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?
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The last time I checked, Reinhardt Cardinals were not known to be inconsistent with ZF. Have you come across an article or link that says otherwise? – Everett Piper May 2 '14 at 1:58
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@Piper: no; I haven't claimed that they were. I'd be very suprised if there are any any as they seem to be defined within the language of ZF; and presumably similar definitions can be made for other set theories like NBG. – Mozibur Ullah May 2 '14 at 3:29
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.
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This possible belongs in meta, but why on earth doesn't this forum support LaTeX? – Neil Barton Apr 22 '14 at 2:55
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I don't know why this forum doesn't support it, but I've edited to look a bit clearer. – Mozibur Ullah Apr 22 '14 at 2:59
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Thanks. Just quickly; the in Delta^1_3, the 1 is a superscript on the Delta and the 3 is a subscript. I'll edit it now (rather than Delta^(1/3) which might be a little confusing). Nice to see people interested in this stuff! – Neil Barton Apr 22 '14 at 3:01
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