Cohn’s Measure Theory is good for measure theory. Rudin’s Real and Complex Analysis is more standard, but less gentle, in my impression.
Measure theory is good to know, but it’s quite different from set theory and logic, and doesn’t require you to know them. You don’t necessarily have to learn it last, and from my perspective it’s easier and you should learn it first. It’s certainly the “foundation of some maths” in the sense that it’s a fundamental tool in analysis, but most of the time I think people use “foundations of math” to refer just to set theory and logic.
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 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 much better by studying a good piece of core math, like number theory. Cox’ “Primes of the form x^2 +n*y^2” or Khinchin’s “Three Pearls of Number Theory” is what I would suggest. If your mind prefers geometry, I’d suggest to browse a good library for some of the great projective geometry textbooks from the early 20th century.
Cohn’s Measure Theory is good for measure theory. Rudin’s Real and Complex Analysis is more standard, but less gentle, in my impression.
Measure theory is good to know, but it’s quite different from set theory and logic, and doesn’t require you to know them. You don’t necessarily have to learn it last, and from my perspective it’s easier and you should learn it first. It’s certainly the “foundation of some maths” in the sense that it’s a fundamental tool in analysis, but most of the time I think people use “foundations of math” to refer just to set theory and logic.
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 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 much better by studying a good piece of core math, like number theory. Cox’ “Primes of the form x^2 +n*y^2” or Khinchin’s “Three Pearls of Number Theory” is what I would suggest. If your mind prefers geometry, I’d suggest to browse a good library for some of the great projective geometry textbooks from the early 20th century.
That was very informative.