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On formulas of one variable in intuitionistic propositional calculus1

Published online by Cambridge University Press:  12 March 2014

Iwao Nishimura
Iwao Nishimura*
Affiliation:
Gifu University, Gifu, Japan
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Extract

McKinsey and Tarski [3] described Gödel's proof that the number of Brouwerian-algebraic functions is infinite. They gave an example of a sequence of infinitely many distinct Brouwerian-algebraic functions of one argument, which means that there are infinitely many non-equivalent formulas of one variable in the intuitionistic propositional calculus LJ of Gentzen [1]. However they did not completely characterize such formulas. In § 1 of this note, we define a sequence of basic formulas P∞(X), P0(X), P1(X), … and prove the following theorems.

Type
Articles
Information
The Journal of Symbolic Logic , Volume 25 , Issue 4 , December 1960 , pp. 327 - 331
Copyright
Copyright © Association for Symbolic Logic 1962

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Footnotes

1

I wish to express my hearty thanks to Prof. K. Ono and Mr. T. Umezawa for their valuable suggestions and criticisms.

References

[1]Gentzen, G., Untersuchungen über das logische Schliessen, Mathematische Zeitschrift, vol. 39. (1934–1935), pp. 176–210, 445–431.CrossRefGoogle Scholar
[2] K. Gödel, Zum intuitionistischen Aussagenkalkül, Ergebnisse eines mathematischen Kolloquiums, Heft 4 (1931–1932, 1933), p. 40.Google Scholar
[3]McKinsey, J. C. C. and Tarski, A., On closed elements in closure algebra, Annals of Mathematics, vol. 47 (1946), pp. 122–162.CrossRefGoogle Scholar
[4]Mostowski, A., Proofs on non-deducibility in intuitionistic functional calculus, this Journal, vol. 13 (1948), pp. 204–207.Google Scholar
[5]Umezawa, T., Über die Zwischensysteme der Aussagenlogik, Nagoya Mathematical Journal, vol. 9 (1955), pp. 181–187.CrossRefGoogle Scholar
[6]Umezawa, T., On intermediate prepositional logics, this Journal, vol. 24 (1959), pp. 20–36.Google Scholar