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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?

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    $\begingroup$ The minimal model satisfies "there is no well-founded model of ZFC". In particular, it has no inaccessible cardinals. (Strong and weak inaccessibility are the same here because the minimal model satisfies GCH.) $\endgroup$ May 13, 2022 at 17:30

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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.)

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  • $\begingroup$ hmmm... when you said they have disagreements about the details, are those details external or internal in the minimal models seen by them. I mean if we have two models M1 and M2 of ZFC + there is a transitive model of ZFC, then is it the case that the minimal models $L^1_{\sf ZFC}; L^2_{\sf ZFC}$ seen by those respectively, would prove different theorems? $\endgroup$ May 13, 2022 at 18:21
  • $\begingroup$ @ZuhairAl-Johar For example, every arithmetic sentence is absolute between $M$ and $(L_\alpha)^M$, so if $M_1$ and $M_2$ "believe" different arithmetical sentences (e.g. one thinks $\mathsf{ZFC}$ + an inaccessible is inconsistent while the other doesn't) then their respective minimal models will also "believe" different arithmetical sentences. $\endgroup$ May 13, 2022 at 18:39
  • $\begingroup$ So $T_0$ is like the intersection of all those minimal theories, I mean the set of all sentences provable in all the minimals, right! $\endgroup$ May 13, 2022 at 19:58
  • $\begingroup$ @ZuhairAl-Johar Yes, but we have to be a bit careful: not every model of $\mathsf{ZFC}$ + "$\mathsf{ZFC}$ has no transitive model" is the minimal transitive model of some other model of $\mathsf{ZFC}$! For example, if $M\models\mathsf{ZFC+\neg Con(ZFC)}$, then no $N\models\mathsf{ZFC}$ has $N\models$ "$M$ is a transitive model of $\mathsf{ZFC}$." $\endgroup$ May 14, 2022 at 5:25
  • $\begingroup$ Ah! I see. I think a minimal needs to be a model seen as minimal by a model of ZFC+ there is a transitive model of ZFC. So $T_0$ is about what is true in all of those kinds of minimals. $\endgroup$ May 14, 2022 at 6:31

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