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A generalization of Gödel's notion of constructibility1

Published online by Cambridge University Press:  12 March 2014

Azriel Lévy
Azriel Lévy*
Affiliation:
Massachusetts Institute of Technology and Hebrew University, Jerusalem
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Extract

The notion of constructibility introduced by Gödel in [2] has been generalized by Hajnal [3], [4] in order to prove the conditional independence of the generalized continuum hypothesis and related axioms. A very similar construction was used independently by Shoenfield [11], [12]2 and the author [7] to prove the conditional independence of V = L and related axioms. Here we shall prove results further in the latter direction than those in Shoenfield [12].

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 25 , Issue 2 , June 1960 , pp. 147 - 155
Copyright
Copyright © Association for Symbolic Logic 1960

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Footnotes

1

This paper forms part of the author's Ph. D. thesis submitted to the Hebrew University. The author wishes to express his gratitude to Prof. A. A. Fraenkel and Prof. A. Robinson for their guidance and kind encouragement. The present version was written while the author was a Sloan Fellow of the School for Advanced Study at the Massachusetts Institute of Technology.

References

[1]Church, A. and Kleene, S. C., Formal definitions in the theory of ordinal numbers, Fundamenta mathematicae, vol. 28 (1937), pp. 11–22.CrossRefGoogle Scholar
[2]Gödel, K., The consistency of the axiom of choice and of the generalized continuum hypothesis with the axioms of set theory, second printing, Princeton (Princeton University Press), 1951.Google Scholar
[3]Hajnal, A., On a consistency theorem connected with the generalized continuum problem (abstract), Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 2 (1956), pp. 131–136.CrossRefGoogle Scholar
[4]Hajnal, A., On a consistency theorem connected with the generalized continuum pvoblem, to appear in Acta mathematica Academiae Scientiarum Hungaricae.Google Scholar
[5]Kreider, D. L., On sets in both two-function-quantifier forms (abstract), Notices of the American Mathematical Society, vol. 6 (1958), p. 364.Google Scholar
[6]Kreider, D. L. and Rogers, H. Jr., Analytic predicates and extensions of the notion of constructive ordinals, to appear.Google Scholar
[7]Lévy, A., Indépendance conditionnelle de V = L et d'axiomes qui se rattachent au système de M. Gödel, Comptes Rendus des séances de l'Académie des Sciences, Paris, vol. 245 (1957), pp. 1582–1583.Google Scholar
[8]Lévy, A., Contributions to the metamathematics of set theory (Hebrew with English summary), Ph. D. thesis, Hebrew University, Jerusalem, 1958.Google Scholar
[9]Robinson, R. M., On finite sequences of classes, this Journal, vol. 10 (1945), pp. 125–126.Google Scholar
[10]Shepherdson, J. C., Inner models for set theory, Part I, this Journal, vol. 16 (1951), pp. 161–190.Google Scholar
[11]Shoenfield, J. R., Constructible sets (abstract), Summaries of talks presented at the Summer Institute of Symbolic Logic in 1957 at Cornell University (mimeographed), vol. II, p. 214.Google Scholar
[12]Shoenfield, J. R., On the independence of the axiom of constructibility. American journal of mathematics, vol. 81 (1959), pp. 537–540.CrossRefGoogle Scholar