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Inspired by this question, I've been trying to figure out for myself the basic properties of definable class well-orderings in transitive models $M$ of ZFC: What is $\omega_1^{CK}(\mathsf{Ord})$?

There were a couple things I couldn't figure out.

  1. Is it necessarily the case that $M$ is correct about well-orderings? I.e., if $M \models ``\varphi(\cdot, \cdot)$ is a total ordering of $Ord" \wedge \forall x(x=\emptyset \vee \exists y \in x(\forall z \in x(\varphi(y, z)))),$ then $\varphi$ truly (in $V$) defines a well-ordering of $Ord^M?$
  2. If $\varphi_1$ and $\varphi_2$ define in $M$ isomorphic well-orderings of $Ord^M,$ is there necessarily a definable isomorphism $\psi(\cdot, \cdot)$ between them? E.g., if $\alpha$ is the first ordinal to exceed a proper class of ordinals under $\varphi_1$ and $\beta$ is the first such ordinal under $\varphi_2,$ then we would have $\psi(\alpha, \beta).$

(1) is easy to prove in the case $Ord^M$ has uncountable cofinality in $V,$ but it isn't obvious to me whether it can fail in models of countable cofinality height. (2) seems unlikely, since I don't really see how to extend transfinite recursion past the ordinals (especially in light of potential failures of (1)), but I have trouble imagining what a counterexample would like.

Elliot Glazer
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1 Answers1

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The answer to question 1 is that no, a transitive model $M$ can be wrong about whether a definable class relation is a well-order.

To see this, consider a transitive model $M$, and let me assume that there is no worldly cardinal in $M$. For example, perhaps we have cut off the universe at the smallest worldly cardinal.

By the reflection theorem, we know that for any given $n$, there are many ordinals $\theta$ in $M$ with $V_\theta^M\prec_{\Sigma_n} M$. In particular, we can make an increasingly elementary chain $$V_{\theta_0}^M\prec_{\Sigma_1} V_{\theta_1}^M\prec_{\Sigma_2}\cdots\prec_{\Sigma_n} V_{\theta_n}^M\prec_{\Sigma_{n+1}}\cdots$$ that unions up to $M$.

Let $T$ be the tree of all finite sequences that obey this increasingly elementary substructure relation with one another, and where also the $n^{th}$ element in the sequence also models the $\Sigma_n$ fragment of ZFC. It follows from our observation above that there are a proper class of such instances in $M$, and so $T$ is a proper class. Elements of $T$ amount to finite sequences of ordinals, which can be coded by single ordinals, and so we may view $T$ as a partial order relation on $\text{Ord}$, if you like.

We order this tree growing downward, and I claim it is well-founded in $M$. That is, $M$ has no $\omega$-sequence that is a descending sequence in this tree, because the union of that chain would be a $V_\theta$ that models ZFC, and so $\theta$ would be worldly, but we assumed there are none in $M$.

So $M$ thinks this tree order is well-founded, and therefore it thinks the Kleene-Brouwer order on the tree is a well-order. But $M$ is wrong about both of these things, since we have already observed that $M$ is the union of an increasingly elementary chain, and this is exactly a descending sequence in the tree order and hence a descending sequence in the Kleene-Brouer order.

So $M$ thinks the relation was a well-order, but it was mistaken.

Joel David Hamkins
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