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).
Best Answer
Axiomatic Set Theory is the term you are looking for. Technically speaking you should really make sure you have a strong background in first-order logic first, as ZFC(Zermelo-Frankel Set Theory with Choice-the "standard" set theory construction) is formulated in FOL. However, you could probably get away without it if you are familiar with the basics of quantifiers and logical symbols and are just looking to take the conceptual approach. If that's the case a good one would be Axiomatic Set Theory by Suppes. A good intro to FOL book is "Computability and Logic" by Boolos. Alternatively, if you search "Axiomatic Set Theory" on amazon a bunch of books will come up that you can read user reviews of.