Are there any standout books on Set Theory and what research is left to be done?

[andytoh]

Any books that really stand out? Currently, I'm reading "Set Theory and Logic" by Stoll. I'm not interested in the axiomatic type of set theory, like Godel's theory and all those unreadable symboic proofs. I'm more interested in stuff like the axiom of choice proofs and such. Also, is there any research left to do in set theory or is it a fully exhausted field?

 

[Son Goku]

I don't know about others, but I quite enjoyed Keith Devlin's "The Joy of Sets" (2nd Edition is much better). It gives a good justification of each of Zermelo's axioms and why they are there and has a very good explanation of the ordinals.
However the unique feature of this book is that it contains a good beginner's explanation of why $$2^{\aleph_{0}} = \aleph_{1}$$ is independant of ZFC and the attempts to resolve this by the addition of new axioms.

 

[MathematicalPhysicist]

well there are plenty of good intro books on set theory.
At my school we're using jech and hrbaceck intro to set theory, after reading this intro I think the next step is reading jech's set theory text, which is more advanced.

As for research in the field, if you search for handbook of set theory in google you'll find a page with the articles from the handbook, they address there their research in the field, so yes there's research in the field.