# Thread: L and inaccessibles as models of ZFC

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

PS, how come, when I wrapped math brackets around ^{L}, it didn't go to superscript?

2. $$\kappa^{L}$$ gives $\displaystyle \kappa^{L}$.

3. Thanks, Plato. I hope the mathematical solution is as simple as the technical one.

4. If I remember correctly, then L is minimal with respect to any universe which has the same ordinals as L. But $\displaystyle L_{\kappa}$ doesn't contain the ordinal $\displaystyle \kappa$.

5. 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?

6. 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.

7. ## Thanks, DrSteve

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

It was, of course, silly to put $\displaystyle \kappa$ instead of $\displaystyle L_{\kappa}$, 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 $\displaystyle \alpha$ smaller than $\displaystyle \kappa$ such that $\displaystyle L_{\alpha}$ is a minimal model of ZFC. You have put me on the right track, so thanks again.

8. Another little tip: double-clicking on "Reply to Thread" takes you to the "Advanced Editing Mode" in one step.

9. Thanks, Ackbeet. Good tip to know.