Solution of a problem of Leon Henkin

Abstract

If Σ is any standard formal system adequate for recursive number theory, a formula (having a certain integer q as its Gödel number) can be constructed which expresses the proposition that the formula with Gödel number q is provable in Σ. Is this formula provable or independent in Σ? [2]. One approach to this problem is discussed by Kreisel in [4]. However, he still leaves open the question whether the formula ( Ex ) ( x, a ), with Gödel-number a, is provable or not. Here ( x, y ) is the number-theoretic predicate which expresses the proposition that x is the number of a formal proof of the formula with Gödel-number y . In this note we present a solution of the previous problem with respect to the system Z μ [3] pp. 289–294, and, more generally, with respect to any system whose set of theorems is closed under the rules of inference of the first order predicate calculus, and satisfies the subsequent five conditions, and in which the function ( k, l ) used below is definable. The notation and terminology is in the main that of [3] pp. 306–326, viz. if is a formula of Z μ containing no free variables, whose Gödel number is a, then ({ }) stands for ( Ex ) ( x, a ) (read: the formula with Gödel number a is provable in Z μ ); if is a formula of Z μ containing a free variable, y say, ({ }) stands for ( Ex ) ( x, g ( y )}, where g ( y ) is a recursive function such that for an arbitrary numeral the value of g ( ) is the Gödel number of the formula obtained from by substituting for y in throughout. We shall, however, depart trivially from [3] in writing ( ), where is an arbitrary numeral, for ( Ex ) { x , ).