Fragments Of Peano Arithmetic Research Articles

Synthetic Undecidability and Incompleteness of First-Order Axiom Systems in Coq

We mechanise the undecidability of various first-order axiom systems in Coq, employing the synthetic approach to computability underlying the growing Coq Library of Undecidability Proofs. Concretely, we cover both semantic and deductive entailment in fragments of Peano arithmetic (PA) as well as ZF and related finitary set theories, with their undecidability established by many-one reductions from solvability of Diophantine equations, i.e. Hilbert’s tenth problem (H10), and the Post correspondence problem (PCP), respectively. In the synthetic setting based on the computability of all functions definable in a constructive foundation, such as Coq’s type theory, it suffices to define these reductions as meta-level functions with no need for further encoding in a formalised model of computation. The concrete cases of PA and the considered set theories are supplemented by a general synthetic theory of undecidable axiomatisations, focusing on well-known connections to consistency and incompleteness. Specifically, our reductions rely on the existence of standard models, necessitating additional assumptions in the case of full ZF, and all axiomatic extensions still justified by such standard models are shown incomplete. As a by-product of the undecidability of set theories formulated using only membership and no equality symbol, we obtain the undecidability of first-order logic with a single binary relation.

Δ2 degrees without Σ1 induction

In this paper, we study the structure of Turing degrees below 0′ in the theory that is a fragment of Peano arithmetic without Σ1 induction, with special focus on proper d-r.e. degrees and non-r.e. degrees. We prove: (1) P− + BΣ1 + Exp ⊦ There is a proper d-r.e. degree. (2) P− +BΣ1+ Exp ⊦ IΣ1 ↔ There is a proper d-r.e. degree below 0′. (3) P− + BΣ1 + Exp ⊬ There is a non-r.e. degree below 0′.

Cofinal elementary extensions

We investigate some properties of ordered structures that are related to their having cofinal elementary extensions. Special attention is paid to models of some very weak fragments of Peano Arithmetic.

Friedberg Numbering in Fragments of Peano Arithmetic andα-Recursion Theory

AbstractIn this paper, we investigate the existence of a Friedberg numbering in fragments of Peano Arithmetic and initial segments of Gödel's constructible hierarchyLα, whereαis Σ1admissible. We prove that(1) OverP−+BΣ2, the existence of a Friedberg numbering is equivalent toIΣ2, and(2) ForLα, there is a Friedberg numbering if and only if the tame Σ2projectum ofαequals the Σ2cofinality ofα.

On nested simple recursion

We give a novel proof that primitive recursive functions are closed under nested simple recursion. This new presentation is supplied with a detailed proof which can be easily formalized in small fragments of Peano Arithmetic.

Subrecursive degrees and fragments of Peano Arithmetic

Let T 0?T 1 denote that each computable function, which is provable total in the first order theory T 0, is also provable total in the first order theory T 1. Te relation ? induces a degree structure on the sound finite Π2 extensions of EA (Elementary Arithmetic). This paper is devoted to the study of this structure. However we do not study the structure directly. Rather we define an isomorphic subrecursive degree structure , and then we study by ubrecursive and computability-theoretic means. Furthermore, we introduce and investigate some operators on the degrees of . These operators corresponds to inferencerules in formal arithmetic. One operator corresponds to the Σ1 collection rule. Another operator corresponds to the Σ1 induction rule.

Σ2 Induction and infinite injury priority argument, Part I: Maximal sets and the jump operator

The study of recursion theory on models of fragments of Peano arithmetic has hitherto been concentrated on recursively enumerable (r.e.) sets and their degrees (with a few exceptions, such as that in [2] on minimal degrees). The reason for such a concerted effort is clear: priority arguments have occupied a central position in post Friedberg-Muchnik recursion theory, and after almost forty years of intensive development in the subject, they are still the essential tools on which investigations of r.e. sets and their degrees depend. There are two possible approaches to the study within fragments of arithmetic: To give a general analysis of strategies, and identify their proof-theoretic strengths (for example in [6] on infinite injury priority methods), or to consider specific theorems in recursion theory, and, if possible, pinpoint the exact levels of induction provably equivalent to the theorems. The work reported in this paper belongs to the second approach. More precisely, we single out two infinitary injury type constructions of r.e. sets—one concerning maximal sets and the other based on the notion of the jump operator—to be the topics of study.

Torre models in the isols

AbstractIn [14] J. Hirschfeld established the close connection of models of the true AE sentences of Peano Arithmetic and homomorphic images of the semiring of recursive functions. This fragment of Arithmetic includes most of the familiar results of classical number theory. There are two nice ways that such models appear in the isols. One way was introduced by A. Nerode in [20] and is referred to in the literature as Nerode Semirings. The other way is called a tame model. It is very similar to a Nerode Semiring and was introduced in [6]. The model theoretic properties of Nerode Semirings and tame models have been widely studied by T. G. McLaughlin ([16], [17], and [18]).In this paper we introduce a new variety of tame model called a torre model. It has as a generator an infinite regressive isol with a nice structural property relative to recursively enumerable sets and their extensions to the isols. What is then obtained is a nonstandard model in the isols of the fragment of Peano Arithmetic with the following property: Let T be a torre model. Let f be any recursive function, and let fΛ be its extension to the isols. If there is an isol A with fΛ(A) ϵ T, then there is also an isol B ϵ T with fΛ(B) = fΛ(A).

A sharpened version of McAloon's theorem on initial segments of models of IΔ0

A generalization is given of McAloon's result (1982) on initial segments ofmodels of GlΔ 0, the fragment of Peano Arithmetic where the induction scheme is restricted to formulas with bounded quantifiers.

Maximal sets and fragments of Peano arithmetic

This work is inspired by the recent paper of Mytilinaios and Slaman [9] on the infinite injury priority method. It may be considered to fall within the general program of the study of reverse recursion theory: What axioms of Peano arithmetic are required or sufficient to prove theorems in recursion theory? Previous contributions to this program, especially with respect to the finite and infinite injury priority methods, can be found in the works of Groszek and Mytilinaios [4], Groszek and Slaman [5], Mytilinaios [8], Slaman and Woodin [10]. Results of [4] and [9], for example, together pinpoint the existence of an incomplete, nonlow r.e. degree to be provable only within some fragment of Peano arithmetic at least as strong as P- + IΣ2. Indeed an abstract principle on infinite strategies, such as that on the construction of an incomplete high r.e. degree, was introduced in [4] and shown to be equivalent to Σ2 induction over the base theory P- + IΣ0 of Peano arithmetic.

Reflection Principles in Fragments of Peano Arithmetic

Reflection Principles in Fragments of Peano Arithmetic

Provably recursive functions in fragments of Peano arithmetic

Provably recursive functions in fragments of Peano arithmetic

Majorizing provably recursive functions in fragments of PA

We prove an interpretation or “majorization” theorem for PA n (n≧1), the fragment of Peano Arithmetic based on ∑ 0 -induction. The theorem relies upon an asymmetric treatment of quantifiers, inspired by Girard's asymmetric rules for ID1, and it implies the well-known classification of provably recursive functions of PA n .