All uncountable cardinals can be singular
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.
Unable to display preview. Download preview PDF.
- 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