Some structural similarities between uncountable sets, powersets and the universe

Abstract

We establish some similarities/analogies between uncountable cardinals or powersets and the class $V$ of all sets. They concern mainly the Boolean algebras ${\cal P}(\kappa)$, for a regular cardinal $\kappa$, and ${\cal C}(V)$ (the class of subclasses of the universe $V$), endowed with some ideals, especially the ideal $[\kappa]^{<\kappa}$ for ${\cal P}(\kappa)$, and the ideal of sets $V$ for ${\cal C}(V)$.