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 :)