A question about the comparability of large cardinals.

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.

• Somehow I lost condition (4). It stated that the sentences of T which assert that C(AX)<C(AY) and that C(AY)<C(AX) are each consistent with T, if T is consistent. – Garabed Gulbenkian Oct 22 '12 at 20:22
• There is quite an issue with using < 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. – Asaf Karagila Oct 22 '12 at 20:23
• Perhaps you should specify what constitutes a large cardinal axiom for you. Otherwise you could take AX to assert the existence of a third strongly inaccessible cardinal and AY to assert the existence of the maximum of the second strong inaccessible and the continuum. Then conditions 1-3 are satisfied and 4 can be forced either way. – Miha Habič Oct 22 '12 at 21:34
• Miha: You meant weakly, not strongly, inaccessible, since all strongly inaccessible cardinals are greater than the continuum. – Andreas Blass Oct 22 '12 at 22:26
• Andreas, you're right, which unfortunately makes my comment inapplicable. – Miha Habič Oct 22 '12 at 22:40