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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.

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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
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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
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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
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1 Answer 1

up vote 14 down vote accepted

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.)

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See also mathoverflow.net/questions/63735/… –  Goldstern Oct 23 '12 at 6:28
    
Thanks for pointing out to me how many examples can be constructed of pairs of large cardinal axioms (which cannot be proved in ZFC) and which satisfy my conditions. I am trying to get a picture of how far up into the set-theoretical hierarchy one can reach with a collection C of large cardinal axioms whose cardinal numbers are totally ordered by size, and whose conjunction is not known to be inconsistent with ZFC. Pairs of axioms which satisfy my conditions should probably not both belong to C. –  Garabed Gulbenkian Oct 23 '12 at 18:00
    
I must also thank you, Asaf, for doing such a wonderful editing job on my post. I had no idea what to do to get out of the mess I had made with the end of it. –  Garabed Gulbenkian Oct 23 '12 at 18:07
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