I'm still studying maths at undergraduate level, but intend to continue exploring topics in pure maths after I have graduated, so am thinking already about what directions I'd like to persue now, (as I like to be several steps ahead of myself!)


One area which seems particularly interesting is 'reverse mathematics'. I'd be interested to learn what the prerequisites would be for understanding it. Presumably it is a part of logic, and is related in some way to proof theory? If say, I wanted to explore questions such as, 'are there a finite number of ways of proving something?', would this be the right direction to take? (In fact, is there already an answer to this question?)


Apologies if the above is a bit broad/vague. This is my first post, and I plan to add to it as I learn more!


**Update 1: Can anyone recommend a good introductory text on reverse mathematics and/or proof theory?**



Understanding the nature and structure of proofs; Reverse Mathematics and Proof Theory. Prerequisites? Good introductory texts?
You probably already know this, from the references you mentioned, but since you didn't explicitly list it, I'll say it here. The "bible" of reverse mathematics is Steve Simpson's book "Systems of Second-Order Arithmetic".


As for the relevant areas of logic, I'd say that recursion theory is (currently) more closely connected to reverse mathematics than proof theory is.


Finally, your question whether there are only a finite number of ways to prove something is an interesting one, but it presupposes that there is a clear and plausible definition of what it means for two proofs to be really different. If one views proofs as just being deductions in some formal system, then, as soon as you have one proof of a result, you can generate infinitely many more by inserting trivial and irrelevant steps, or by other silly reformulations of your original proof. For your question to be meaningful, one has to exclude such trivialities and ask how many genuinely different proofs there are for a given result. And it's not easy (maybe not even possible) to draw a clear line between genuinely different proofs and silly variations.


