Solution of a Problem of Leon Henkin

If Σ is any standard formal system adequate for recursive number theory, a formula (having a certain integer q as its Godel number) can be constructed which expresses the proposition that the formula with Godel 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 Godel-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 Godel-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 Godel number is a, then ({ }) stands for ( Ex ) ( x, a ) (read: the formula with Godel 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 Godel 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 , ).