So I've heard that picture is shooped, and that's cool and all, but I gotta know who that lady is.

It's not that it's inconsistent, it's that it is either inconsistent or incomplete.

You could prove the inconsistency of ZFC within ZFC by showing a contradiction. Similarly, you could show the incompleteness by proving that a theorem is unable to be proven true or false.

No one has been able to do the first; as far as we know ZFC is consistent. If it weren't, that would be a really, really big deal. Like, rewriting the foundations of mathematics big.

On the other hand, incompletenesses have been shown. There are statements in ZFC that could be true or false without violating any of the axioms.

The only way to get around Goedel's incompleteness theorem is to lose one of the conditions. Either your system is incapable of expressing arithmetic on natural numbers (meaning that counting is impossible) or it is inconsistent (meaning that some statement is both true and false).

Systems unable to express arithmetic on the natural numbers are probably very uninteresting, but you might be able to come up with something novel; that's something I don't know much about. Breaking consistency, on the other hand, means that everything is true and false. Those systems are very uninteresting.

>>346
Unless I misunderstand the premise of what you are saying, it sounds a little like you are trying to reinvent type theory. There are plenty of other, more abstract languages that can be used to reason about ZFC in the way you are describing. Type theory, category and (the nascent but probably most interesting) homotopy type theory among others.

I recommend a text on abstract algebra if you are just interested in learning about reasoning about algebraic structures, such as sets, in general or a text on category theory or something if you want to know more about higher level mathematical languages.

>>581

Can you recommend me a good category or type theory text? I've become fairly comfortable (still learning of course) with the fundamentals of set theory and logic since the OP, but I know nothing about those two topics.

The big question is this: why should I bother learning type theory or category theory if (nearly) the whole of the mathematical edifice has been done in the context of set theory? I don't mean to be argumentative at all, but what's the point of having an entirely different system that is for all intents and purposes equivalent? What do you mean by reasoning about ZFC with category or type theory?

>>581

Are you talking about propositions as types?