I am seeking answers from experts in mathematical logic about the amount (if any) of university mathematics I need to know in order to understand mathematical logic and later hopefully do meaningful (independent) research on the subject in general and Godel's Theorems in particular.
I am proficient in high school math and have a bachelor's degree in Physics. I have also recently taught myself some calculus, linear algebra, and parts of real analysis as I assumed you must need at least undergrad math to eventually get proficient in a certain math discipline.
Earlier I had decided to learn up to grad level math but after I glanced through some logic books it appears they make close to zero use of even undergrad math. Also, I have come to know that philosophers too do research in mathematical logic, and as far as I know, they don't study any university math.
So, my question is should I first teach myself undergrad (and grad math) or just dive into mathematical logic as I don't want to later find myself in a position where I have to study all that university math before I can make further progress in logic? If that is the case I would consider enrolling myself in a math program first and doing the research later in the conventional way.
I would actually disagree, to a certain extent anyways, with the comments. I think that logic is a bit deceptive; there absolutely are some things you should be familiar with before diving into logic, even if they're not routinely used to prove results.
Specifically, I would recommend that anyone interested in logic first master the following topics:
Standard arguments by mathematical induction, and in particular $(i)$ the proof of the fundamental theorem of arithmetic and $(ii)$ the construction of a number system where it fails.
The $\epsilon$-$\delta$ definition of continuity, the construction of the real numbers via Dedekind cuts and via Cauchy sequences, and the proof that these yield the same mathematical structure.
Building off of the previous bulletpoint, basic point-set topology is also fundamental. Specifically, I have in mind comfort with compactness and arguments with open covers.
The basics of either group or ring theory, up to the definition of quotients and the proof of the first isomorphism theorem (basically, the part of the subject which is "universal-algebraic").
This is a fair amount of material, essentially what I think of as the first year of a math major at most institutions. The issue is that without this grounding, lots of material in logic will be unmotivated or overly complicated. While it's true that e.g. computability theory - one of the five main branches of logic (the others being set theory, model theory, proof theory, and a catch-all called "nonclassical logics") - can technically be leapt into without any preparation, that's not true of model theory and even for computability theory I think that would result in the student not getting nearly as much out of the subject as they could.