Proving consistency of ZF in ZFC Is it possible to prove the consistency of ZF theory using the ZFC theory (assuming ZF is consistent)? What's the best way to approach this kind of proofs where we want to show consistency of a theory?
I'm looking more for a general idea of the proof rather than the entire proof (if it exists) if not, why is it that it is not possible?
 A: No, this cannot be done (we hope!).
The key fact is the following (due to Goedel): that the consistency of ZF implies the consistency of ZFC, and that moreover this can be proved inside of (much less than) ZFC. More on this below.
By Godel's second incompleteness theorem, if ZFC is consistent then it can't prove its own consistency - so by the above fact, ZFC can't prove that ZF is consistent either (unless they are both inconsistent).

So how does the consistency of ZF imply the consistency of ZFC?
Well, let's think semantically at first for simplicity. Within any model $M$ of ZF sits an "inner model" called $L^M$ - the "constructible universe" of $M$. $L^M$ is defined within $M$ as the collection of sets gotten by iterating the definable powerset $DefP(X)=\{A: $ $A$ is a definable subset of $X\}$, just as $M$ thinks of itself as gotten by iterating the actual powerset. It turns out that $L^M$ is always a model of ZFC, even if $M$ doesn't satisfy the axiom of choice! So from any model of ZF, we can build a model of ZFC; hence if ZF is consistent, so is ZFC.
This is a long and technical proof, but it boils down to using the fact that $L^M$ comes with a nice well-ordering - order it level-by-level, and within each level order the sets by the Godel numbers of the formulas defining them from the previous level together with the already-defined order on the parameters used in these formulas. From this well-ordering $\prec$, the axiom of choice trivial follows: simply pick the $\prec$-least element of each set! 
The key point, of course, is even stronger: that the above argument can  proved inside ZFC (indeed much less than even ZF!). And of course I haven't really done anything to indicate why the above is true, I just said some vague words - which is appropriate, since the proof is fairly technical. But it can be found in any of the good books on serious set theory, like Jech or Kunen.
