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The generalized continuum hypothesis is equivalent to the generalized maximization principle1

Published online by Cambridge University Press:  12 March 2014

Joel I. Friedman
Joel I. Friedman*
Affiliation:
University of California, Davis, Davis, California95616
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Extract

In spite of the work of Gödel and Cohen, which showed the undecidability of the Generalized Continuum Hypothesis (GCH) from the axioms of set theory, the problem still remains to decide GCH on the basis of new axioms. It is almost 100 years since Cantor first conjectured the Continuum Hypothesis, yet we seem to be no closer to determining its truth (or falsity). Nevertheless, it is a sound methodological principle that given any undecidable set-theoretical statement, we should search for “other (hitherto unknown) axioms of set theory which a more profound understanding of the concepts underlying logic and mathematics would enable us to recognize as implied by these concepts” (see Gödel [7, p. 265]).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 36 , Issue 1 , March 1971 , pp. 39 - 54
Copyright
Copyright © Association for Symbolic Logic 1971

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Footnotes

1

A major part of the work on this paper was done in the summer of 1968 under a Faculty Fellowship of the University of California, Davis. Also, the author is greatly indebted to the referee for motivating a complete revision and expansion of a previous version of this paper and for making the following contributions: (i) suggesting a simpler definition of “local universe”. The author previously defined a local universe as a transitive class distinct from Zermelo's ωz and closed under union and replacement, (ii) streamlining the proof of (GCH ↔ GMP), especially by showing that the characterization of the local universes given in L3.2 in no way depends on GMP or on their being maximized, (iii) inventing L3.9, which is the key lemma in showing T3.10, MTII, and MTVII (previously, the author proved T3.10 without using L3.9, but the proof was much less elegant), (iv) suggesting the kind of result that led to MTII and MTVII, (v) suggesting a simplification which helped to simplify the proof of MTIV (b), and (vi) simplifying considerably the Zermelo-Shepherdson type result given in MTV, as well as its proof, by showing that again it in no way depends on GMP or MP.

References

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