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I'm looking for a book - more likely, books - that could take me from the axioms of mathematical logic up to the major subjects of mathematics, like analysis, algebra, geometry, etc.

For example, a book that starts from first principles in propositional calculus... a book that takes the logic proved therein as axioms to do set theory. And then a book that takes set theory and uses it to construct the real numbers. From there, a book to prove the real number properties and basic analysis. This last book could probably be some combination of Rudin and Spivak. (I'm not really sure where geometry fits in with this - as I haven't had any geometry at the college level yet, other than topology.)

So who has the best list?

The reason I ask is that I'm making a sort of outline of mathematics for myself to study from, and I want it to be as rigorous as possible, which means I want to cite specific theorems and definitions and axioms in all my proofs. This may seem over the top to some, but I'm a bit obsessive about everything being super-logical, and perhaps I can publish some kind of axiomatic book(s) one day if it doesn't already exist.

EDIT: I've decided on a few books that I'm going to try to use.

Mathematical Logic by George Tourlakis

The Bourbaki Theory of Sets


Jech's Set Theory

I'd still love to hear the opinions of real mathematicians and/or more experienced students who might have some insight here. Until that happens, I will be journeying through the above books.

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This is not a good way to study mathematics. But I think the Bourbaki books are supposed to do this. – Qiaochu Yuan May 9 '13 at 21:04
Yuan's comment above implies a certain universality about his statement. It might not be a good way for him to study mathematics, nor for most people, but it could be for some. In retrospective I wish I had been taught the way the OP wants to learn. So many things confused me that only got answers after three yers of studying... @QiaochuYuan – Git Gud May 9 '13 at 21:08
@Git: It can be overdone. e.g. if you went so far as to try and make a library of formal proofs that could be mechanically verified, then I vaguely recall people estimating it would take years (decades?) of person effort even for an elementary subject. – Hurkyl May 9 '13 at 21:13
I don't know of any books on propositional calculus that aim to achieve your goal. However when you start doing real numbers, I can recommend the books Analysis I and Analysis II by Terence Tao. Analysis I starts by defining the piano axioms and then constructs the natural numbers from this. Then the integers are constructed, and after this the rational numbers and then finally the real numbers are constructed. The author spends about 5 or 6 chapters on this, and then he starts doing calculus with focus on the theorems rather than computations. The second book studies real analysis. – Oliver E. Anderson May 11 '13 at 10:37
Well, open up Russell and Whitehead's Principia Mathematica, and when you finally prove (*110.643, in volume II) that $1 +_c 1 = 2$ (that's cardinal sum) you might wish you hadn't. – inactive... for now May 11 '13 at 11:08
up vote 2 down vote accepted

From my personal experience, an historical approach can be useful.

Why not try with :

Morris Kline, Mathematics: The Loss of Certainty, Oxford University Press, 1980 .

It explains the road to modern math, including the big isue regarding Foundations of Math (and the birth of Mathematical Logic).

For an understanding of Math Log and Set Theory, I would suggest you the 2 volumes set :

George Tourlakis, Lectures in Logic and Set Theory, Cambridge UP, 2010.

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