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The various large cardinal axioms are usually described in terms of some roughly linear hierarchy of varying consistency strengths. However, some cardinal axioms potentially contradict one another. As an example, asserting the existence of unboundedly many strong limit cardinals implies AC. However, asserting the existence of Reinhardt cardinals or Berkeley cardinals implies the negation of AC (all in ZF). This seems to imply different mutually exclusive hierarchies of large cardinal axioms in ZF.

Are there other mutually exclusive LCA's in ZFC? If so, what are some of the exact statements that they conflict on? If not, why wouldn't we expect this in ZFC when it seems to be the case in ZF?

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    $\begingroup$ We don't really know. Also the statement that a class of strong limit cardinals implies the axiom of choice is only true under certain definitions of "strong limit", which---in the spirit of maximality---is therefore probably not the "right" definition if you want to work in ZF. $\endgroup$
    – Asaf Karagila ♦
    Jun 21, 2016 at 5:24

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The answer to your question depends on what counts as a large cardinal axiom, and there is no agreed-upon official definition for that term.

On the one hand, it is easy to formulate incompatible theories involving large cardinals. If these theories themselves count as large cardinal axioms, then they provide the answer.

  1. There is an inaccessible cardinal with CH versus there is an inaccessible cardinal with $\neg\text{CH}$.

  2. The least supercompact cardinal is Laver-indestructible versus the least supercompact cardinal $\kappa$ has $V_\kappa\not\subset\text{HOD}$.

  3. The least measurable cardinal is strongly compact versus every strongly compact cardinal is supercompact.

  4. There is an inaccessible cardinal and V=L versus $0^\sharp$ exists.

  5. $V=L[\mu]$ versus $0^\dagger$ exists.

Some researchers prefer to adopt a narrow sense of what counts officially as a large cardinal axiom, and on that perspective, these theories are not large cardinal axioms themselves, but theories about large cardinals, and so they do not constitute examples.

Other researchers use a broader sense for what counts as a large cardinal axiom, and with that perspective, there are abundant examples.

So it may come down to an issue of semantics.

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