It's known, for example in the answer to this question: [Is there a computable model of ZFC?](https://mathoverflow.net/questions/12426/is-there-a-computable-model-of-zfc) that ZFC has no computable model. My questions is: is there a model of ZFC for which the order relation on the class of ordinals is computable?



Computable models of the ordinal numbers
Yes, this can happen: if $M$ is a countable $\omega$-model of ZF whose well-founded part has ordertype $\omega\_1^{CK}$ (that is: has the shortest well-founded part possible for $\omega$-models), then $Ord^M$ as a linear order is just the **Harrison order**: $$\omega\_1^{CK}+(\omega\_1^{CK}\cdot\eta),$$ where $\eta$ is the ordertype of the rationals. This linear order does in fact have a computable copy, and is one of the basic examples/counterexamples in computable structure theory: a computable linear order which is illfounded but has no *hyperarithmetic* descending sequence (it has other nice properties too).


The key to seeing that this must be the ordertype is the following pair of observations:


* Given such an $M$ and an $\alpha\in Ord^M$, there must be an interval in $Ord^M$ beginning with $\alpha$ and isomorphic to $\omega\_1^{CK}$.
* No interval in $Ord^M$ can be isomorphic to $\omega\_1^{CK}+1$.


As to why such a model exists in the first place, this is trickier; depending how you phrase it, it's an application of either the **Gandy basis theorem** or the **Barwise(-Kreisel) compactness theorem**. Unfortunately, this doesn't have a one-line explanation.


