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Math Help - L and inaccessibles as models of ZFC

  1. #1
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    L and inaccessibles as models of ZFC

    Three facts:
    (1)The constructible universe L is the minimal model for ZFC;
    (2) L is a model of "there exists an inaccessible cardinal \kappa", and
    (3) if V=L,an inaccessible cardinal with the membership relation \epsilon is a model of ZFC.
    So, what is confusing me is: if the universe of L contains \kappa ^{L} , then how can L be the minimal model? Wouldn't < \kappa", \epsilon > be a model that is smaller?

    PS, how come, when I wrapped math brackets around ^{L}, it didn't go to superscript?
    Last edited by nomadreid; February 1st 2011 at 06:30 AM. Reason: problems with Latex
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  2. #2
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    [tex]\kappa^{L} [/tex] gives  \kappa^{L} .
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  3. #3
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    Thanks, Plato. I hope the mathematical solution is as simple as the technical one.
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  4. #4
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    If I remember correctly, then L is minimal with respect to any universe which has the same ordinals as L. But L_{\kappa} doesn't contain the ordinal \kappa.
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    Thanks, DrSteve. This is a key point: I did not know the bit about
    which has the same ordinals as L
    Thanks.

    Also you implicitly pointed out that my question should not have been "isn't < k,epsilon> a smaller model?" but "isn't <L_k, epsilon> a smaller model?" Again, thanks.

    Hm, on the "Quick Reply" mode the possibility to use LaTex seems to have disappeared. But DrSteve used LaTex in his reply. What is going on?
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  6. #6
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    If you hit the "go advanced button" after replying you will get the tex button back.

    Of course an ordinal can never be a model of set theory. For example the pairing axiom fails (most pairs of ordinals aren't ordinals).

    Please just note my statement "if I remember correctly." I haven't studied the constructable universe in a while, so just make sure you confirm that what I said regarding "having the same ordinals" is correct.
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  7. #7
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    Thanks, DrSteve

    Thanks, DrSteve. I got the LaTex back. I will try it out on this post.

    It was, of course, silly to put \kappa instead of L_{\kappa}, you're right.

    Rephrasing your suggestion about the minimal model, it does indeed make sense: it seems that L is the minimal inner model of ZFC, but there is an ordinal \alpha smaller than \kappa such that L_{\alpha} is a minimal model of ZFC. You have put me on the right track, so thanks again.
    Last edited by nomadreid; February 2nd 2011 at 04:50 AM. Reason: erased something incorrect
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    Another little tip: double-clicking on "Reply to Thread" takes you to the "Advanced Editing Mode" in one step.
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  9. #9
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    Thanks, Ackbeet. Good tip to know.
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