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Interpretations of Kleene's metamathematical predicate Γ ∣ a in intuitionistic arithmetic1

Published online by Cambridge University Press:  12 March 2014

T. Thacher Robinson
T. Thacher Robinson*
Affiliation:
University of Illinois
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Extract

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).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 30 , Issue 2 , June 1965 , pp. 140 - 154
Copyright
Copyright © Association for Symbolic Logic 1965

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Footnotes

1

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.

References

REFERENCES

[1]Church, A., Introduction to mathematical logic, Princeton University Press, Princeton, N. J., 1956, X + 376 pp.Google Scholar
[2]Harrop, R., Concerning formulas of the types A → B ∨ C, A → (Ex)B(x) in intuitionistic formal systems, this Journal, vol. 25 (1960), pp. 27–32.Google Scholar
[3]Kleene, S. C., Introduction to metamathematics, Amsterdam (North Holland), Groningen (Noordhoff), New York and Toronto (Van Nostrand), 1952, X + 550 pp.Google Scholar
[4]Kleene, S. C., Disjunction and existence under implication in elementary intuitionistic formalisms, this Journal, vol. 27 (1962), pp. 11–18.Google Scholar
[5]Robinson, T. T., Disjunction under implication in the intuitionistic predicate calculus Pd, American Mathematical Society Notices, vol. 10, no. 2, issue 66, p. 197.Google Scholar
[6]Robinson, T. T., Interpretations of Kleene's metamathematical predicate Γ ∣ A in intuitionistic arithmetic N, Dissertation, Princeton University, 1963.Google Scholar