I am undergraduate student in physics with not that many mathematics courses done in my career. Nevertheless, I am very interested in mathematical physics. I started studying from the lectures "Geometrical anatomy of theoretical physics" which can be found at https://youtu.be/V49i_LM8B0E.
From the lectures I have gained some sort of intuition but I feel that for my satisfaction I want to learn a little bit more. For you to better understand my interests, I want to mention what kind of things I would like to learn next. The general idea is that I do not want to go very deep in these exciting and interesting topics but I want to know enough so that I can learn topics like topology, manifolds, bundles and so on that are necessary for physics without stumbling across things that come from logic or set theory. For example, while I can prove basic things in topology (which basically just use set-theoretical arguments), I am sometimes dissatisfied because I can prove things "intuitively" but if some day I wanted to make my proofs formal, I am afraid that I couldn't. This is certainly something I would like to change.
- Propositional and predicate logic. Proofs and rules of inference.
I feel satisfied with classical logic which defined proposition and predicate in an intuitive way and which defined operators using truth tables. This is enough rigor for me and I am willing to accept this. I am also willing to accept the philosophy that logic goes first and then I use it to tell what set theory is. The thing I have the most problems with in logic is how to make formal proof. In the lectures I heard the recipe (proof is sequence of propositions which are either axioms, tautologies or "modus ponens"), but I have trouble proving many things as I am unsure if I am doing it right. I would be happy for some reference that could show some examples on how to make formal proofs, so that I would have a nice feeling that if someone in the future would not believe my proof then I could go back to logic and write a nice, billion lines long formal proof.
- ZFC set theory.
My whole life I have been using "naive set theory" without a doubt but once I got introduced to Russell's paradox I feel uneasy about it. That is indeed possible that in my whole physics career I will always be able to prove and write everything in terms of naive set theory, but something just feels wrong. That being said, I would like to receive some reference about ZFC set theory for beginners. I am not interested in some deep topics in the set theory - I just want to be sure that I know what axioms are (that are now considered to be consistent) so that I could consult them whenever I get too confused in my mathematical journey.
I would appreciate any suggestions and/or comments from your personal experience!