Abstract
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.
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References
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.
J. Dodd and R. Jensen,The core model, handwritten notes.
U. Felgner,Comparison of the axioms of local and universal choice, Fund. Math.71 (1971), 73–62.
T. J. Jech,Lectures in set theory with particular emphasis on the method of forcing, Lecture Notes in Mathematics217, Springer-Verlag, Berlin, 1971, p. 137.
T. J. Jech,The Axiom of Choice, North-Holland, Amsterdam, 1977.
A. Levy,Independence results in set theory by Cohen's Method IV (abstract), Notices Amer. Math. Soc.10 (1963), 592–593.
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.
K. Prikry,Changing measurable, Dissertationes Math.68 (1970), 5–52.
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.
E. Speker,Zur Axiomatik der Mengenlehre, Z. Math. Logik Grundlagen Math.3 (1957), 173–210.
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Gitik, M. All uncountable cardinals can be singular. Israel J. Math. 35, 61–88 (1980). https://doi.org/10.1007/BF02760939
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DOI: https://doi.org/10.1007/BF02760939