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Undecidable sentences generated by semantic paradoxes

Published online by Cambridge University Press:  12 March 2014

Hao Wang
Hao Wang*
Affiliation:
Burroughs Research Center, Paoli, Pennsylvania
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Extract

In applying the method of arithmetization to a proof of the completeness of the predicate calculus, Bernays has obtained a result which, when applied to set theories formulated in the predicate calculus, may be stated thus.

1.1. By adding an arithmetic sentence Con(S) (expressing the consistency of a set theory S) as a new axiom to the elementary number theory Zμ (HB II, p. 293), we can prove in the resulting system arithmetic translations of all theorems of S.

It then follows that things definable or expressible in S have images in a simple extension of Zμ, if S is consistent. Since S can be a “strong” system, this fact has interesting consequences. Some of these are discussed by me and some are discussed by Kreisel. Kreisel finds an undecidable sentence of set theory by combining 1.1 and the Cantor diagonal argument. I shall prove below, using similar methods, a few further results, concerned with the notions of truth and designation. The method of numbering sets which I use (see 3.1 below) is different from Kreisel's. While the method used here is formally more elegant, Kreisel's method is much more efficient if we wish actually to calculate the numerical values.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 20 , Issue 1 , March 1955 , pp. 31 - 43
Copyright
Copyright © Association for Symbolic Logic 1955

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References

REFERENCES

[1]Bernays, P., A system of axiomatic set theory, Part II, this Journal, vol. 6 (1941), pp. 1–17.Google Scholar
[2]Hilbert, D. and Bernays, P., Grundlagen der Mathematik, vol. 2, Berlin (Springer) 1939, xii + 498 pp.Google Scholar
[3]Kreisel, G., Note on arithmetic models lor consistent formulae of the predicate calculus, Part I, Fundamenta Mathematicae, vol. 37 (for 1950, pub. 1951), pp. 265–285; Part II, Proceedings of the XIth International Congress of Philosophy (Brussels, August 20-26, 1953), vol. 14, 1953, pp. 39–49.CrossRefGoogle Scholar
[4]Kreisel, G., Some concepts concerning formal systems of number theory, Mathematische Zeitschrift, vol. 57 (1952), pp. 1–12.CrossRefGoogle Scholar
[5]Wang, H., Arithmetic translations of axiom systems, Transactions of the American Mathematical Society, vol. 71 (1951), pp. 283–293.CrossRefGoogle Scholar
[6]Wang, H., Arithmetic models of formal systems, Methodos, vol. 3 (1951), pp. 217–232.Google Scholar
[7]Wang, H., Truth definitions and consistency proofs, Transactions of American Mathematical Society, vol. 73 (1952), pp. 243–275.CrossRefGoogle Scholar