Are there any large cardinals that are inconsistent with ZF?

Question

Reinhard cardinals are the largest cardinals yet defined that are consistent with ZF. This suggests that there are large cardinals that are inconsistent with ZF - are there in fact any?

Answer 1

So, there is a very facetious answer to your question; it depends what you mean by large cardinal axiom.

For instance 0-sharp is something like a large cardinal axiom without asserting the existence of a cardinal explicitly. It's rather about sets of indiscernibles that code the construction of L, and is very naturally viewed as a (Delta^1)_3 non-constructible real. However, 0-sharp fits very nicely into the large cardinal hierarchy, so is something like a large cardinal axiom; it's a principle of strength.

If we're talking about principles of strength then, there is a very obvious axiom that is inconsistent with ZF, namely 0=1. However, this is obviously not what you're looking for, rather you want an axiom that is in some sense "natural" or "conceivable" or "contentful" (or some other similarly philosophically problematic notion; cans of worms everywhere!), but that is inconsistent with ZF. Hugh Woodin and Peter Koellner have been working on thinking up such a large cardinal notion inconsistent with ZF; the hierarchy is roughly outlined in the Wikipedia article you cite (I would be somewhat careful of that, however; there are a couple of errors in the text). No inconsistency has yet been found, however.