Some results on the length of proofs

ABSTRACT. Given a theory T, let \-^A mean "A has a proof in T of at most k lines". We consider a formulation PA* of Peano arithmetic withfull induction but addition and multiplication being ternary relations. We showthat \-k A is decidable for PA* and hence PA* is closed under a weak enrule. Ananalogue of Godel's theorem on the length of proofs is an easy corollary. 1. Introduction. In this paper we shall consider questions regarding thelengthOjof proofs. Now the length of (the shortest) proof of a given formulain a formal system depends strongly on the way in which the system is pre-sented. E.g. adjoining one of the theorems as an axiom reduces the lengthof some proofs. Thus in order to get significant results, we have either toconfine ourselves to particular formalisations of particular theories or elseto tormulate a criterion which distinguishes "nice" and "not so nice"formalisations of the same theory. We shall take here the second approach.In particular we shall consider theories formalised in some language of thelower predicate calculus by means of a finite number of axioms, axiomschemata and schematic rules of inference. Formalisations of this kind willinclude Hilbert type and Gentzen type formalisations of classical and in-tuitionistic arithmetic.2. Schematic systems. Since axiom schemata and rules of inference aregenerally explained in the literature with the help of formula variables, inorder to define the notion of a schematic system we do the obvious, namely,we expand the notation of the predicate calculus to include metamathematicalsymbols and emphasize substitution as the central idea. Precise details