L and inaccessibles as models of ZFC

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In summary: What is happening is that the automatic function is not correctly inferring the TeX symbol for \kappa. When you put in \kappa, the [tex] automatically appears. To get the subscripts, you need to type in [itex]\kappa.
  • #1

nomadreid

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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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  • #2
Use [itex] instead of [tex] for latex inlined in a paragraph.
 
  • #3
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
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
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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1. What are L and inaccessibles?

L and inaccessibles are mathematical models used in set theory to study the Zermelo-Fraenkel set theory with the Axiom of Choice (ZFC). They are sets that satisfy certain properties and are used to understand the consistency of ZFC.

2. How are L and inaccessibles related to ZFC?

L and inaccessibles are specifically designed to be models of ZFC. This means that they satisfy all the axioms of ZFC, and can be used to prove the consistency of ZFC.

3. What is the significance of studying L and inaccessibles?

Studying L and inaccessibles allows researchers to gain a better understanding of the foundations of mathematics. It also helps to answer important questions about the consistency and completeness of ZFC, which is the most commonly used set theory.

4. How are L and inaccessibles constructed?

L and inaccessibles are constructed using the concept of transfinite recursion, which is a method of defining a sequence of sets by specifying the next set in the sequence in terms of the previous ones. This process continues until a certain limit is reached, resulting in the desired model.

5. Are there any limitations to using L and inaccessibles as models of ZFC?

Yes, there are some limitations to using L and inaccessibles as models of ZFC. For example, since they are constructed using transfinite recursion, they are only able to prove statements that can be expressed in terms of this process. Additionally, some mathematical statements may not be provable within these models.

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