I was playing around with a theorem in combinatorics about finite sets and was trying to find a meaningful generalization of it to non-finite sets. At one point without thinking I wrote that:
$$\text{ The power-set of }X=\mathcal{P}(X)=\bigcup_{n=0}^{\infty}\{S\subseteq X:|S|=n\}$$
Then about five seconds later I realized it was false, because my set $X$ might contain a subset whose cardinality is not a natural number. So I tried to salvage this partially, using my essentially zero knowledge on ordinal numbers and more advanced set theory, basically I wrote something akin to: $$\mathcal{P}(X)=\bigcup_{\substack{\gamma\leq |X|\\\gamma\text{ a cardinal number}}}\{S\subseteq X:|S|=\gamma\}$$
However I'm not even sure if this makes sense. To make it formal I would have to reform that lower index to something like "the set of cardinals", but I don't even know if that's a set?? I'm starting to feel really stupid, I mean throughout most of my time studying I've never really made use of any sets with cardinality say greater then $|\mathbb{R}|$ maybe $|\mathcal{P}(\mathbb{R})|$ when working with function spaces, though even then I never really made use of it.
Also I've tried to play around with stuff like this before, where I see something cool and want to make a generalization for non-finite sets and sometimes I can manage by manipulating things weirdly and finding bijections to establish equinumerosity. But I imagine if I had some more tools up my sleeve like a familiarity with cardinal arithmetic etc. I could probably do this much faster and likely find results I couldn't have got before because I lacked the ability to manipulate sets with arbitrary cardinality in a nice way.
So in short can someone recommend me some reading on "advanced set theory" (no idea what to call it, just want to make sure its not a book on naive set theory etc. which I'm fine with).