Published online by Cambridge University Press: 12 March 2014
We shall say that a logic is “simply consistent” if there is no formula A such that both A and ∼ A are provable. “ω-consistent” will be used in the sense of Gödel. “General recursive” and “primitive recursive” will be used in the sense of Kleene, so that what Gödel calls “rekursiv” will be called “primitive recursive.” By an “Entscheidungsverfahren” will be meant a general recursive function ϕ(n) such that, if n is the Gödel number of a provable formula, ϕ(n) = 0 and, if n is not the Gödel number of a provable formula, ϕ(n) = 1. In specifying that ϕ must be general recursive we are following Church in identifying “general recursiveness” and “effective calculability.”
First, a modification is made in Gödel's proofs of his theorems, Satz VI (Gödel, p. 187—this is the theorem which states that ω-consistency implies the existence of undecidable propositions) and Satz XI (Gödel, p. 196—this is the theorem which states that simple consistency implies that the formula which states simple consistency is not provable). The modifications of the proofs make these theorems hold for a much more general class of logics. Then, by sacrificing some generality, it is proved that simple consistency implies the existence of undecidable propositions (a strengthening of Gödel's Satz VI and Kleene's Theorem XIII) and that simple consistency implies the non-existence of an Entscheidungsverfahren (a strengthening of the result in the last paragraph of Church).
2 See p. 187 of Gödel, K., Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatshefte für Mathematik und Physik, vol. 38 (1931), pp. 173–198Google Scholar. We shall assume familiarity with this paper, to which we shall refer as “Gödel,” and with the paper to which we shall refer as “Kleene,” namely, Kleene, S. C., General recursive functions of natural numbers, Mathematische Annalen, vol. 112 (1936), pp. 727–742Google Scholar. The notations used throughout will be those used in these two papers.
3 Church, A., An unsolvable problem of elementary number theory, American journal of mathematics, vol. 58 (1936), pp. 345–363Google Scholar. Cf. p. 356. We shall refer to this paper as “Church”.
4 This lemma is a generalization of three lemmas which appeared in an earlier draft of this paper. Upon reading this earlier draft, S. C. Kleene suggested this lemma and furnished the proof of it which is given here.