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The consistency of the ramified Principia

Published online by Cambridge University Press:  12 March 2014

Frederic B. Fitch
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Extract

When the axioms of infinity and choice are added to the system of logic of the second edition of Whitehead and Russell's Principia mathematica, and when, as in the second edition, the axiom of reducibility is omitted (so that the ramified nature of Russell's theory of types, especially his distinction between “orders,” is not in effect obliterated), the resulting system will hereafter be referred to as “the ramified Principia.” It will be shown that a certain system S, slightly stronger than the ramified Principia, is consistent, so that the ramified Principia itself is consistent. From Gödel's theorem it will then follow that the sort of mathematical induction employed in the consistency proof of S cannot be adequately handled within S or within the ramified Principia.

The method of proving the consistency of S will be roughly as follows: A consistent non-constructive system S′ will be defined by means of induction with respect to a serial well-ordering of all the propositions of S. It will then be shown that every true proposition of S is a true proposition of S′, so that S must also be consistent.

1. Preliminary conventions.

1.1. Definition. By a “symbol” is to be understood any typographical expression which does not consist of a horizontal collinear sequence of expressions. Expressions formed by adding indices to either of the two upper or two lower corners of a symbol will also be considered symbols.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 3 , Issue 4: Including an Update to A Bibliography of Symbolic Logic , December 1938 , pp. 140 - 149
Copyright
Copyright © Association for Symbolic Logic 1938

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References

1 Gödel, K., Über formal unentscheidbare Sätze der Principia Mathemaiica und verwandter Systeme I, Monatshefte für Mathematik und Physik, vol. 38 (1931), pp. 173–198CrossRefGoogle Scholar.

2 This consistency proof is non-finitary. It may be compared with finitary consistency proofs of certain systems which do not have an axiom of infinity. See: Tarski, A., Einige Betrachtungen über die Begriffe der ω-Widerspruchsfreiheit und der ω-Vollständigkeit, Monatshefte für Mathematik und Physik, vol. 40 (1933), pp. 97–112CrossRefGoogle Scholar; Gentzen, G., Die Widerspruchsfreiheit der Stufenlogik, Mathematische Zeitschrift, vol. 41 (1936), pp. 357–366CrossRefGoogle Scholar; Beth, E. W., Une démonstration de la non-contradiction de la logique des types au point de vue fini, Nieuw archief voor unskunde, 2 s. vol. 19 nos. 1–2 (1936), pp. 59–62Google Scholar.

3 Professor Alonzo Church pointed out to the writer that the commutativity of order of precedence of universal quantification is deducible by use of 4.8 and need not be separately assumed.

4 Loc. cit.

5 A system of logistic, Cambridge, Mass., 1934Google Scholar; also Set-theoretic foundations for logic, this Journal, vol. 1 (1936), pp. 45–57Google Scholar, and Logic based on inclusion and abstraction, this Journal, vol. 2 (1937), pp. 145–152Google Scholar.

6 The logical syntax of language, New York and London, 1937Google Scholar.

7 Loc. cit.

8 Theorems 6.20 and 6.21, as here proved, were suggested to the writer by Professor Alonzo Church.

9 Loc. cit., Satz XI.

10 In other words, S remains consistent if the set of S-theorems is extended just enough to be closed with respect to “Carnap's rule,” in the sense of Rosser, , Gödel theorems for non-constructive logics, this Journal, vol. 2 (1937), pp. 129–137Google Scholar. Because of complications introduced by the ramified theory of types, it is not clear to the writer how Rosser's methods could be used in proving Gödel's theorem for S as thus extended.