Reddit reviews Set Theory and the Continuum Problem (Dover Books on Mathematics)

I second Hrbáček and Jech followed by Kunen for a thorough, rigorous treatment. But Set Theory and the Continuum Problem by Smullyan and Fitting is another interesting, self-contained exposition that concentrates on consistency and independence proofs, the axiom of choice, and the continuum hypothesis. It covers both Gödel's and Cohen's proofs. It says it does not have any prerequisites, but that does not make it easy. It also has interesting philosophical asides.

You can ask for a reading course from that professor.

Here's two inexpensive references 1 , 2

I would recommend reading further into this topic. This is a book I am reading: http://www.amazon.com/Theory-Continuum-Problem-Dover-Mathematics/dp/0486474844