Cantor’s theorem states that $|\mathcal{P}(A)| > |A|$ for any set $A$.
As a special case of this, we have $|\mathcal{P}(\mathbb{N})| > |\mathbb{N}|$. If we denote the power set of the naturals by $\mathcal{P_1}$, then $|\mathcal{P}(\mathcal{P_1})| > |\mathcal{P_1}|$. Now denote the power set of $\mathcal{P_1}$ by $\mathcal{P_2}$. Applying the theorem again, we have $|\mathcal{P}(\mathcal{P_1})| > |\mathcal{P_2}|$. We can repeat this process forever, with each power set, $\mathcal{P_n}$, having strictly greater cardinality than the last, $\mathcal{P_{n-1}}$.
Clearly this shows that there is an infinite number of different cardinalities of infinite sets (an infinity of infinities). My first thought is then to wonder about the nature of this infinite quantity.
I have two questions:
- What is the cardinality of the set of different cardinalities of infinite sets? In other words, what kind of infinity is the number of kinds of infinity? It seems like it would be countable infinity to me, as we clearly have a bijection between the naturals and these cardinalities. However, my question below may cause a problem with this argument.
- In performing this iterative process on $\mathbb{N}$, are we actually covering all possible kinds of infinity? Can we be sure that there don’t exist any kinds of infinity that won’t at some point be generated by this process?
A side question regarding the definition of ‘uncountable’: Does uncountable infinity refer to any infinity that isn’t $|\mathbb{N}|$, or is it strictly $|\mathbb{R}|$?
I hope this question makes sense and that I’m not asking something pointless or unanswerable.
