It can be argued that only the equational theories of some sub-elementary function algebras are finitistic or intuitive according to a certain interpretation of Hilbert's conception of intuition. The purpose of this paper is to investigate the relation of those restricted forms of equational reasoning to classical quantifier logic in arithmetic. The conclusion reached is that Edward Nelson's ‘predicative arithmetic’ program, which makes essential use of classical quantifier logic, cannot be justified finitistically and thus requires a different philosophical foundation, possibly (...) as a restricted form of logicism. (shrink)

We investigate the bimodal logics sound and complete under the interpretation of modal operators as the provability predicates in certain natural pairs of arithmetical theories . Carlson characterized the provability logic for essentially reflexive extensions of theories, i.e. for pairs similar to . Here we study pairs of theories such that the gap between and is not so wide. In view of some general results concerning the problem of classification of the bimodal provability logics we are particularly interested in such (...) pairs that is axiomatized over by ∏1-sentences only, and, for each n 1, proves the n-times iterated consistency of . A complete axiomatization, along with the appropriate Kripke semantics and decision procedures, is found for the two principal cases: finitely axiomatizable extensions of this sort, like e.g. ), ), etc., and reflexive extensions, like ¦n 1\s}), etc. We show that the first logic, ICP, is the minimal and the second one, RP, is the maximal within the class of the provability logics for such pairs of theories. We also show that there are some provability logics lying strictly between these two. As an application of the results of this paper, in the last section the polymodal provability logics for natural recursive progressions of theories based on iteration of consistency are characterized. We construct a system of ordinal notation , which gives exactly one notation to each constructive ordinal, such that the logic corresponding to any progression along coincides with that along natural Kalmar elementary well-orderings. (shrink)

In this paper, we characterize the strength of the predicative Frege hierarchy, , introduced by John Burgess in his book [J. Burgess, Fixing frege, in: Princeton Monographs in Philosophy, Princeton University Press, Princeton, 2005]. We show that and are mutually interpretable. It follows that is mutually interpretable with Q. This fact was proved earlier by Mihai Ganea in [M. Ganea, Burgess’ PV is Robinson’s Q, The Journal of Symbolic Logic 72 619–624] using a different proof. Another consequence of the our (...) main result is that is mutually interpretable with Kalmar Arithmetic . The fact that interprets EA was proved earlier by Burgess. We provide a different proof. Each of the theories is finitely axiomatizable. Our main result implies that the whole hierarchy taken together, , is not finitely axiomatizable. What is more: no theory that is mutually locally interpretable with is finitely axiomatizable. (shrink)

In this paper we study proofs of some general forms of the Second Incompleteness Theorem. These forms conform to the Feferman format, where the proof predicate is fixed and the representation of the set of axioms varies. We extend the Feferman framework in one important point: we allow the interpretation of number theory to vary.

In this paper we study the idea of theories with containers, like sets, pairs, sequences. We provide a modest framework to study such theories. We prove two concrete results. First, we show that first-order theories of finite signature that have functional non-surjective ordered pairing are definitionally equivalent to extensions in the same language of the basic theory of non-surjective ordered pairing. Second, we show that a first-order theory of finite signature is sequential (is a theory of sequences) iff it is (...) definitionally equivalent to an extension in the same language of a system of weak set theory called WS. (shrink)

In his paper "Undecidability without arithmetization," Andrzej Grzegorczyk introduces a theory of concatenation $\mathsf{TC}$. We show that pairing is not definable in $\mathsf{TC}$. We determine a reasonable extension of $\mathsf{TC}$ that is sequential, that is, has a good sequence coding.

This paper develops the philosophy and technology needed for adding a supremum operator to the interpretability logic ILM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {ILM}$$\end{document} of Peano Arithmetic. It is well-known that any theories extending PA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {PA}$$\end{document} have a supremum in the interpretability ordering. While provable in PA\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {PA}$$\end{document}, this fact is not reflected in the theorems of the modal (...) system ILM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {ILM}$$\end{document}, due to limited expressive power. Our goal is to enrich the language of ILM\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {ILM}$$\end{document} by adding to it a new modality for the interpretability supremum. We explore different options for specifying the exact meaning of the new modality. Our final proposal involves a unary operator, the dual of which can be seen as a provability predicate satisfying the axioms of the provability logic GL\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathsf {GL}$$\end{document}. (shrink)

In this paper we study local induction w.r.t. Σ1‐formulas over the weak arithmetic. The local induction scheme, which was introduced in, says roughly this: for any virtual class that is progressive, i.e., is closed under zero and successor, and for any non‐empty virtual class that is definable by a Σ1‐formula without parameters, the intersection of and is non‐empty. In other words, we have, for all Σ1‐sentences S, that S implies, whenever is progressive. Since, in the weak context, we have (at (...) least) two definitions of Σ1, we obtain two minimal theories of local induction w.r.t. Σ1‐formulas, which we call Peano Corto and Peano Basso.In the paper we give careful definitions of Peano Corto and Peano Basso. We establish their naturalness both by giving a model theoretic characterization and by providing an equivalent formulation in terms of a sentential reflection scheme.The theories Peano Corto and Peano Basso occupy a salient place among the sequential theories on the boundary between weak and strong theories. They bring together a powerful collection of principles that is locally interpretable in. Moreover, they have an important role as examples of various phenomena in the metamathematics of arithmetical (and, more generally, sequential) theories. We illustrate this by studying their behavior w.r.t. interpretability, model interpretability and local interpretability. In many ways the theories are more like Peano arithmetic or Zermelo Fraenkel set theory, than like finitely axiomatized theories as Elementary Arithmetic, and. On the one hand, Peano Corto and Peano Basso are very weak: they are locally cut‐interpretable in. On the other hand, they behave as if they were strong: they are not contained in any consistent finitely axiomatized arithmetical theory, however strong. Moreover, they extend, the theory of parameter‐free Π1‐induction. (shrink)

This paper will introduce the notion of a naming convention and use this paradigm to both develop a new version of the Second Incompleteness Theorem and to describe when an axiom system can partially evade the Second Incompleteness Theorem.

In this paper we formulate a version of Second Incompleteness Theorem. The idea is that a sequential sentence has ‘consistency power’ over a theory if it enables us to construct a bounded interpretation of that theory. An interpretation of V in U is bounded if, for some n , all translations of V -sentences are U -provably equivalent to sentences of complexity less than n . We call a sequential sentence with consistency power over T a pro-consistency statement for T (...) . We study pro-consistency statements. We provide an example of a pro-consistency statement for a sequential sentence A that is weaker than an ordinary consistency statement for A . We show that, if A is $${{\sf S}^{1}_{2}}$$ , this sentence has some further appealing properties, specifically that it is an Orey sentence for EA . The basic ideas of the paper essentially involve sequential theories. We have a brief look at the wider environment of the results, to wit the case of theories with pairing. (shrink)

We prove that any proof of a [Formula: see text] sentence in the theory [Formula: see text] can be translated into a proof in [Formula: see text] at the cost of a polynomial increase in size. In fact, the proof in [Formula: see text] can be obtained by a polynomial-time algorithm. On the other hand, [Formula: see text] has nonelementary speedup over the weaker base theory [Formula: see text] for proofs of [Formula: see text] sentences. We also show that for (...) [Formula: see text], proofs of [Formula: see text] sentences in [Formula: see text] can be translated into proofs in [Formula: see text] at a polynomial cost in size. Moreover, the [Formula: see text]-conservativity of [Formula: see text] over [Formula: see text] can be proved in [Formula: see text], a fragment of bounded arithmetic corresponding to polynomial-time computation. For [Formula: see text], this answers a question of Clote, Hájek and Paris. (shrink)

In this paper we study the interpretations of a weak arithmetic, like Buss’s theory $\mathsf{S}^{1}_{2}$, in a given theory $U$. We call these interpretations the arithmetics of $U$. We develop the basics of the structure of the arithmetics of $U$. We study the provability logic of $U$ from the standpoint of the framework of the arithmetics of $U$. Finally, we provide a deeper study of the arithmetics of a finitely axiomatized sequential theory.

A theory T is trustworthy iff, whenever a theory U is interpretable in T, then it is faithfully interpretable. In this paper we give a characterization of trustworthiness. We provide a simple proof of Friedman’s Theorem that finitely axiomatized, sequential, consistent theories are trustworthy. We provide an example of a theory whose schematic predicate logic is complete Π20.

In 1981, Paris and Wilkie raised the open question about whether and to what extent the axiom system did satisfy the Second Incompleteness Theorem under Semantic Tableaux deduction. Our prior work showed that the semantic tableaux version of the Second Incompleteness Theorem did generalize for the most common definition of appearing in the standard textbooks.However, there was an alternate interesting definition of this axiom system in the Wilkie–Paris article in the Annals of Pure and Applied Logic 35 , pp. 261–302 (...) which we did not examine in our year-2002 article in the Journal of Symbolic Logic. Our first goal is to show that the incompleteness results of our prior paper can generalize in this alternate context. We will also develop a formal analysis, using a new technique called Passive Induction, that is simpler than the formalism we had used before.A further reason our results are of interest is that we have shown in a companion paper published in Electronic Notes in Theoretical Computer Science 165 , pp. 213–226 that some very unorthodox axiomizations for are anti-thresholds for the Herbrandized version of the Second Incompleteness Theorem. Thus, different axiomizations for have nearly fully opposite incompleteness properties.This paper is self-contained. It will not require a knowledge of our earlier results. (shrink)

In this paper we show that the degrees of interpretability of finitely axiomatized extensions-in-the-same-language of a finitely axiomatized sequential theory—like Elementary Arithmetic EA, IΣ1, or the Gödel–Bernays theory of sets and classes GB—have suprema. This partially answers a question posed by Švejdar in his paper (Commentationes Mathematicae Universitatis Carolinae 19:789–813, 1978). The partial solution of Švejdar’s problem follows from a stronger fact: the convexity of the degree structure of finitely axiomatized extensions-in-the-same-language of a finitely axiomatized sequential theory in the degree (...) structure of the degrees of all finitely axiomatized sequential theories. In the paper we also study a related question: the comparison of structures for interpretability and derivability. In how far can derivability mimic interpretability? We provide two positive results and one negative result. (shrink)

In this paper we carry out a comparative study of $\mathrm{I}\Sigma_1$ and PRA. We will in a sense fully determine what these theories have to say about each other in terms of provability and interpretability. Our study will result in two arithmetically complete modal logics with simple universal models.