Set theoretic realism What are the main contemporary arguments for and against realism about set theory?
 A: One difficulty here is that it isn't clear what counts as "realism" about set theory. 
Compare, just for a start, these two views:


*

*There is One True Universe of Sets, out there in Plato's heaven, and our aim as mathematicians is to explore that universe as best we can. And a claim like the Continuum Hypothesis, for example, is just plain true or plain false about that One True Universe (the snag is our best attempts to settle the matter by making plausible-seeming assumptions about that universe haven't got us in reach of an answer). Still, there is a Real Fact of the Matter about which way it goes with the Continuum Hypothesis, or with any other coherent question we can ask about sets.

*A theory like ZFC has lots of models, there's lots of different set theoretic universes (a "multiverse", if you like, not One True Universe). These different set theoretic universes are all as good as each other -- and in some CH is true, and in some CH isn't. These universes of ZFC-governed sets are all equally "real" -- and then there are other worlds of sets where, for example, New Foundations rules (assuming that theory is consistent). 


Now, both views might be called species of realism. The first may have been Gödel's view (and there are still some who think, yes, there is a Fact of the Matter about whether CH is true, we just haven't yet found a way of settling which). The second view is in one way, you might think, a stronger form of realism (instead of believing in One True Universe of sets, it believes in lots of different set universes); but in other ways it makes weaker claims -- CH lacks a determinate truth value, but only is true or false relative to a particular model. 
But anyway, the present point is that when one asks about arguments for and against set-theoretic realism, it is obviously going to matter which kind of realism or anti-realism is in question (and we've only touched on two varieties).

A footnote. Mauro mentions some key readings. Maddy's book is probably not an easy read for non-philosophers: but for an account of what's at stake in that book -- she explores in particular a positions she calls  Thin Realism, which is different again from both the positions mentioned above -- you could look at  the review I wrote with Luca Incurvati, available here. And I'd add that it could be worth looking too at the opening of Michael Potter's justly admired book Set Theory and Its Philosophy
A: You can see :

Penelope Maddy, Defending the Axioms: On the Philosophical Foundations of Set Theory (2013)
Richard Tieszen, After Godel: Platonism and Rationalism in Mathematics and Logic (2011)

and some chapters of :

George Boolos, Logic, Logic, and Logic (1998), mainly Ch.8 : Must We Believe in Set Theory ? (page 120-on).

