Can you recommend a popular science book that deals with axiomatic foundation of mathematical areas and consequences? The areas could be geometry, algebra/numbers, set theory, etc. It should be exact and not too hand-wavy, but by "popular" I mean a book that just gives a nice overview. Basically I'd like to find out about these approaches, but I don't have the time to go into all details.
EDIT: I do prefer mathematical notation and reasoning. I just want to avoid many lemmas and proofs :)
On
I first learned about these things as a teenager by reading the third volume of James R. Newman's anthology, World of Mathematics. It contains essays by many of the original workers in the field, as well as commentary by Newman. This may be at a lower level than what you're looking for, but I highly recommend it nonetheless.
You might try "Naive Set Theory" by Paul Halmos. It would seem to meet your requirements in that it strives to be exact while remaining "popular" by using ordinary language and avoiding the deep logic foundations of mathematics. As set theory can generally be used to formulate all other mathematics, this would seem to be a good place to start.
Note that it does not meet the popular definition of "popular" in the sense that a book by Greene or Hawking would. Halmos' book still requires a certain mathematical mind.