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I am Master of Science in ICT, and I had always been in loved in math. On University we haven't been doing any Foundational Mathematics, the closest being Automata Theory and mention of Church-Turing thesis which blew my mind back then and left huge impression. After graduating I took closer look to thesis and discovered Computation, which led me to discovering Mathematical Logic, which led me to discovering Set Theory.

I think I have good background in undergraduate Arithmetic, Algebra and Analysis, so I understand concept of "set" in sense that is needed to conquer this fields on undergraduate level - I would say this is most basic understanding of set.

I have been reading about history of Set Theory, and now I would like recommendation of book which from the standpoint of Naive Set Theory gives introduction to Axiomatic Set Theory and explains motivation for each axiom. Basically it gives me insight how you think while you are trying to set some new axiom and why you want to do it.

Maybe some of books that are already suggested:

What are good books/other readings for elementary set theory?

Study of Set theory: Book recommendations?

are just good enough to give me insight that I want, only that is not evident from their Content page, and I never started to read :)

Thank you for your suggestions!

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I recommend the Introduction and Chapters 1 and 3 of Set Theory:An Introduction To Independence Proofs, by Kenneth Kunen. It includes discussions about general motivation, alternate set theories,and independnce of the axioms, and clearly shows the roles of each of the axioms, and in some cases, the effect of negating them. With much extra material in the Exercises and Problems.

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