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?