I looked up the wiki entry for Alain Badiou
http://en.wikipedia.org/wiki/Alain_BadiouThis situation of being and the rupture which characterizes the event are thought in terms of set theory, and specifically Zermelo–Fraenkel set theory (with the axiom of choice), to which Badiou accords a fundamental role in a manner quite distinct from the majority of either mathematicians or philosophers.
and clicked the link for "Zermelo–Fraenkel set theory"
The Metamath project sounded interesting so I clicked on that
http://us.metamath.org/Metamath Proof Explorer - Constructs mathematics from scratch, starting from ZFC set theory axioms. Over 6,000 proofs. Updated 26-Jul-2006.
...
clicked on "Metamath Proof Explorer" and noticed the line:
clicked on "Reading Suggestions"
http://us.metamath.org/mpegif/mmset.html#readReading Suggestions Logic and set theory provide a foundation for all of mathematics. One possible way to start to learn about them is to use the Metamath Proof Explorer in conjunction with one or more textbooks listed in the Bibliography of the next section. The textbooks provide a motivation for what we are doing, whereas the Metamath Proof Explorer lets you see in detail all hidden and implicit steps. Most standard theorems are accompanied by citations. Some closely followed texts in the Metamath Proof Explorer are listed below. While these aren't always the best books for beginners (if you have better suggestions, let me know), it will be easier to acquire a "high-level" understanding of some of the Metamath Proof Explorer proofs if you consult them while studying the proofs.
To start with, I would suggest Margaris (now available in a Dover edition, it is inexpensive and reasonably readable for beginners, once you learn the archaic notation that uses dots in place of parentheses) and Quine (it also uses the archaic dot notation, but it is wonderfully written and a pleasure to read; the first part on virtual classes is a must-read if you want to understand the class variables we use).
# Axioms of propositional calculus - Margaris.
# Theorems of propositional calculus - Whitehead and Russell.
# Axioms of predicate calculus - Megill (System S3' in the article referenced).
# Theorems of pure predicate calculus - Margaris.
# Theorems of equality and substitution - Monk2, Tarski, Megill.
# Axioms of set theory - Bell and Machover.
# Virtual classes in set theory (our class builder notation and our purple class variables) - Quine.
# Development of set theory - Takeuti and Zaring.
# Construction of real and complex numbers - Gleason
# Theorems about real numbers - Apostol
Then I went to the Dover link posted by Davis_X_Machina above,
and searched for Margaris:
That's not exactly the same title as the one on the Metamath list,
but it sounds like it covers exactly the areas you want:
http://store.doverpublications.com/0486662691.htmlWell-written undergraduate-level introduction begins with symbolic logic and set theory, followed by presentation of statement calculus and predicate calculus. First-order theories are discussed in some detail, with special emphasis on number theory. After a discussion of truth and models, the completeness theorem is proved. "...an excellent text."—Mathematical Reviews. Exercises. Bibliography.