Can anyone recommend a book that explains math from the ground up?

[theneedtoknow]

I mean, a book that states the basic axioms of contemporary mathematics and then proceeds to prove/derive things based on those axims. Something like a contemporary mathematics version of Euclid's Elements?

[Nabeshin]

This may or may not be what you're looking for, but the first thing that absolutely jumps to my mind is Russel and Whitehead's Principia Mathematica. The first of three volumes is listed here:
http://www.amazon.com/gp/product/1603861823/ref=pd_lpo_k2_dp_sr_2?pf_rd_p=486539851&pf_rd_s=lpo-top-stripe-1&pf_rd_t=201&pf_rd_i=0521626064&pf_rd_m=ATVPDKIKX0DER&pf_rd_r=0WCEMY9F30J8MQ1JVPK5

And you can get a general idea from the wiki article:
http://en.wikipedia.org/wiki/Principia_Mathematica

[theneedtoknow]

That actually sounds like exactly what I am looking for! I am gonna order me a copy, thank you very much!

[qspeechc]

No! You would be making a huuuuuuuuuuge mistake by buying that book. Almost no one reads that book, not even mathematicians! It is dense and irrelevant to any mathematics you will want to do. Don't waste your money on it.
Rather buy books that deal with the specific topic you want to learn.

[Reedeegi]

A book on Category/Topos Theory could serve as a mathematical foundation. From it you can create various logic systems (including intuitionist logic) as well as locally-true set theories. Two inexpensive books that cover this are

Topoi: The Categorial Analysis of Logic by Richard Goldblatt
Toposes and Local Set Theories: An Introduction by John Bell

Both are published by Dover, a publisher that produces rigorous, high-quality texts for a very affordable price. I highly recommend Topoi, but unless you've had an extensive training in rigorous thinking it may come off as pointlessly abstract and useless. However, if you approach it with no expectations you will gain a deeply profound view of mathematics.

Good luck.