Truth and infinity

Author:
R. v. B. Rucker

Journal:
Proc. Amer. Math. Soc. **59** (1976), 138-143

MSC:
Primary 02K15; Secondary 02A05

DOI:
https://doi.org/10.1090/S0002-9939-1976-0424565-5

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 universe. We then make precise the extent to which unbound quantifiers can be taken to range only over ordinals in , 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.

**[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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DOI:
https://doi.org/10.1090/S0002-9939-1976-0424565-5

Keywords:
Gödel's conjecture,
axioms of infinity,
class of all sets,
reflection principle,
power-set axiom

Article copyright:
© Copyright 1976
American Mathematical Society