Equivalence of the Induction Schema and the Least Number Principle for Open Formulas (original) (raw)

A Note on Induction Schemas in Bounded Arithmetic

Arxiv preprint cs/0210011, 2002

Abstract: As is well known, Buss' theory of bounded arithmetic $ S^{1} _ {2} $ proves Sigma0b(Sigma1b)−LIND\ Sigma_ {0}^{b}(\ Sigma_ {1}^{b})-LIND Sigma0b(Sigma1b)−LIND; however, we show that Allen's $ D_ {2}^{1} $ does not prove Sigma0b(Sigma1b)−LLIND\ Sigma_ {0}^{b}(\ Sigma_ {1}^{b})-LLIND Sigma0b(Sigma1b)−LLIND unless $ P= NC .Wealsogivesomeinterestingalternativeaxiomatisationsof. We also give some interesting alternative axiomatisations of .Wealsogivesomeinterestingalternativeaxiomatisationsof S^{1} _ {2} $.

On the induction schema for decidable predicates

The Journal of Symbolic Logic, 2003

We study the fragment of Peano arithmetic formalizing the induction principle for the class of decidable predicates, IΔ1. We show that IΔ1 is independent from the set of all true arithmetical Π2-sentences. Moreover, we establish the connections between this theory and some classes of oracle computable functions with restrictions on the allowed number of queries. We also obtain some conservation and independence results for parameter free and inference rule forms of Δ1-induction.An open problem formulated by J. Paris (see [4, 5]) is whether IΔ1 proves the corresponding least element principle for decidable predicates, LΔ1 (or, equivalently, the Σ1-collection principle BΣ1). We reduce this question to a purely computation-theoretic one.

Intuitionistic open induction and least number principle and the Buss operator

1998

In "Intuitionistic validity in T-normal Kripke structures," Buss asked whether every intuitionistic theory is, for some classical theory T, that of all T-normal Kripke structures H (T) for which he gave an r.e. axiomatization. In the language of arithmetic Iop and Lop denote PA − plus Open Induction or Open LNP, iop and lop are their intuitionistic deductive closures. We show H (Iop) = lop is recursively axiomatizable and lop i c iop, while i∀ 1 lop. If iT proves PEM atomic but not totality of a classically provably total Diophantine function of T, then H (T) ⊆ iT and so iT ∈ range(H). A result due to Wehmeier then implies i 1 ∈ range(H). We prove Iop is not ∀ 2-conservative over i∀ 1. If Iop ⊆ T ⊆ I∀ 1 , then iT is not closed under MR open or Friedman's translation, so iT ∈ range (H). Both Iop and I∀ 1 are closed under the negative translation. 1 Iop-normal Kripke structures vs. models of iop or lop We begin with a version of the Kripke semantics for (arithmetic in) intuitionistic predicate logic. The language

Shepherdson's theorems for fragments of open induction

arXiv (Cornell University), 2017

By a well-known result of Shepherdson, models of the theory IOpen (a first order arithmetic containing the scheme of induction for all quantifier free formulas) are exactly all the discretely ordered semirings that are integer parts of their real closures. In this paper we prove several analogous results that provide algebraic equivalents to various fragments of IOpen.

Notes on bounded induction for the compositional truth predicate

We prove that the theory of the extensional compositional truth predicate for the language of arithmetic with ∆ 0-induction scheme for the truth predicate and the full arithmetical induction scheme is not conservative over Peano Arithmetic. In addition, we show that a slightly modified theory of truth actually proves the global reflection principle over the base theory.

On Wright's inductive definition of coherence truth for arithmetic

Analysis, 2003

As the first illustration of a potential satisfier for the 'platitudes for truth' in the appendix to his engaging recent discussion of the concept of truth (Wright 1999), Crispin Wright has proposed a notion of 'truth conceived as coherence' for arithmetic.

Existentially Closed Models and Conservation Results in Bounded Arithmetic

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.

A Proof Theory for the Logic of Provability in True Arithmetic

STUDIA LOGICA, vol.108(4), pp 857-875, 2020

[Attached is an accepted version.] In a classical 1976 paper, Solovay proved the arithmetical completeness of the modal logic GL; provability of a formula in GL coincides with provability of its arithmetical interpretations in Peano Arithmetic. In that paper, he also provided an axiomatic system GLS and proved arithmetical completeness for GLS; provability of a formula in GLS coincides with truth of its arithmetical interpretations in the standard model of arithmetic. Proof theory for GL has been studied intensively up to the present day. However, it might sound somewhat strange that no proof theory for GLS was ever developed nor even suggested thus far, except for the axiomatic system offered by Solovay. In this paper, we develop a proof theory for GLS based on the sequent calculus method. We provide a sequent calculus for GLS and prove the cut- elimination and some standard consequences of it: the interpolation and de- finability theorems. As another consequence of cut-elimination, we also prove the equivalence of GL and GLS with respect to a special form of formulas which we call G ̈odel sentences, using a purely proof-theoretical method.