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Interpretations of set theory and ordinal number theory

Published online by Cambridge University Press:  12 March 2014

Mariko Yasugi
Mariko Yasugi*
Affiliation:
University of Illinois
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Extract

In [3], Takeuti developed the theory of ordinal numbers (ON) and constructed a model of Zermelo-Fraenkel set theory (ZF), using the primitive recursive relation ∈ of ordinal numbers. He proved:

(1) If A is a ZF-provable formula, then its interpretation A0 in ON is ON-provable;

(2) Let B be a sentence of ordinal number theory. Then B is a theorem of ON if and only if the natural translation B* of B in set theory is a theorem of ZF;

(3) (V = L)° holds in ON.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 32 , Issue 2 , August 1967 , pp. 145 - 161
Copyright
Copyright © Association for Symbolic Logic 1967

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References

[1]Gödel, K., The consistency of the axiom of choice and of the generalized continuum hypothesis with the axiom of set theory, Princeton Univ. Press, Princeton, N.J., 1951.Google Scholar
[2]Lévy, A., The theory of transfinite effectivity, Technical report No. 12, The Hebrew University of Jerusalem, 1963, pp. 1–110.Google Scholar
[3]Takeuti, G., A formalization of the theory of ordinal numbers, this Journal, vol. 31 (1966), pp. 1–23.Google Scholar
[4]Takeuti, G., Recursive functions and arithmetical functions of ordinal numbers, Proceedings of the 1964 International Congress for Logic, Methodology and Philosophy of Science, pp. 179–196.Google Scholar