I have studied real analysis, linear algebra, and how number sets are constructed from N, but now, i want to learn the foundations of math and some more advanced set theory (cardinals, ordinals), because my brain is full of questions such as "what is a property?, how does logic work in math?, What is ZFC?" and things like that. Can you recommend any books that could help me with that?
Book Recommendation – Logic and Set Theory Books
book-recommendation
Best Answer
Axiomatic Set Theory by Patrick Suppes is an easy intro to the basics. Last year it was available as a free PDF. Maybe still is.
Lectures In Set Theory. Various authors. Edited by Morley. I found the essay on the definition of L (Godel's constructible class) to be the easiest and clearest intro to L that I've seen.
Introduction To Set Theory And Modern Analysis by Simmons.
Set Theory: An Introduction To Independence Proofs by K. Kunen. A thorough axiomatic development from the bottom up.
You will need to learn about Godel's incompleteness theorems. These are like the axiomatic foundation of the properties of the reals, in that you study it once and then take it for granted. Do NOT read Godel, Escher,Bach: An Eternal Golden Braid by Hofstader. For a long time Godel's Proof by Nagy & Newman was the only "popular" exposition in English.
Something on Model Theory and on Logic. Sorry I can't name a book.
Stories About Sets by V'Lenkin (Vilenkin). Good fun.
50 years ago Dover Publications (formerly Dover Press) was an excellent source of cheap re-prints of math & science books. It still is.
BTW. You will meet the Schroeder-Bernstein, Cantor-Bernstein, and Cantor-Schroeder-Bernstein theorems. These are all the same theorem. The Simmons book has a nice presentation of the short proof. There is also a long proof, which I saw & ran away from.