Let's take the minimal transitive model of $\sf ZFC$ which, I came to know, is some minimal $L_\kappa$ for a countable $\kappa$, that models $\sf ZFC$, and since its minimal so no subset of it can be a transitive model of $\sf ZFC$ and at the same time not isomorphic to it, call this stage $L_{\sf ZFC}$.
What kind of cardinalities $Th(L_{\sf ZFC})$ prove? I know it proves $\sf GCH$ because it's a model of $\sf ZFC + [V=L]$, but what theorems about cardinal existence it proves? Since this theory is complete, then it must decide on for example whether inaccessible cardinals exist or not?
A which large cardinal, $Th(L_{\sf ZFC})$ starts to prove its non-existence?
I tend to think that because it is minimal so it must prove the non-existence of inaccessibles whether strong or weak. Is that correct? And even if so, then what's the exact argument for that? If no, then at which large cardinal it starts proving its absence?
Best Answer
I don't like using the same notation to denote theories and ordinals, so I'll just use "$\alpha$" for the smallest ordinal such that $L_\alpha\models\mathsf{ZFC}$ (and assume that there is one). We have $L_\alpha\models$ "There is no transitive model of $\mathsf{ZFC}$." Consequently, $L_\alpha\models$ "There are no weakly inaccessible cardinals," since $\mathsf{ZFC}$ (which $L_\alpha$ satisfies!) proves "If $\kappa$ is weakly inaccessible then $L_\kappa$ is a transitive model of $\mathsf{ZFC}$."
In general, nothing deserving to be called a large cardinal principle will be consistent with even the fragment $T_0$ of (the "Platonic") $Th(L_\alpha)$ consisting of sentences which $\mathsf{ZFC}$ proves are satisfied by the least transitive model of $\mathsf{ZFC}$ if such exists. (Each model of $\mathsf{ZFC}$ + "There is a transitive model of $\mathsf{ZFC}$" will have something it thinks is $Th(L_\alpha)$, but different models may have disagreements about the details; however, they'll all agree about basic things like "$Th(L_\alpha)\models$ "There are no weakly inaccessible cardinals,"" and this is what I'm trying to capture with $T_0$ above.)