Measurable cardinals and the cofinality of the symmetric group
Volume 207 / 2010
Fundamenta Mathematicae 207 (2010), 101-122
MSC: Primary 03E35; Secondary 03E55, 03E99.
DOI: 10.4064/fm207-2-1
Abstract
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 ${\it Sym}(\kappa)$ 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. Symbolic Logic 73 (2008)].
