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 Zμ [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 Zμ 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 Zμ); if is a formula of Zμ 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, ).

Reflection principles and provability algebras in formal arithmetic

“…The following generalization of Gödel's famous second incompleteness theorem is known as Löb's theorem [55]. Because of its importance for the present paper we present a short proof.…”

Interpretability suprema in Peano Arithmetic

“…The direction from left to right follows by the fact that the Hilbert-Bernays-Löb derivability conditions for Pr T are verifiable in IΔ 0 +exp [11,18]. The proof of the other direction with T = PA is due to Solovay [27].…”

“…The system GL of propositional provability logic contains a unary modality , interpreted as formalized provability in PA. It follows from the work of Hilbert and Bernays [11], Löb [18], and Solovay [27], that the theorems of GL are exactly the propositional schemata involving formalized provability that are provable in PA.…”

Self-Reference and Validity Revisited

“…13 See Geach 1954-5 andLöb 1955. In the words of my colleague Roger Squires, sentence γ is full-proof.…”