The cardinality of reduced power set algebras

Abstract:

We prove a general result on the cardinality of reduced powers of structures via filters that has several consequences including the following: if $I$ is a uniform, countably complete ideal on the real line $\mathcal {R}$ and $\mathcal {B}$ is the Boolean algebra of subsets of $\mathcal {R}$ modulo $I$, then $\left | \mathcal {B} \right | > {2^{{\aleph _0}}}$ and if ${2^\nu } \leq {2^{{\aleph _0}}}$ for all $\nu < {2^{{\aleph _0}}}$ then $\left | \mathcal {B} \right | = {2^{{2^{{\aleph _0}}}}}$. This strengthens some results of Kunen and Pelc [7] and Prikry [8] obtained by Boolean ultrapower techniques. Our arguments are all combinatorial and some applications are included.