I am studying Philosophy but most of my interests have to do with the philosophy of Maths and Logic. I would like to be able to have a very high level of competence in the topics mentioned in the title, and I was wondering, given that I don't have a mathematical background beyond basic school level maths, what particular branches of pure mathematics will help me to go deeper in my study of Logic, Model theory and Set theory? Calculus? Group theory? I hope you can give me some suggestions.
[Math] The Maths necessary to understand Logic, Model theory and Set theory to a very high level
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It is not necessary to write your Ph.D. dissertation as a direct continuation of your masters thesis. I will not write my Ph.D. as such continuation.
You could study more model theory on the side, or you could study more pure logic, or you could expand into another area. Then when the time comes to write your Ph.D. you could make a much better decision. Furthermore, I know several people who were set to solve one problem in their Ph.D. and gave up halfway only to switch to an unrelated problem.
Some universities even support external advising (especially for Ph.D. students) which means that you have a local advisor, and another advisor (often the actual advisor) to work with on your problems. You might also find it easier just to switch universities, if that's a viable option.
Besides that, it is true that it is the easiest thing to just continue your masters research into a Ph.D. dissertation, but the main use of a masters thesis is like "research training wheels" which give you a taste of doing mathematical (or otherwise) research. In the university I did my M.Sc. you are not even expected to do original research or publish papers at the end of your masters. You are only expected to write a thesis which shows that you know, a bit more, how to research a problem in mathematics.
The important thing is to do what you love. Writing a thesis, especially a good one, takes a lot of effort and time. Spending so much energy on something you dislike is not a good advice.
Let me share one experience from my masters degree. I was set to research into axiom of choice related topics, and I actually dragged my advisor into the topic. I came up with most of the questions and problems, and I made him curious about things so we studied together. Certainly if I would stay there for a Ph.D. with him we would continue to study together, even though my advisor's main interest is proper forcing, and order theory.
Elementary number theory, i.e., modular congruence, linear Diophantine equations, quadratic residues and quadratic reciprocity can be easily studied without experience in higher algebra or calculus, and there a probably several books on the topic accessible to you (in almost every mother tongue). Really, it is a question of taste. One of my favorites is Elementary Number Theory, by Gareth and Josephine Jones. Also, it will cover basic and central concepts of number theory you'll need in any advanced study.
From there, number theory breaks in two major branches (with a great deal of overlap between them, and, of course, not comprehensive branches): analytic and algebraic number theory. For analytic, some calculus may come in hand. As for algebraic, basic higher algebra will be expected.
The standard reference of analytic number theory is Apostol's Introduction to Analytic Number Theory. How much calculus you'll need depends on how deep will you go (it can be single variable, multi variable, complex variable, you'll may also need some general topology, who knows?).
I've never read something specific about algebraic number theory, but the Internet seems to recommend Rosen's & Ireland's A Classical Introduction to Modern Number Theory. From the summary, it appears to cover the basics of algebraic number theory (also, with a lot of overlap with elementary stuff). I've had a good experience with Serre's A Course in Arithmetic, which covers both analytic and algebraic aspects, though it's a very hard book to digest (also, the french original is superior).
From there, you'll probably already have very specific interests from which you will be able to get better references. And for the calculus, algebra and topology, you can find introductory references here on the site. Expect to need at least some knowledge of groups, rings and fields, derivatives, infinite series, topology and complex variables to delve in the most advanced (yet central) areas.
Best Answer
Apart from actually learning logic, set theory and model theory you would probably benefit from some basic understanding in
While these things are not necessary per se in order to gain understanding in logic, or set theory (although model theory deals a lot with actual mathematics, so you can't escape it there); in order to fully understand set theory I think that one has to see "usual" mathematical theories and understand them at a basic level. If not for anything else, then in order to understand what makes set theory special.
It seems, if so, that the better part of an undergrad degree in mathematics is needed. But then again, it is needed if you wish to understand any mathematical theory in depth.