SIZE OF PIECES IN DECOMPOSITIONS INTO THE FIRST UNCOUNTABLE CARDINAL MANY PIECES

Abstract Within the determinacy setting, ${\mathscr {P}({\omega _1})}$ is regular (in the sense of cofinality) with respect to many known cardinalities and thus there is substantial evidence to support the conjecture that ${\mathscr {P}({\omega _1})}$ has globally regular cardinality. However, there is no known information about the regularity of ${\mathscr {P}(\omega _2)}$ . It is not known if ${\mathscr {P}(\omega _2)}$ is even $2$ -regular under any determinacy assumptions. The article will provide the following evidence that ${\mathscr {P}(\omega _2)}$ may possibly be ${\omega _1}$ -regular: Assume $\mathsf {AD}^+$ . If $\langle A_\alpha : \alpha < {\omega _1} \rangle $ is such that ${\mathscr {P}(\omega _2)} = \bigcup _{\alpha < {\omega _1}} A_\alpha $ , then there is an $\alpha < {\omega _1}$ so that $\neg (|A_\alpha | \leq |[\omega _2]^{<\omega _2}|)$ .