Definability of isomorphisms between class well-orderings 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.


*

*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 $Ord^M?$

*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.
 A: 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.
