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.
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.