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The generalized continuum hypothesis is equivalent to the generalized maximization principle1

  • Joel I. Friedman (a1)

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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]).

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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.

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[1]Bachmann, H., Transfinite Zahlen, Springer-Verlag, Berlin, 1955.
[2]Benacerraf, P. and Putnam, H., Philosophy of mathematics: selected readings, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1964.
[3]Cohen, P. J., The independence of the continuum hypothesis, Proceedings of the National Academy of Sciences. I, vol. 50 (1963), pp. 11431148; II, vol. 51 (1964), pp. 105–111.
[4]Cohen, P. J., Set theory and the continuum hypothesis, W. A. Benjamin, Inc., New York, 1966.
[5]Friedman, J. I., Proper classes as members of extended sets, Mathematischen Annalen, vol. 83 (1969), pp. 232240.
[6]Gödel, K., The consistency of the continuum hypothesis, Princeton Univ. Press, Princeton, 1940.
[7]Gödel, K., What is Cantor's continuum problem?, American Mathematical Monthly, vol. 54 (1947), pp. 515525 (revised in Benacerraf [2]).
[8]Montague, R. and Vaught, R. L., Natural models of set theories, Fundamenta Mathematicae, vol. XLVII (1959), pp. 219242.
[9]Shepherdson, J. C., Inner models for set theory, this Journal. II, vol. 17 (1952), pp. 225237.
[10]Takeuti, G., Hypotheses on power set, Proceedings of the 1967 Set Theory Conference, Los Angeles, American Mathematical Society, Providence, R.I., pp. 439446 (to appear).
[11]Tarski, A., Über unerreichbare Kardinalzahlen, Fundamenta Mathematicae, vol. XXX (1938), pp. 6889.
[12]van Heijenoort, J., From Frege to Gödel, Harvard Univ. Press, Cambridge, Mass., 1967.
[13]von Neumann, J., Eine Axiomatisierung der Mengenlehre, Journal für die reine und angewandte Mathematik, vol. 154 (1925), pp. 219240 (translated in [12]).
[14]von Neumann, J., Über eine Widerspruchsfreiheitsfrage in der axiomatischen Mengenlehre, Journal für die reine und angewandte Mathematik, vol. 160 (1929), pp. 227241.

The generalized continuum hypothesis is equivalent to the generalized maximization principle1

  • Joel I. Friedman (a1)

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