When I was a teenager, I always thought that mathematical results/theorems constitute absolute truths. However after having studied maths in college, I’ve came across axioms, and things like the continuum hypothesis. When I first read that the continuum hypothesis is independent of ZFC: it can be neither proven nor disproven within the context of that axiom system. It blew my mind. I always thought that there was only one theory of maths. And that any proposition can be shown to be true or false. I also encountered the axiom of choice quite a bit in my studies. And learned that it is also independent from ZF theory.
I have finished college now, and I think I have more time to delve deeper into this subject. So I have compiled a list of things that I guess are kinda related, and that I want to learn more about :
What are the different axioms behind arithmetic, real analysis, topology, algebra, measure theory, probability, geometry (I know a little about this one : Euclide’s axioms).
Logical / non logical axioms.
axiomatic systems/ formal systems
ontology / epistemology of mathematics
philosophy of mathematics
I remember vaguely that there are two school of thoughts about mathematical objects/concepts : They exist independently of the human mind, and all we do is discover them/ They exist solely in the human mind, they are a creation of the mind. I am interested about this as well.
maths and metaphysics
decidability/undecidability in logic
mathematical “paradoxes” like the Banach Tarski theorem.
godel’s completeness theorem
I also remember reading that Bertrand Russel has discovered some inconsistencies in earlier set theories which led to their refinement. And then Kurt Godel proving that ZFC is a consitent theory (how on earth can you prove that no matter what you try you won’t get inconsistencies ?).
These things deeply fascinate me. And I would like to know where to start to learn about them. If you can suggest a list of courses/ books ranked in increasing difficulty, that would be great.
PS : I have studied the basics in these theories : arithmetic, real analysis, topology, algebra, measure theory, probability, geometry.
