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

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.

If $T$ denotes the theory $ZFC+AX+AY$, then $T$ has not yet been proved inconsistent.

$T$ proves that each of $C(AX)$ and $C(AY)$ is larger that the smallest strongly inaccessible cardinal number.

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.

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adjacent to another letter (without spacing), the parser usually truncates the post at that point. I edited that and added LaTeX, let me know if I did something wrong. Also, for future reference: line breaks are good; copy paste from a post on some other board/email/etc without any editing: not as good. $\endgroup$ – Asaf Karagila Oct 22 '12 at 20:23