All uncountable cardinals can be singular
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Assuming the consistency of the existence of arbitrarily large strongly compact cardinals, we prove the consistency with ZF of the statement that every infinite set is a countable union of sets of smaller cardinality. Some other statements related to this one are investigated too.
KeywordsForce Condition Finite Subset Generic Subset Countable Union Regular Cardinal
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- 1.K. Devlin and R. Jensen,Marginalia to a theorem of Silver, in Proc. Internat. Summer Institute and Logic Colloq., Lecture Notes in Math.499, Springer-Verlag, Berlin, 1975, pp. 115–143.Google Scholar
- 2.J. Dodd and R. Jensen,The core model, handwritten notes.Google Scholar
- 5.T. J. Jech,The Axiom of Choice, North-Holland, Amsterdam, 1977.Google Scholar
- 6.A. Levy,Independence results in set theory by Cohen's Method IV (abstract), Notices Amer. Math. Soc.10 (1963), 592–593.Google Scholar
- 7.A. Levy,Definability in Axiomatic Set Theory II, inMath Logic and Formulations of Set Theory, Proc. Internat. Colloq. (J. Bar-Hillel, ed.), North-Holland, Amsterdam, 1970, pp. 129–145.Google Scholar
- 9.J. R. Shoenfield,Unramified forcing in ‘Axiomatic Set Theory’, Proc. Symposia in Pure Math.13 (D. Scott, ed.), Providence, Rhode Island, 1971, pp. 351–382.Google Scholar