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Gödel Theorems for Non-Constructive Logics

Published online by Cambridge University Press:  12 March 2014

Barkley Rosser
Barkley Rosser*
Affiliation:
Cornell University
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Extract

Introduction. In setting up his definition of analyticity, Carnap uses various non-constructive rules, of which we shall be concerned with only one. This one we shall call simply “Carnap's rule.” It can be roughly stated thus: If f(0), f(1), f(2), … are all provable, then (x)f(x) shall be provable.

In this paper will be briefly considered the logics got by starting with the system of Principia plus Peano's axioms and allowing one, two, …, ω, ω+l, and so on up to, but not including ω uses of Carnap's rule (interspersed with uses of the ordinary rules), as well as the system which includes Principia and Peano's axioms and is closed under application of Carnap's rule and the ordinary rules. Under suitable assumptions as to consistency, it is shown that in each of these logics there occur undecidable propositions and that a formula which states the consistency of the logic exists in the logic but is not provable in the logic. An interesting side result is that the logic got by allowing co applications of Carnap's rule is not closed under Carnap's rule.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 2 , Issue 3 , September 1937 , pp. 129 - 137
Copyright
Copyright © Association for Symbolic Logic 1937

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References

2 Carnap, R., Ein Gültigkeitskriterium für die Sätze der klassischen Mathematik, Monatshefte für Mathematik und Physik, vol. 42 (1935), pp. 163–190CrossRefGoogle Scholar. See p. 173, bottom.

3 This same rule appeared in a paper by Hilbert, , Mathematische Annalen, vol. 104 (1930–1931), p. 491, lines 13–18. However, Hilbert did practically nothing with the ruleGoogle Scholar.

4 The notations in this paper shall be same as those in the paper by Gödel, K., Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I, Monatshefte für Mathematik und Physik, vol. 38 (1931), pp. 173–198CrossRefGoogle Scholar. We shall refer to this paper as “Gödel.”

5 In the sense of Kleene, S. C., General recursive functions of natural numbers, Mathematische Annalen, vol. 112 (1936), pp. 727–742. Cf. p. 729CrossRefGoogle Scholar. Under this interpretation, “primitive recursive” means identically the same thing as Gödel's “rekursiv.”

6 We adopt, with obvious amplifications, Gödel's convention that “x is provable” shall mean “the formula with the number x is provable.”

7 Here, as elsewhere throughout this paper, we use Kleene's modified definition of n Gl x (Kleene, p. 732 of the paper cited in footnote 5) instead of Gödel's original definition (Gödel, p. 182).

8 is inserted here because its presence facilitates the proof of Thm. 2.

9 In the sense of Rosser, B., Extensions of some theorems of Gödel and Church, this Journal, vol. 1 (1936), pp. 87–91Google Scholar. Cf. the first paragraph. We shall refer to this paper as “Rosser.”

10 In the sense defined in Rosser, second paragraph after Lemma I, Corollary I.

11 In the spring of 1935, Kleene suggested rules of this type to the author. Kleene also sketched the outlines of a proof which he had that if a logic is ω-consistent and strong enough for a certain amount of number theory, then the addition of a rule of Kleene's type will enable one to prove formulas not provable in the original logic. For certain special logics this theorem is proved in this paper by a totally different method and with simple consistency assumed instead of ω-consistency (cf. the corollaries to Thm.13, Thm.14, and Thm.16).