Measurable cardinals and the cofinality of the symmetric group

Sy-David Friedman ¹   ; Lyubomyr Zdomskyy ¹

¹ Kurt Gödel Research Center for Mathematical Logic University of Vienna Währinger Strasse 25 A-1090 Wien, Austria

Fundamenta Mathematicae, Tome 207 (2010) no. 2, pp. 101-122

Résumé

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)].

Détail

DOI : 10.4064/fm207-2-1

Keywords: assuming existence kappa hypermeasurable cardinal construct model set theory measurable cardinal kappa kappa kappa group sym kappa permutations kappa cannot written union chain proper subgroups length kappa proof involves iteration suitably defined uncountable version miller forcing poset tuning fork argument introduced first author nbsp thompson nbsp symbolic logic