TITLE: How much of ZFC does Quine's New Foundations prove?
QUESTION [20 upvotes]: Main Question: Does anyone know of a reference that can tell me which axioms of ZFC Quine's New Foundations prove, disprove, and leave undecided?  
Secondary Question: I've read that diagonal arguments don't go through in NF and thus can't be used to prove that the reals are uncountable.  Does NF manage to prove the uncountability of the reals by some other means or does that fact (normally rendered as "$P _1(\mathbb{N}) < P(\mathbb{N})$" in order to make sense in NF) turn out to be undecidable in NF?

REPLY [6 votes]: I didn't see anyone specifically address the secondary question, so I thought I'd pipe in. In NF the natural numbers are provably Cantorian (see above for definition). Rosser's "Logic for Mathematicians" (available from Dover and a pretty nice resource) has a detailed proof on p.437. Basically, if $a \in n$ and $b \in m$ and $|a| = |\mathscr{P}_1(b)|$ then the relation pairing $n$ and $\{m\}$ turns out to be stratified and is a bijection between $\mathbb{N}$ and $\mathscr{P}_1(\mathbb{N})$. When you have such a bijection, Cantor's proof goes through in the usual fashion.