number: uncountably infinite of some sort or infinite in the sense of satisfying the Generalized Continuum Hypothesis.

An infinite cardinality which is not countable (aleph-0); if the Generalized Continuum Hypothesis (GCH) is rejected, meaning that the only infinite cardinalities are those of aleph-n form (for nonnegative integer n), then this word would incorporate all uncountable infinities (aleph-m for any strictly positive integer m); else, it incorporates those and any infinity which is strictly larger than aleph-0.