Learning the foundations of math

jsalvatier

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.

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.

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 theory. I used Rudin, but I'm not partisan in favor of it. But you've got to learn analysis first. Or you will get confused. Do not pass Go, do not collect $200.

The thing about foundations is that they're mysterious, even at the research level, but when you're working on classical special cases it doesn't matter. In the same way that you can do arithmetic even though you can't prove the axioms of arithmetic are consistent. Measure spaces are very much nicer than sets. And often it's safe to think of the reals as a guiding example. In set theory, thinking of familiar sets as guiding examples is dangerous, which is why I think it's a harder subject, but that may be my idiosyncrasy. The point is, measure theory and set theory are independent from the point of view of the learner -- learn whichever you like first, but one isn't a prerequisite for the other.

A very excellent recent book, with fascinating new ideas and superior readable intros into many themes, is the new edition of Manin's "course in mathematical logic". So I'd recommend that. But: Why "foundations"? Like "foundational themes" in th. physics, "foundations" are not an appropriate place to start, they are a bundle of very advanced research areas whose intuitions and ideas come from core fields of research. "Foundations" in the sense of "what is it, really?" can be exprerienced probably ... (read more)

0jsalvatier16y

That was very informative.

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Learning the foundations of math

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Learning the foundations of math

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