# Foundations of mathematics

## Main Question or Discussion Point

I am trying to decide whether to study the foundations of mathematics i.e. formal logic, Godel's Incompleteness Theorem, set theory, etc.

The reason is that recently I have been taking advanced math courses such as topology, analysis, and abstract algebra and I am fascinated by this seemingly infallible system of logic that mathematicians have developed. There is always a right answer in mathematics (at least for my homework problems). I love this concept of correctness and things being true or false.

As I get deeper into advanced math, I find that almost everything is constructed or proved or defined unlike in high school calculus for example where theorems are just given to you. So, I keep uncovering that there are actually reasons why the things I learned are true. So, now I have started to ask for the proof of everything I have learned and that has gotten me into the foundations of mathematics. I have realized that even though it appears that Rudin goes back and rigorously proves everything I learned in calculus, that is not really true since Rudin like most authors implicitly assumes tons of very subtle things about set theory that seem intuitive but really should come from somewhere in a system of formal logic. So, there is probably another textbook that goes back and rigorously proves everything Rudin assumes from even more basic principles. And so on.

But where does it end? That is what throttles my imagination! At some point there must a book that says, "We assume nothing. We lay down an ω-consistent recursive class κ of formulas and prove everything formally we need from our own axioms." I am debating whether I should actually try to understand what an "ω-consistent recursive class κ of formulas" and how everything I have learned in mathematics comes from one. The benefit would be that my appetite for logic would be satisfied i.e. I could classify much of my mathematical knowledge into theorems and axioms and I would find proofs of theorems that rely only on formal logic not at all on intuition (like when Munkres says "clearly these two curves are homotopic" or "we shall assume that what is meant by a set is intuitively clear").

One of the things that made me consider studying foundations was http://en.wikipedia.org/wiki/Mizar_system. I was shocked that a computer could actually understand abstract mathematical proofs and at first that drove me away from mathematics but eventually I came back wanting to understand things in the same way a computer does. It is hard to explain why I would want to do that, but having things be consistent in my brain is just something I like a lot.

So, anyway, I wanted someone who has studied foundations to tell me whether it has actually made them more or less impressed with mathematics and whether they think it is something worthwhile for me to do. Part of me thinks that it will be just a massive waste of time because that kind of math has no real applications and maybe foundations is just like the "dirty work" that some mathematician has to do but no one wants to because it is just so unelegant and tedious.

I don't know...I am really just making this thread for people to discuss their choice to study or to not study the foundations of mathematics.

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Well, i have almost no experience in formal set theory including logic, which i think is the foundation of everything in math. However, i really would love one day to work on such stuff. For me it is really important to know where and how comese every bit of information i am presented in math. I really cannot take for granted anything at all, so i always try to find its roots to the best of my capabilities of doing that. But yeah, I will defenitely end up one day studying the very foundations of mathematics, and i am sure i will be loving every single moment while doing it.

I'd say at least an undergraduate class on set theory or logic is probably a good idea. In fact, to get the B.A. at my school, this is a requirement.

Rather on topic: A professor of mine said that knowing the compactness and incompleteness theorems of logic is something that most mathematicians could benefit from. You've probably noticed that well-roundedness is praised by many top mathematicians.

Rather off topic: I started out wanting to go into logic, but I moved away from it into mainstream math.

Definitely. I studied recursion theory, turing machines, 1st and 2nd order logic, the incompleteness theorems, etc. as an independent study my senior year. Probably the most fascinating area of all of mathematics IMO. The lines between mathematics and philosophy become extremely blurry once you study some formal logic.

Lately, I've been finding this very interesting also.

I have been in discussion with stupidmath about his schedule plans where this topic arose. I am curious as to what level "logic" is taught at an undergraduate level.

My school offers a general 'logic' course for those who do not wish to take another math class but need to fulfill that part of our core system.

My math dept. offers an upper division class called Mathematical Logic periodically which covers propositional logic then moves to first order predicate logic, and usually ends dealing with compactness, completeness, and incompleteness theorem

Is this standard for the level of formal logic taught in undergrad curricula ? Is logic ever an introduction course? like that of number theory? where you do not need to take just about anything as a pre-req?