Are there any examples of two large cardinal axioms $AX$ and $AY$, in the language of first order $ZFC$, which satisfy the following conditions.


1. Each of them defines a unique cardinal number - $C(AX)$ for $AX$
and $C(AY)$ for $AY$ - not like the axiom of measurable cardinals which defines a whole collection of cardinal numbers.
2. If $T$ denotes the theory $ZFC+AX+AY$, then $T$ has not yet been proved inconsistent.
3. $T$ proves that each of $C(AX)$ and $C(AY)$ is larger that the smallest strongly inaccessible cardinal number.
4. The sentences of $T$ stating that $C(AX) < C(AY)$ and that $C(AY) < C(AX)$ are each consistent with $T$, if $T$ is consistent.



A question about the comparability of large cardinals.
Let AX = "there exists a least strongly compact cardinal", and let AY be "there exists a least whatever cardinal" (there are many possibilities for "whatever", e.g., "inaccessible with a measurable below it" if you want to be modest). 


C(AX) = the least strongly compact, C(AY) = the least whatever, say the least inaccessible past the least measurable.


It is well known (Magidor) that the least strongly compact can be the least supercompact or the least measurable. In the first case, C(AX) is much larger than C(AY), in the second case, slightly smaller. 


(Arthur Apter has written several papers about this "identity crisis of strongly compacts". Many more examples in the same spirit can be obtained from his results.) 


