First, let me clear up a misunderstanding. Question: Does 2 ω = ℵ 1 ? More generally, does 2 ℵ α = ℵ α + 1 ? The answer of "yes" to the first question is known as the continuum hypothesis (CH), while an answer of "yes" to the second is known as the generalized continuum hypothesis (GCH). read more
Math 114S and Phil 134 Supplement 3: Cardinal Numbers A (von Neumann) cardinal number is an ordinal number α such that (∀ β < α) β 6 = c α. Theorem (ω and Cardinals). Every natural number (i.e., every member of ω) is a cardinal number, and ω is a cardinal number. read more