Tableau systems for first order number theory and certain higher order theories

Date

1970

Journal Title

Journal ISSN

Volume Title

Publisher

ProQuest Dissertations & Theses

YU Faculty Profile

Abstract

This work will examine various topics in higher order logic and proof theory ·from the point of view of tableau systems similar to those developed by Smuilyan in [1]. __Chapter I presents constructive consistency proofs for first: order number theory that are closely related to those of Gentzen [l] ancl Schütte [2]. The development follows that of Schütte. __Chapter II considers topics in pure second order logic and the theory, with an emphasis on the former for the sake of convenience. Here no constructive consistency proofs arc known. I have found, however, a theorem similar to the Hauptsatz hut with a more elegant classical proof, a constructive proof of which would yield a constructive consistency proof for the formal systems discussed. (On the other hand there is at present no reason to assume that this theorem would be likely to have a constructive proof that would be simpler than one for a Hauptsstz.) The higher order logics are considered briefly within the context of a generalized abstract framework similar to those considered by Smullyan in [2]. In particular, a Henkin completeness proof is given which is simultaneously a completeness proof for first-order logic, the usual higher order logics and type theory. __Chapter III completes the proof-theoretic treatment of systems equivalent to those considered by Schütte in [2] that was begun in chapter I. __The first appendix explores further the constructivity of the constructive cut-elimination proof for first order logic. It shows that when we eliminate cuts from a first order proof, we form a new proof which preserves the "arguments" of the first proof although these arguments may be intertwined and some may be deleted, The second appendix illustrates translation procedures for going from a proof in a Schütte system to one in a (Smullyan) tableau system and vice-versa. Such a procedure is presented only for· the first order systems since the modifications for higher order systems ·are easily made.

Description

Doctoral dissertation, PhD / YU only

Keywords

Mathematics.

Citation

Walker, S. A. (1970). Tableau systems for first order number theory and certain higher order theories (Publication No.302559759) [Doctoral dissertation, Yeshiva University]. Source: Dissertation Abstracts International, Volume: 31-04, Section: B, page: 2137.