Measurable cardinals and the cofinality of the symmetric group
Friedman, Sy-David ; Zdomskyy, Lyubomyr, 1983-
Assuming the existence of a ▫$P_2\kappa$▫-hypermeasurable cardinal, we construct a model of set theory with a measurable cardinal ▫$\kappa$▫ such that ▫$2^\kappa=\kappa^{++}$▫ and the group ...▫$\text{Sym}(\kappa)$▫ of all permutations of ▫$\kappa$▫ cannot be written as the union of a chain of proper subgroups of length ▫$<\kappa^{++}$▫. The proof involves iteration of a suitably defined uncountable version of the Miller forcing poset as well as the "tuning fork" argument introduced by the first author and K.Thompson [J. Symb. Log. 73, No. 3, 906-918 (2008)].
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