To the axioms of Peano arithmetic formulated in a language with an additional unary predicate symbol T we add the rules of necessitation φ/Tφ and conecessitation T φ/φ and axioms stating that T commutes with the logical connectives and quantifiers. By a result of McGee this theory is w-inconsistent, but it can be approximated by models obtained by a kind of rule-of-revision semantics. Furthermore we prove that FS is equivalent to a system already studied by Friedman and Sheard and give (...) an analysis of its proof theory. (shrink)

Stephen George Simpson. with definition 1.2.3 and the discussion following it. For example, taking 90(n) to be the formula n §E Y, we have an instance of comprehension, VYEIXVn(n€X<—>n¢Y), asserting that for any given set Y there exists a ...

This comprehensive monograph is a cornerstone in the area of mathematical logic and related fields. Focusing on Gentzen-type proof theory, the book presents a detailed overview of creative works by the author and other 20th-century logicians that includes applications of proof theory to logic as well as other areas of mathematics. 1975 edition.

For several subsystems of second order arithmetic T we show that the proof-theoretic strength of T + (bar rule) can be characterized in terms of T + (bar induction) □ , where the latter scheme arises from the scheme of bar induction by restricting it to well-orderings with no parameters. In addition, we demonstrate that ACA + 0 , ACA 0 + (bar rule) and ACA 0 + (bar induction) □ prove the same Π 1 1 -sentences.

This paper is divided into two parts. Part I provides a resumé of the evolution of the notion of predicativity. Part II describes our own work on the subject.Part I§1. Conceptions of sets.Statements about sets lie at the heart of most modern attempts to systematize all (or, at least, all known) mathematics. Technical and philosophical discussions concerning such systematizations and the underlying conceptions have thus occupied a considerable portion of the literature on the foundations of mathematics.

The starting point of this article is an old question asked by Feferman in his paper on Hancock's conjecture [6] about the strength of ${\rm ID}_{1}^{\ast}$. This theory is obtained from the well-known theory ID₁ by restricting fixed point induction to formulas that contain fixed point constants only positively. The techniques used to perform the proof-theoretic analysis of ${\rm ID}_{1}^{\ast}$ also permit to analyze its transfinitely iterated variants ${\rm ID}_{\alpha}^{\ast}$. Thus, we eventually know that $|\widehat{{\rm ID}}_{\alpha}|=|{\rm ID}_{\alpha}^{\ast}|$.

We present a theory VF of partial truth over Peano arithmetic and we prove that VF and ID 1 have the same arithmetical content. The semantics of VF is inspired by van Fraassen's notion of supervaluation.

To the axioms of Peano arithmetic formulated in a language with an additional unary predicate symbol T we add the rules of necessitation and conecessitation T and axioms stating that T commutes with the logical connectives and quantifiers. By a result of McGee this theory is -inconsistent, but it can be approximated by models obtained by a kind of rule-of-revision semantics. Furthermore we prove that FS is equivalent to a system already studied by Friedman and Sheard and give an analysis (...) of its proof theory. (shrink)

A subtheory of the theory of self-referential truth known as FS is shown to be weak as a theory of truth but equivalent to full FS in its proof-theoretic strength.

We give a new elementary proof of the comparison theorem relating $\sum^1_{n + 1}-\mathrm{AC}\uparrow$ and $\Pi^1_n -\mathrm{CA}\uparrow$ ; the proof does not use Skolem theories. By the same method we prove: a) $\sum^1_{n + 1}-\mathrm{DC} \uparrow \equiv (\Pi^1_n -CA)_{ , for suitable classes of sentences; b) $\sum^1_{n+1}-DC \uparrow$ proves the consistency of (Π 1 n -CA) ω k, for finite k, and hence is stronger than $\sum^1_{n+1}-AC \uparrow$ . a) and b) answer a question of Feferman and Sieg.

We define a class of functions, the descent recursive functions, relative to an arbitrary elementary recursive system of ordinal notations. By means of these functions, we provide a general technique for measuring the proof-theoretic strength of a variety of systems of first-order arithmetic. We characterize the provable well-orderings and provably recursive functions of these systems, and derive various conservation and equiconsistency results.

The proof-theoretic results on axiomatic theories oftruth obtained by different authors in recent years are surveyed.In particular, the theories of truth are related to subsystems ofsecond-order analysis. On the basis of these results, thesuitability of axiomatic theories of truth for ontologicalreduction is evaluated.

We present a theory VF of partial truth over Peano arithmetic and we prove that VF and ID 1 have the same arithmetical content. The semantics of VF is inspired by van Fraassen's notion of supervaluation.

This paper deals with: (i) the theory ID # 1 which results from $\widehat{\mathrm{ID}}_1$ by restricting induction on the natural numbers to formulas which are positive in the fixed point constants, (ii) the theory BON(μ) plus various forms of positive induction, and (iii) a subtheory of Peano arithmetic with ordinals in which induction on the natural numbers is restricted to formulas which are Σ in the ordinals. We show that these systems have proof-theoretic strength φω 0.

The larger project broached here is to look at the generally sentence “if X is well-ordered then f is well-ordered”, where f is a standard proof-theoretic function from ordinals to ordinals. It has turned out that a statement of this form is often equivalent to the existence of countable coded ω-models for a particular theory Tf whose consistency can be proved by means of a cut elimination theorem in infinitary logic which crucially involves the function f. To illustrate this theme, (...) we prove in this paper that the statement “if X is well-ordered then εX is well-ordered” is equivalent to . This was first proved by Marcone and Montalban [Alberto Marcone, Antonio Montalbán, The epsilon function for computability theorists, draft, 2007] using recursion-theoretic and combinatorial methods. The proof given here is principally proof-theoretic, the main techniques being Schütte’s method of proof search [Kurt Schütte, Proof Theory, Springer-Verlag, Berlin, Heidelberg, 1977] and cut elimination for a fragment of. (shrink)