Decidability and Specker sequences in intuitionistic mathematics (original) (raw)

Decidable Kripke models of intuitionistic theories

Annals of Pure and Applied Logic, 1998

The introduction of computable (alternately, recursive) function theory by Post, Church, Kleene, Godel, Turing, Malcev made it possible to analyse the computability of mathematical notions and constructions within the context of classical mathematics. Quite separately, the constructiveness of algebra was a principal concern of Kronecker in the late nineteenth century, and the constructiveness of analysis was a principle concern of Brouwer in the early twentieth century. Brouwer's work motivated the denition of rst order intuitionistic logic as introduced by his disciple Heyting. Kroneckerian eld theory was reworked as computable eld theory by F r/"oelich and Shepherdson in the 1950's, [4]. Systematic study of recursive algebra and recursive

On the Hierarchy of Intuitionistic Bounded Arithmetic

Journal of Logic and Computation, 2007

In this paper we are concerned with cuts in models of Samuel Buss' theories of bounded arithmetic, i.e. theories like S i 2 and T i 2. In correspondence with polynomial induction, we consider a rather new notion of cut that we call p-cut. We also consider small cuts, i.e. cuts that are bounded above by a small element. We study the basic properties of pcuts and small cuts. In particular, we prove some overspill and underspill properties for them.

THE POLYNOMIAL HIERARCHY AND INTUITIONISTIC BOUNDED ARITHMETIC

Intuitionistic theories IS: of Bounded Arithmetic a r e introduced and i t is shown t h a t the definable functions of IS: a r e precisely the 0: functions of t h e polvnomial hierarchy. This is an extension of earlier work on t h e classical Bounded Arithmetic and was first conjectured by S. Cook. In contrast t o t h e classical theories of Bounded b Arithmetic where Ci-definable functions are of interest, our results for intuitionistic theories concern all the definable functions. The method of proof uses 0;-realizability which is inspired by t h e recursive realizability of S.C. Kleene 131 and D. Nelson 151. I t also involves polynomial hierarchy functionals of finite type which a r e introduced in this paper. * Research supported in part by NSF Grant DMS 85-11465. In general, 0: is P The theories Si a r e most advantageously viewed a s Gentzen-style natural deduction systems. A formal proof in a natural deduction system contains sequents of t h e form where each A. and B. is a formula. The meaning of such a sequent is J J In addition t o t h e usual inference rules for natural deduction. the Z:-PIND inference is b

Intuitionistically provable recursive well-orderings

Annals of Pure and Applied Logic, 1986

We consider intuitionistic number theory with recursive infinitary rules (HA*). Any primitive recursive binary relation for which transfinite induction schema is provable is in fact well founded. Its ordinal is less than .sa if the transfinite induction schema is intuitionistically provable in elementary number theory. These results are provable intuitionistically. In fact, it suffices to consider transfinite induction with respect to one particular number-theoretic property.

Continuity and comprehension in intuitionistic formal systems

Pacific Journal of Mathematics, 1977

CONTINUITY AND COMPREHENSION IN INTUITIONISTIC FORMAL SYSTEMS 31 KLS(JR, R) in various arithmetic theories was proved in [2]. The proof carries over easily to the case of KLS(X, R), as soon as one discusses how to formalize complete separable metric spaces, as is done in [3]. Thus the independence results of this paper apply equally to KLS(X, R) as to KLS.

An Independence Result for Intuitionistic Bounded Arithmetic

Journal of Logic and Computation, 2006

It is shown that the intuitionistic theory of polynomial induction on positive Π b 1 (coNP) formulas does not prove the sentence ¬¬∀x, y∃z ≤ y(x ≤ |y| → x = |z|). This implies the unprovability of the scheme ¬¬PIND(Σ b+ 1) in the mentioned theory. However, this theory contains the sentence ∀x, y¬¬∃z ≤ y(x ≤ |y| → x = |z|). The above independence result is proved by constructing an ω-chain of submodels of a countable model of S 2 + Ω 3 + ¬exp such that none of the worlds in the chain satisfies the sentence, and interpreting the chain as a Kripke model.

Intuitionistic computability logic

Cornell University - arXiv, 2004

Computability logic (CL) is a systematic formal theory of computational tasks and resources, which, in a sense, can be seen as a semantics-based alternative to (the syntactically introduced) linear logic. With its expressive and flexible language, where formulas represent computational problems and "truth" is understood as algorithmic solvability, CL potentially offers a comprehensive logical basis for constructive applied theories and computing systems inherently requiring constructive and computationally meaningful underlying logics. Among the best known constructivistic logics is Heyting's intuitionistic calculus INT, whose language can be seen as a special fragment of that of CL. The constructivistic philosophy of INT, however, just like the resource philosophy of linear logic, has never really found an intuitively convincing and mathematically strict semantical justification. CL has good claims to provide such a justification and hence a materialization of Kolmogorov's known thesis "INT = logic of problems". The present paper contains a soundness proof for INT with respect to the CL semantics. * This material is based upon work supported by the National Science Foundation under Grant No. 0208816 1 Using Blass's [2] words, 'Supplementary rules governing repeated attacks and defenses were devised by Lorenzen so that the formulas for which P [proponent] has a winning strategy are exactly the intuitionistically provable ones'. Quoting [6], 'Lorenzen's approach describes logical validity exclusively in terms of rules without appealing to any kind of truth values for atoms, and this makes the semantics somewhat vicious ... as it looks like just a "pure" syntax rather than a semantics'.