Reading on Mathematical Logic I am looking for books to read, so as to dive into mathematical logical and related disciplines like set theory, model theory, and topos theory. 
I have a decent background in category theory and algebra, analysis, topology, etc. but little in explicit logic or set theory aside from the first chapter of Munkres.
Any suggestions? I am interest in topics such as para-consistency, computability, and ill-founded logics. 
 A: The place to start, perhaps, is here:
http://www.logicmatters.net/resources/pdfs/TeachYourselfLogic9-2.pdf
This is a detailed annotated guide to a wide range of logic literature, at different levels of sophistication. You will be able to choose entry points to suit your background.
On non-paraconsistent logic, see in particular
http://plato.stanford.edu/entries/logic-paraconsistent/
which gives pointers to the literature. On computability, there is much to be said for Boolos, Burgess and Jeffrey,
http://www.amazon.co.uk/Computability-Logic-George-S-Boolos/dp/0521701465
I'm not sure what is meant by "ill-founded logics". For non-wellfounded set theory, see
http://plato.stanford.edu/entries/nonwellfounded-set-theory/
which again gives more pointers. On topos theory, I still think it is worth starting with Robert  Goldblatt's book:
http://www.amazon.co.uk/Topoi-Categorial-Analysis-Logic-Mathematics/dp/0486450260
A: I fondly own the following books.
Logic


*

*Peter Smith's An Introduction To Godel's Theorems is an excellent first introduction to logic and computability. Yes, that's the same Peter Smith whose answered your question.


Set theory


*

*Goldrei's Classic Set Theory is a clear and well-motivated first introduction to the subject.

*Jech's Set Theory is very highly regarded, and would be excellent for a second exposure to set theory (this is the book I really need to work through).

*Also, you mentioned an interest in topos theory. I've never studied topos theory, but I think a good starting would be Lawvere's Sets for Mathematics which describes basic set theory from a categorial perspective. I think this will get you "in the mindset for topoi," so to speak.
