A Proof of the Cut-Elimination Theorem in Simple Type Theory

In [4], I introduced a quasi-Boolean algebra, and showed that in a formal system of simple type theory, from which the cut rule is omitted, wffs form a quasi-Boolean algebra, and that the cut-elimination theorem can be formulated in algebraic language. In this paper we use the result of [4] to prove the cut-elimination theorem in simple type theory. The theorem was proved by M. Takahashi [2] in 1967 by using the concept of Schutte's semivaluation. We use maximal ideals of a quasi-Boolean algebra instead of semivaluations. The logical system we are concerned with is a modification of Schutte's formal system of simple type theory in [1] into Gentzen style. Inductive definition of types . 0 and 1 are types. If τ 1 , …, τ n are types, then ( τ 1 , …, τ n ) is a type. Basic symbols . a 1 τ , a 2 τ , … for free variables of type τ . x 1 τ , x 2 τ , … for bound variables of type τ . An arbitrary number of constants of certain types. An arbitrary number of function symbols with certain argument places.