also obtained as corollaries of these more general results.

A new principle in the interpretability logic of all reasonable arithmetical theories

arXiv (Cornell University), 2020

The interpretability logic of a mathematical theory describes the structural behavior of interpretations over that theory. Different theories have different logics. This paper revolves around the question what logic describes the behavior that is present in all theories with a minimum amount of arithmetic; the intersection over all such theories so to say. We denote this target logic by IL(All). In this paper we present a new principle R in IL(All). We show that R does not follow from the logic ILP0W * that contains all previously known principles. This is done by providing a modal incompleteness proof of ILP0W * : showing that R follows semantically but not syntactically from ILP0W *. Apart from giving the incompleteness proof by elementary methods, we also sketch how to work with so-called Generalized Veltman Semantics as to establish incompleteness. To this extent, a new version of this Generalized Veltman Semantics is defined and studied. Moreover, for the important principles the frame correspondences are calculated. After the modal results it is shown that the new principle R is indeed valid in any arithmetically theory. The proof employs some elementary results on definable cuts in arithmetical theories. 1 Technically speaking the property of so-called essential reflexivity is sufficient. A theory is

Completeness theorems, incompleteness theorems and models of arithmetic

Transactions of the American Mathematical Society, 1978

Let & be a consistent extension of Peano arithmetic and let 6EJJ denote the set of TL°" consequences of &. Employing incompleteness theorems to generate independent formulas and completeness theorems to construct models, we build nonstandard models of SP"+2 m which the standard integers are A°+1-definable. We thus pinpoint induction axioms which are not provable in éE¡¡+2; in particular, we show that (parameter free) A?-induction is not provable in Primitive Recursive Arithmetic. Also, we give a solution of a problem of Gaifman on the existence of roots of diophantine equations in end extensions and answer questions about existentially complete models of 3^. Furthermore, it is shown that the proof of the Gödel Completeness Theorem cannot be formalized in 6E § and that the MacDowell-Specker Theorem fails for all truncated theories (£¡¡.