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New foundation of formal metamathematics

Published online by Cambridge University Press:  12 March 2014

L. Chwistek  and
W. Hetper
L. Chwistek
Affiliation:
Instytut Logiki, Lwów
W. Hetper
Affiliation:
Instytut Logiki, Lwów
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Extract

Our method is based on the same principles which we have assumed in a paper about the foundations of metamathematics which we published in collaboration with J. Herzberg some years ago. Nevertheless we have introduced some important modifications which enable us to use a very restricted preliminary language and to avoid any artificial construction.

The problem of a restricted language in mathematics appears to be a very important one, as our ordinary language is really ambiguous and obscure. On the other hand there is the problem of restricting the intuitive logic to be used in our construction. We shall see that for this purpose we need only substitutions and modus ponens.

If we avoided any abbreviation, we could reduce our preliminary language to propositions of the following forms:

X is a theorem.

If P, then Q.

But we have to do with very long symbolic expressions and as there is not yet any machine to build up such expressions we must use abbreviations. For this purpose we shall use propositions of the following forms in the preliminary language:

X is an expression.

X is an abbreviation of Y.

Any proposition which does not belong to one of our four forms will be excluded from our construction.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 3 , Issue 1 , March 1938 , pp. 1 - 36
Copyright
Copyright © Association for Symbolic Logic 1938

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References

1 Chwistek, L., Hetper, W., and Herzberg, J., Fondements it la métamethématiqiie rationnelle, Bulletin de l'Académie Polonaise des Sciences et des Lettres, 1933Google Scholar.

2 Chwistek, L., Granice nauki, Lwów-Warszawa 1935Google Scholar.

3 Bentley, A. F., Linguistic analysis of mathematics, Bloomington, Ind., 1932Google Scholar.

4 Granice nauki.

5 See Hetper, W., Rola schematów niezależnych w budowie systemu semantyki, Archiwum Towarzystwa Naukowego we Lwowie, 1938Google Scholar.

6 In the directions of §§II, III, V, VI, the hypothesis, “If E is an expression, if F is an expression, …,” will be tacitly assumed.

7 The theorems 5.21, 5.22, 5.23 are used as axioms in Hetper's, Podstawy semantyki, Wiadomości matematyczne, vol. 43 (1936), pp. 57–86Google Scholar.

8 Hetper, W., Arytmetyka semantyczna, Comptes rendus des séances de la Société des Sciences et des Lettres de Varsovie, Classe III, vol. 27 (1934), pp. 9–26Google Scholar.

9 Loc. cit.

10 Arytmetyka semantyczna.

11 Chwistek, L., The theory of constructive types, Annales de la Société Polonaise de Mathématique, vol. 2 (1924), pp. 9–48, and vol. 3 (1925), pp. 92–141Google Scholar.

12 Cf. L. Chwistek, A formal proof of Gödel's theorem (to appear).