Semantics for Intuitionistic Arithmetic Based on Tarski Games with Retractable Moves

Abstract. We define an effective, sound and complete game semantics for HAinf, Intuitionistic Arithmetic with ω-rule. Our semantics is equivalent to the original semantics proposed by Lorentzen [6], but it is based on the more recent notions of ”backtracking ” ([5], [2]) and of isomorphism between proofs and strategies ([8]). We prove that winning strategies in our game semantics are tree-isomorphic to the set of proofs of some variant of HAinf, and that they are a sound and complete interpretation of HAinf. 1 Why game semantics of Intuitionistic Arithmetic? In [7], S.Hayashi proposed the use of an effective game semantics in his Proof Animation project. The goal of the project is ”animating” (turning into algorithms) proofs of program specifications, in order to find bugs in the way a specification is formalized. Proofs are formalized in classical Arithmetic, and the method chosen for “animating ” proofs i

Ten Misconceptions from the History of Analysis and Their Debunking

The widespread idea that infinitesimals were "eliminated" by the "great triumvirate" of Cantor, Dedekind, and Weierstrass is refuted by an uninterrupted chain of work on infinitesimal-enriched number systems. The elimination claim is an oversimplification created by triumvirate followers, who tend to view the history of analysis as a pre-ordained march toward the radiant future of Weierstrassian epsilontics. In the present text, we document distortions of the history of analysis stemming from the triumvirate ideology of ontological minimalism, which identified the continuum with a single number system. Such anachronistic distortions characterize the received interpretation of Stevin, Leibniz, d'Alembert, Cauchy, and others.Comment: 46 pages, 4 figures; Foundations of Science (2012). arXiv admin note: text overlap with arXiv:1108.2885 and arXiv:1110.545

In the beginning was game semantics

This article presents an overview of computability logic -- the game-semantically constructed logic of interactive computational tasks and resources. There is only one non-overview, technical section in it, devoted to a proof of the soundness of affine logic with respect to the semantics of computability logic. A comprehensive online source on the subject can be found at http://www.cis.upenn.edu/~giorgi/cl.htmlComment: To appear in: "Games: Unifying Logic, Language and Philosophy". O. Majer, A.-V. Pietarinen and T. Tulenheimo, eds. Springer Verlag, Berli