A question about the number of infinities.

Question

if there are infinite whole numbers, and there are infinite decimals between 0 and 1, and there are infinite decimals between 0.1 and 0.12, and there are infinite decimals between 0.1111111 and 0.1111112, and (etc.) does that mean that there are infinitely infinite infinitely infinite infinitely infinite infinitely infinite infinitely… (etc.) infinities?

Answer 1

Yes. One can see this via Cantor's theorem: given a set $A$, the set of subsets of $A$ (the power set, $\mathcal{P}(A)$) has a strictly larger cardinality. (Of course, understanding what this means and why it holds requires some background in set theory.)

Hence, you achieve an infinite chain of infinite cardinals: if $|S|$ denotes the cardinality of $S$, and if $S$ is some infinite set (e.g. the integers or real numbers), then $$\newcommand{\P}[1]{\mathcal{P}(#1)} |S| < | \P{S} | < | \P{\P{S}} |< | \P{\P{\P{S}}} |< | \P{\P{\P{\P{S}}}} | < \cdots $$