I have recently become interested in the foundations of math. I am interested in tracing the fundamentals of math in a path such as: propositional logic -> first order logic -> set theory -> measure theory. Does anyone have any resources (books, webpages, pdfs etc.) they would like to recommend?
This seems like it would be a popular activity among LWers, so I thought this would be a good place to ask for advice.
My criteria (feel free to post resources which you think others who stumble across this might be interested in):
- The more basic the starting point the better: I would prefer a resource that defines propositional logic in terms of a context free grammar and an evaluation procedure (don't know if that is possible, but that's the sort of thing I am interested in) to one that just describes propositional logic in English; I would prefer a resource which builds first order logic from propositional logic + some definitions to one that just describes how first order logic works; etc.
- The fewer axioms (perhaps that's not quite the right word) the better. I prefer a resource defines describes propositional logic with just two operators (say negation and conjugation) and then builds the other operators of interest to one that defines it with 5 or 6 operators (I've seen many resources which do this).
- I expect that there are multiple ways to build math from basic building blocks. I am more interested in standard ways than than non-standard ways.
Can you say why you think it should be learned first? The measure theory I have seen seems to always involve sets (measurable functions, sigma-algebras etc), maybe I am just confused about something.
Of course it involves sets, but the kind of set theory you need for that is rather limited, because measure theory deals with very special cases, compared to which set theory proper looks pathological. So, it's a prerequisite to know about countable and uncountable, union and intersection, open and closed and compact, but not the contents of a course in set theory, which gets quite a bit more complicated. I know very little set theory and never studied logic.
Edit: nhamann is right, you need a first course in analysis before you'll understand measure theo... (read more)