Height of minimal model of ZFC

Question

What do we know about the height of the minimal (transitive) model of ZFC, that is, about the least α such that L_α is a model of ZFC? Call this ordinal μ. It is countable, since otherwise we could build L inside the collapse of a countable elementary submodel of L_μ to obtain a lesser such α. Moreover, as I learned here on MO, μ is Δ¹₂ definable in V.

Answer 1

I claim that the ordinal $\mu$, if it exists, is actually $\Pi^1_1$ definable, which improves on your $\Delta^1_2$ claim. What I mean by this is that the set of reals coding a relation on $\omega$ having order type $\mu$ is a $\Pi^1_1$ set of reals. Even more, I claim that it is a $\Delta^1_1$ property about reals, among those that do code well-orders. That is, the set of codes for $\mu$ is the intersection of WO with a hyperarithmetic set.

First, let's show that the set of reals $x$ coding a relation $\lhd$ of order type $\mu$ has complexity $\Pi^1_1$. It is well-known to be a $\Pi^1_1$ property of $x$ to assert that the relation $\lhd$ it codes on $\omega$ is a well-order, of some order type $\alpha$. I claim that $x$ is coding $\mu$ if and only if, in addition, for every countable structure $M$, if $M$ satisfies V=L and the ordinals of $M$ have an initial segment isomorphic to $\alpha+1$, then $M$ thinks that the $\alpha$-th ordinal is the height of the minimal model of ZFC.

This additional property is indeed $\Pi^1_1$. To see this, note first that the assertion that $M$ satisfies a certain theory is an arithmetic property about $M$, and the existence of the order-isomorphism from $\lhd$ to an initial segment of $M$ is a $\Sigma^1_1$ assertion, but it appears in the hypothesis of an implication here, so altogether the assertion is $\Pi^1_1$. The critical fact we are using here is that we don't assert that $M$ is fully well-founded, but only that it is well-founded beyond the height of the order coded by $x$, which is sufficient to determine whether $x$ codes $\mu$ or not.

Let us now argue that this property does indeed define the codes of $\mu$. If $x$ does code $\mu$, then any model $M$ satisfying V=L and well-founded up to $\mu+1$ will agree that $\mu$ is the height of the minimal model of ZFC, so $x$ will pass this definition. Conversely, if $x$ codes a well-order of order type $\alpha$ and any model $M$ thinks that $\alpha$ is the height of the minimal model, then it will be right.

What the argument shows is that the set of codes is a $\Delta^1_1$ property, that is, a hyperarithmetic property, on the set WO of codes for well orders. Given that $x$ codes a well-order, then we can say that $x$ codes $\mu$ if and only if, every countable $M$ satisfying V=L and well-founded beyond the height of the ordinal $\alpha$ coded by $x$ agrees that $\alpha$ is $\mu$; and also, if and only if there is such an $M$. So we've got a $\Pi^1_1$ and a $\Sigma^1_1$ characterization, provided that we already know that $x$ is coding a well-order.

We can get a unique code, rather than a set of codes, by observing that if $\mu$ exists, then not only is it countable, but it is countable in $L$, and so there is an $L$-least code for $\mu$. This code is also $\Pi^1_1$-definable, since $z$ is this $L$-least code if and only if it does code a well-order of type $\alpha$, and all countable models of V=L that are well-founded at least that high and that think that that ordinal $\alpha$ is countable, think that $z$ is the $L$-least code of $\alpha$. The point again is that such models, even if ill-founded up high, will be right about this information.