Published online by Cambridge University Press: 12 March 2014
Let Pp, Pd, and N be the intuitionistic systems of prepositional calculus, predicate calculus, and elementary arithmetic, respectively, described in [3].
Kleene [4] introduces a metamathematical predicate Γ ∣ A for each of the systems Pp, Pd, and N, where Γ ranges over finite sequences of wffs, and A ranges over wffs, of that system. In the case of N, if Γ is consistent, then ‘ Γ ∣ A’ is essentially the result of deleting all references to recursive functions from the metamathematical predicate ‘A is realizable-(Γ ⊦)’ described in [3], pp. 502–503.
Through use of this predicate, Kleene [4] obtains elegant constructive proofs of the following results for N:
Metatheorem 0.1. If B ∨ C is a closed theorem of N, then ⊦ B or ⊦ C.
Metatheorem 0.2. If (∃a)D(a) is a closed theorem of N, then there is a numeral n such that ⊦D(n).
Metatheorem 0.3. If A is a closed wff of N, then A ∣ A is a necessary and sufficient condition that, for all closed B, C, (∃a)D(a) inN:
(0.3.1) ⊦ A ⊃ B ∨ C implies ⊦ A ⊃ B or ⊦A ⊃ C
and
(0.3.2) ⊦A ⊃ (∃)D(a) implies there is a numeral n such that ⊦ A ⊃ D(n).
This paper presents in a condensed form the principal results of the author's doctoral dissertation [6]. The author is indebted to Alonzo Church for the suggestion to use realizability of a formalization in N of Γ ∣ A as the characterization of the intuitionistic validity of Γ ∣ A; and to S. C. Kleene for bringing [4] to the author's attention before its publication. Appreciation is also due to the referee for several suggestions which have helped to simplify the original exposition to its present form; and for a careful reading of the manuscript which uncovered certain minor errors.
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