Mathematical logic and foundations of mathematics in the 20th century I would like some references regarding the foundations of mathematics in the 20th century, and mathematical logic, e.g. (1) I want to understand what happened to the foundation, what originated the transition from Russell's logicism to the ZFC system, and other aspects that characterize modern mathematics; (2) When does logic start to became formal in mathematical aspects ? I know it originated with Boole's Laws of Thought, but I wanted a deeper insight. I think I'm looking for history books. I just want to know what happened, that's all. Have a nice day.
 A: Bertrand Russell's attempt to place everything under one, unified, logic based system of thought came to a screeching halt when Kurt Godel published his incompleteness theorem(s). Godel's work was done largely to answer two of David Hilbert's questions, so you may want to start with a look at the problems he posed. 
While ZFC performs well against the several paradoxes that affect other set theories, I wouldn't say that ZFC is necessarily the main set theory system in use today, nor is Russell's logic system unappreciated. Mathematicians use and study many different set theories.
Although the work has its inaccuracies, misleadings, and general flaws, Morris Klein's "Mathematics: the Loss of Certainty" provides a relatively detailed history of the branching of mathematics, including discussions on Godel, Russell, and the like. I don't think you'll find too many mathematicians recommending it on here, so I wouldn't quote it as scripture or anything. It can be appreciated for its compact history and, if nothing else, will lead you to other sources. Best wishes.
A: It sounds like the following book is what you're looking for:
Ivor Grattan-Guinness, The Search for Mathematical Roots, 1870-1940. Logics, Set Theories and the Foundations of Mathematics from Cantor through Russell to Godel, Princeton University Press, 2000, xiv + 690 pages.
Reviewed by: James W. Van Evra, Isis 94 #2 (June 2003), 387-388.
A: The references provided by other editors are excellent but since you seem to want a brief summary, here it goes: 
(1) "what originated the transition from Russell's logicism to the ZFC system" -- I would say there was no such transition in fact, since Russell's logicism never took root to such an extent that it could have been considered a commonly used system. There were several competing schools of thought in the foundations of math at the beginning of the 20th century, and logicism certainly lost out in the end, largely because of its fiendish complications.
(2) "When does logic start to became formal in mathematical aspects ?" This is a bit ambiguous but I would say relational logic (the logic we express today using quantifiers) started with Frege, was popularized by Peano and others, and soon became commonly accepted.
