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May 24th, 2007, 04:44 AM   #1
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Good book on proving the fundamentals of mathematics

During courses I've taken, I've proved pretty much every theorem I have ever used. But recently, while doing physics, I came to think of something: Every little mathematical step I made, was very logical, but were they possible to prove? Could you prove that 2+2=4? Or that a+0=a? Or even more fundamental; could you prove the existence of the number 1? I found the idea quite compelling, so I set out to find what the most basic components of mathematics is.

After some googling, I came across that Zermelo-Fraenkel set theory was able to prove pretty much everything from a set of axioms (yeah, I know about about the fact that strictly speking there's an infinite number of axioms, and the Gödel's Incompleteness theorem and such "problems"), so I wanted to find out if this theory was able to provide an answer to the above mentioned questions.

What I'm looking for is a book that sets out to prove/justify the assumptions most are taking for granted, like the properties of the numbers (real numbers, integers, negative numbers, complex numbers etc.), mathematical operations (addition, subtraction, multiplication, square roots, exponentials, logarithms) at a very basic level and starting with the beginning and going from there (justifying every assumption you make by logic or mathematics). I understand that to do everything entirely from ZFC would be probably impossible to do for a human being, but that's not my goal either. I just want to be able to justify the fundamentals that my entire mathematical knowledge is built upon.

So basically I'm looking for a good introductory book that proves the most fundamental mathematical concepts, by using logic and/or set theory. Any suggestions?
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May 24th, 2007, 05:06 AM   #2
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I own Mr. Suppe's Set Theory book, and it is quite good indeed, and does not presuppose any background in logic or in set theory. It is written in an intuitive way, but the proofs are complete (it's just that it is not completely formal). You can find it published by the Dover editions, which have the advantage of coming at a very cheap price.
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