Existentially Closed Models in the Framework of Arithmetic

The Journal of Symbolic Logic, 2005

We investigate theories of initial segments of the standard models for arithmetics. It is easy to see that if the ordering relation is definable in the standard model then the decidability results can be transferred from the infinite model into the finite models. On the contrary we show that the Σ2-theory of multiplication is undecidable in finite models. We show that this result is optimal by proving that the Σ1-theory of multiplication and order is decidable in finite models as well as in the standard model. We show also that the exponentiation function is definable in finite models by a formula of arithmetic with multiplication and that one can define in finite models the arithmetic of addition and multiplication with the concatenation operation.

Annals of Mathematical Logic, 1976

Theoretical Computer Science, 2004

In this paper, we introduce and study some syntactical fragments of monadic second-order and ÿrst-order (PEANO) arithmetic which we will prove the connection to famous complexity classes. Starting from descriptive complexity results, and giving an e ective method for translating formulas between di erent logical structures representing encodings of integers, we give some new arithmetical characterizations of NP, PH, NL, and P.

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.

Logic Journal of the IGPL, Volume 32, Issue 5, Pages 880-908, 2024

In this work, we aim at understanding incompleteness in an abstract way via metamathematical properties of formal theories. We systematically examine the relationships between the following twelve important metamathematical properties of arithmetical theories: Rosser, EI (effectively inseparable), RI (recursively inseparable), TP (Turing persistent), EHU (essentially hereditarily undecidable), EU (essentially undecidable), Creative, 0 (theories with Turing degree 0), REW (all RE sets are weakly representable), RFD (all recursive functions are definable), RSS (all recursive sets are strongly representable), RSW (all recursive sets are weakly representable). Given any two properties P and Q in the above list, we examine whether P implies Q.

Bollettino dell'Unione Matematica Italiana

In 1994 Jech gave a model-theoretic proof of Gödel’s second incompleteness theorem for Zermelo–Fraenkel set theory in the following form: $${{\,\mathrm{\mathrm {ZF}}\,}}$$ ZF does not prove that $${{\,\mathrm{\mathrm {ZF}}\,}}$$ ZF has a model. Kotlarski showed that Jech’s proof can be adapted to Peano Arithmetic with the role of models being taken by complete consistent extensions. In this note we take another step in the direction of replacing proof-theoretic by model-theoretic arguments. We show, without the need of formalizing the proof of the completeness theorem within $${{\,\mathrm{\mathrm {PA}}\,}}$$ PA , that the existence of a model of $${{\,\mathrm{\mathrm {PA}}\,}}$$ PA of complexity $$\Sigma ^0_2$$ Σ 2 0 is independent of $${{\,\mathrm{\mathrm {PA}}\,}}$$ PA , where a model is identified with the set of formulas with parameters which hold in the model. Our approach is based on a new interpretation of the provability logic of Peano Arithmetic where $$\Box \phi $$ □ ϕ is ...

Theoretical Computer Science, 2001

We survey results and problems concerning subsystems of Peano Arithmetic. In particular, we deal with end extensions of models of such theories. First, we discuss the results of Paris

Notre Dame Journal of Formal Logic, 1981

A countable recursively saturated model of arithmetic has, up to isomorphism, a unique countable recursively saturated elementary end extension. As one might guess from the countability restriction, this isomorphism is not canonical. Indeed, if 1 is a recursively saturated model of arithmetic and 31 is an isomorphic copy thereof into which we want to embed 1 as an elementary initial segment, then there are continuum many positions in 31 in which to place 1. In this paper we note that, in fact, 1 can be so embedded in 31 in continuum many decidedly distinct ways. We are concerned with structures of the form, (R;H) = (lRl;l«l;+, Λ0,...), where 1 < e 31 are countable recursively saturated models of PA. Our main theorem asserts great variety: for fixed 1 (or, equivalently, for fixed 3^) there are continuum many elementarily inequivalent such structures. If, however, we assume (ϋίt l) to be recursively saturated, the situation becomes more tractable: there is only a countable infinity of elementarily inequivalent such structures. Following the introduction, we get down to business in Section 1 where we first discuss cofinal extensions instead of end extensions. The proofs of the corresponding results are simpler and, besides, somewhat cute and deserving of display. End extensions are then discussed in Section 2. This paper is not self-contained and the reader is advised to have copies of [1] and [6] at hand. Our notation is that of the latter and is reasonably standard. The merely near standard alphabetical exceptions are: Gothic capitals, 1, 31, denote models of arithmetic, which we take to mean nonstandard models of arithmetic. Lower case Latin letters are generally integers of various kinds:

Notre Dame Journal of Formal Logic, 1981

A relatively neglected aspect of the study of nonstandard models of arithmetic is the study of their cofinal extensions.* These extensions certainly do not present themselves to the intuition as readily as do their more popular cousins the end extensions; but they are not exactly shrouded in mystery or unnatural objects of study either. They are equal partners with end extensions in the construction of general extensions of models; they offer both special advantages and disadvantages worthy of our interest; and, occasionally, they are useful in understanding the generally more simply behaved end extensions. Cofinal extensions deserve more attention than they have traditionally received. The fundamental theorem on cofinal extensions is Gaifman's Splitting Theorem, which not only establishes their existence but also reveals one of their most basic properties. Briefly, the Splitting Theorem asserts that every extension of nonstandard models splits into an elementary cofinal extension followed by an end extension. In particular, it follows that cofinal extensions are always elementary. Unfortunately, the Splitting Theorem is language dependent. If we add a few new relations to the language of arithmetic, the theorem could well become false. For this reason, there are two additional versions of the theorem. The simplest form it takes, valid for any language (provided full induction is assumed), is the following: Let 1 < 5R be models of arithmetic. There is another model WL C ? of arithmetic satisfying *The present paper was originally intended as a lecture to be presented at the 1980 ASL Summer Meeting in Praha. Due to the cancellation of this meeting it is appearing here.