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GENERIC LARGE CARDINALS AS AXIOMS

Published online by Cambridge University Press:  14 May 2019

MONROE ESKEW
MONROE ESKEW*
Affiliation:
Kurt Gödel Research Center, University of Vienna
*
*KURT GÖDEL RESEARCH CENTER UNIVERSITY OF VIENNA WÄHRINGER STRASSE 25 1090 WIEN, AUSTRIA E-mail: [email protected]
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Abstract

We argue against Foreman’s proposal to settle the continuum hypothesis and other classical independent questions via the adoption of generic large cardinal axioms.

Keywords

03A0503E5503E65generic large cardinalscontinuum hypothesismutual inconsistency
Type
Research Article
Information
The Review of Symbolic Logic , Volume 13 , Issue 2 , June 2020 , pp. 375 - 387
Copyright
Copyright © Association for Symbolic Logic 2019 

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References

BIBLIOGRAPHY

Cantor, G. (1878). Ein Beitrag zur Mannigfaltigkeitslehre. Journal für die Reine und Angewandte Mathematik, 84, 242–258.Google Scholar
Cohen, P. (1963). The independence of the continuum hypothesis. Proceedings of the National Academy of Sciences of the United States of America, 50, 1143–1148.CrossRefGoogle ScholarPubMed
Cummings, J. (2010). Iterated forcing and elementary embeddings. In Foreman, M. and Kanamori, A., editors. Handbook of Set Theory, Vol. 2. Dordrecht: Springer, pp. 775–883.CrossRefGoogle Scholar
Eisworth, T. (2010). Successors of singular cardinals. In Foreman, M. and Kanamori, A., editors. Handbook of Set Theory, Vol. 2. Dordrecht: Springer, pp. 1229–1350.CrossRefGoogle Scholar
Eskew, M. (2015). Some mutually inconsistent generic large cardinals. Research Institute for Mathematical Sciences Kokyuroku, 1949, 23–33.Google Scholar
Eskew, M. (2016). Dense ideals and cardinal arithmetic. The Journal of Symbolic Logic, 81(3), 789–813.CrossRefGoogle Scholar
Fischer, A., Goldstern, M., Kellner, J., & Shelah, S. (2017). Creature forcing and five cardinal characteristics in Cichoń’s diagram. Archive for Mathematical Logic, 56(7–8), 1045–1103.CrossRefGoogle Scholar
Foreman, M. (1983). More saturated ideals. In Kechris, A. S., Martin, D. A., and Maschovakis, Y. N., editors. Cabal Seminar 79–81. Lecture Notes in Mathematics, Vol. 1019. Berlin: Springer-Verlag, pp. 1–27.CrossRefGoogle Scholar
Foreman, M. (1986). Potent axioms. Transactions of the American Mathematical Society, 294(1), 1–28.CrossRefGoogle Scholar
Foreman, M. (1998). An ${\aleph _1}$-dense ideal on ${\aleph _2}$.. Israel Journal of Mathematics, 108, 253–290.CrossRefGoogle Scholar
Foreman, M. (1998). Generic large cardinals: New axioms for mathematics? Documenta Mathematica, II, 11–21.Google Scholar
Foreman, M. (2006). Has the continuum hypothesis been settled? Lecture Notes in Logic, 24, 56–75.Google Scholar
Foreman, M. (2010). Ideals and generic elementary embeddings. In Foreman, M. and Kanamori, A., editors. Handbook of Set Theory, Vol. 2. Dordrecht: Springer, pp. 885–1147.CrossRefGoogle Scholar
Foreman, M. & Magidor, M. (1995). Large cardinals and definable counterexamples to the continuum hypothesis. Annals of Pure and Applied Logic, 76(1), 47–97.CrossRefGoogle Scholar
Gitik, M. and Shelah, S. (1997). Less saturated ideals. Proceedings of the American Mathematical Society, 125(5), 1523–1530.CrossRefGoogle Scholar
Gödel, K. (1940). The Consistency of the Continuum Hypothesis. Annals of Mathematics Studies, Vol. 3. Princeton, NJ: Princeton University Press.CrossRefGoogle Scholar
Gödel, K. (1947). What is Cantor’s continuum problem? American Mathematical Monthly, 54(9), 515–525.CrossRefGoogle Scholar
Hamkins, J. D. (2003). Extensions with the approximation and cover properties have no new large cardinals. Fundamenta Mathematicae, 180(3), 257–277.CrossRefGoogle Scholar
Hilbert, D. (1902). Mathematical problems. Bulletin of the American Mathematical Society, 8(10), 437–479.CrossRefGoogle Scholar
Jech, T. (1972/73). Some combinatorial problems concerning uncountable cardinals. Annals of Mathematical Logic, 5, 165–198.CrossRefGoogle Scholar
Kanamori, A. (2003). The Higher Infinite. Berlin: Springer-Verlag.Google Scholar
Lévy, A. & Solovay, R. (1967). Measurable cardinals and the continuum hypothesis. Israel Journal of Mathematics, 5, 234–248.CrossRefGoogle Scholar
Maddy, P. (1988). Believing the axioms I. Journal of Symbolic Logic, 53(2), 481–511.CrossRefGoogle Scholar
Sakai, H. (2005). Semiproper ideals. Fundamenta Mathematicae, 186(5), 251–267.CrossRefGoogle Scholar
Solovay, R., Reinhardt, W., & Kanamori, A. (1978). Strong axioms of infinity and elementary embeddings. Annals of Mathematical Logic, 13(1), 73–116.CrossRefGoogle Scholar
Specker, E. (1949). Sur un problème de Sikorski. Colloquium Mathematicum, 2, 9–12.CrossRefGoogle Scholar
Usuba, T. (2014). The approximation property and the chain condition. Research Institute for Mathematical Sciences Kokyuroku, 1895, 103–107.Google Scholar
Woodin, W. H. (2010). The Axiom of Determinacy, Forcing Axioms, and the Non-Stationary Ideal. Second revised edition. De Gruyter Series in Logic and its Applications, Vol. 1. Berlin: Walter de Gruyter GmbH & Co. KG.CrossRefGoogle Scholar
Zermelo, E. (1904). Beweis, daß jede Menge wohlgeordnet werden kann. Mathematische Annalen, 59(4), 514–516.CrossRefGoogle Scholar