Andrea Cantini

A theory of formal truth arithmetically equivalent to ID 1

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 Fraassens notion of supervaluation.

Paradoxes, Self-Reference and Truth in the 20th Century

Here is a crude list, possibly summarizing the role of paradoxes within the framework of mathematical logic: 1. directly motivating important theories (e.g. type theory, axiomatic set theory, combinatory

Polytime, combinatory logic and positive safe induction

Abstract. We characterize the polynomial time operations as those which are provably total in a first order system, which comprises (untyped) combinatory logic with extensionality, together with positive “safe induction” on the set of binary strings. The formalization of safe induction is inspired by Leivants idea of ramification. We also show how to replace ramification by means of modal logic.

The Undecidability of Grisin's Set Theory

We investigate a contractionless naive set theory, due to Grisin [11]. We prove that the theory is undecidable.

On the relation between choice and comprehension principles in second order arithmetic

On donne une nouvelle demonstration elementaire du theoreme de comparaison reliant les principes de choix et de comprehension en arithmetique du second ordre

Asymmetric Interpretations for Bounded Theories

We apply the method of asymmetric interpretation to the basic fragment of bounded arithmetic, endowed with a weak collection schema, and to a system of “feasible analysis”, introduced by Ferreira and based on weak Konigs lemma, recursive comprehension and NP-notation induction. As a byproduct, we obtain two conservation results. Mathematics Subject Classification: 03F30, 03F35.

Extending constructive operational set theory by impredicative principles

We study constructive set theories, which deal with (partial) operations applying both to sets and operations themselves. Our starting point is a fully explicit, finitely axiomatized system ESTE of constructive sets and operations, which was shown in 10 to be as strong as PA. In this paper we consider extensions with operations, which internally represent description operators, unbounded set quantifiers and local fixed point operators. We investigate the proof theoretic strength of the resulting systems, which turn out to be (except for the description operator) impredicative (being comparable with full second-order arithmetic and the second-order μ–calculus over arithmetic).

LEVELS OF IMPLICATION AND TYPE FREE THEORIES OF CLASSIFICATIONS WITH APPROXIMATION OPERATOR

We investigate a theory of Frege structures extended by the Myhill-Flagg hierarchy of implications. We study its relation to a property theory with an approximation operator and we give a proof theoretical analysis of the basic system involved. MSC: 03F35, 03D60.

Elementary Constructive Operational Set Theory

We introduce an operational set theory in the style of [5] and [17]. The theory we develop here is a theory of constructive sets and operations. One motivation behind constructive operational set theory is to merge a constructive notion of set ([1], [2]) with some aspects which are typical of explicit mathematics [14]. In particular, one has non-extensional operations (or rules) alongside extensional constructive sets. Operations are in general partial and a limited form of self–application is permitted. The system we introduce here is a fully explicit, finitely axiomatised system of constructive sets and operations, which is shown to be as strong as HA.

Remarks on applicative theories

Abstract We deal with applicative theories which are based on combinatory logic with total application, extensionality and natural numbers. We prove a conservative extension theorem and two consistency results, involving principles of uniformity, reflection, enumeration and choice.