L and inaccessibles as models of ZFC

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  • #1
nomadreid
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Main Question or Discussion Point

Three facts:

(1)The constructible universe L is the minimal model for ZFC;

(2) L is a model of "there exists an inaccessible cardinal [tex]\kappa[/tex]"; and

(3) if V=L, an inaccessible cardinal [tex]\kappa[/tex] with the membership relation [tex]\epsilon[/tex] is a model of ZFC.

So, what is confusing me is: if the universe of L contains[tex]\kappa[/tex] [tex]^{L}[/tex] , then how can L be the minimal model? Wouldn't <[tex]\kappa[/tex],[tex]\epsilon[/tex]>" be a model that is smaller?

P.S. Why are my Greek letters all getting superscripted? I only asked for L in (3) to be superscripted.
 
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Answers and Replies

  • #2
Hurkyl
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Use [itex] instead of [tex] for latex inlined in a paragraph.
 
  • #3
nomadreid
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Thanks, Hurkyl. I wanted to try that, but in the Quick Reply mode, the list of LaTex is no longer there. I don't know how to access it in Quick Reply. When I did have it in posting the first time, I have my list of LaTex symbols to the side, and I just click on a symbol. The [tex] then appears automatically. In order to change this to [itex], is it sufficient just to type in the "i" into the brackets provided to me by the automatic function?

Back to the original question: my point #3 should have read
"L_k with the epsilon function is a model of ZFC. Since L_k is a subset of L, why isn't <L_k, epsilon> minimal, not <L, epsilon>?"
A possible explanation is that the statement that (1) should read not "L is a minimal model of ZFC", but "L is a minimal inner model of ZFC", and that "L_k" does not contain all the ordinals. Is this correct?
 
  • #4
Hurkyl
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Yes, you only need to insert the i. You can even remove the superfluous closing/opening tag between adjacent latex symbols.


Are you sure you have the details right? e.g. should fact (1) be something like "L is minimal among all models containing the ordinals that L contains"?
 
  • #5
nomadreid
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Hurkyl, thanks. First, for the LaTex hint.

Now, for the mathematics: More exactly, (1) is
(1') Given a model <M, [itex]\epsilon[/itex] ,...> of ZFC , then there exists a model <N,[itex]\epsilon_{M}[/itex] ,...> which is a model of ZFC such that N is a subclass of M and N contains all the ordinals of M. Since <V,[itex]\epsilon[/itex] ,...> is a model of ZFC, this means that there is a minimal model which contains all the ordinals. Before going on, I should also mention that (3) should have been <L_[itex]{\kappa}[/itex] , [itex]\epsilon[/itex]>, not < [itex]\kappa,\epsilon[/itex]>, which changes things. So, it appears to me that the solution is that < L, [itex]\epsilon[/itex]> is the minimal inner model (that is, containing all ordinals), but not the minimal model (since L_[itex]L_{\kappa}[/itex] is a proper subclass of L). However, any confirmation of this would be appreciated.

P.S. For some reason I can't get the subscripts. But I will keep trying.
 
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