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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Truth and infinity

Author: R. v. B. Rucker
Journal: Proc. Amer. Math. Soc. 59 (1976), 138-143
MSC: Primary 02K15; Secondary 02A05
MathSciNet review: 0424565
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Abstract: This paper formalizes Gödel's 1946 conjecture that every set-theoretic sentence is decidable from the present axioms plus some true axioms of infinity; and we prove a weak variant of this conjecture to be true for every $ {\text{ZF}}$ universe. We then make precise the extent to which unbound quantifiers can be taken to range only over ordinals in $ {\text{ZF}}$, obtaining a sort of normal-form theorem. The last section relates these results to the problem of how wide the class of all sets is.

References [Enhancements On Off] (What's this?)

  • [1] K. Gödel, Remarks before the Princeton Bicentennial Conference on Problems in Mathematics, 1946, The Undecidable, M. Davis (editor), Raven Press, Hewlett, N.Y., 1965, pp. 84-88.
  • [2] R. Rucker, Truth and infinity, Notices Amer. Math. Soc. 20 (1973), A-444. Abstract #73TE45.
  • [3] -, On Cantor's continuum problem (to appear).
  • [4] G. Takeuti, Formalization principle, Logic, Methodology and Philos. Sci. III (Proc. Third Internat. Congr., Amsterdam, 1967) North-Holland, Amsterdam, 1968, pp. 105–118. MR 0252220

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Keywords: Gödel's conjecture, axioms of infinity, class of all sets, reflection principle, power-set axiom
Article copyright: © Copyright 1976 American Mathematical Society

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