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

## Date

## Authors

## Journal Title

## Journal ISSN

## Volume Title

## Publisher

## 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.