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?



Are There Mutually Exclusive Large Cardinal Axioms in ZFC?
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.


