TITLE: Is there a computable model of ZFC?
QUESTION [35 upvotes]: Background
Assuming ZFC is consistent, then by downward Löwenheim–Skolem, there is a countable model (M,$\in$) of ZFC. Since the universe M is countable, we may as well think of it as actually being the set of natural numbers, so $\in$ will be some binary relation on the natural numbers.

Can such a relation ever be computable?

Partial results
One can show that the class of binary relations $R$ on the natural numbers such that $(\mathbb{N},R) \models ZFC$ forms a $\Pi_0^1$ class, and will be nonempty so long as ZFC is consistent. This already gives us some interesting results. For example, by the low basis theorem, there is a low $R$ such that $(\mathbb{N},R) \models ZFC$. But I have been unable to determine whether such a function can be made computable; the best I can do is show that if such a function is computable, then there is no effective way of finding, given a finite set D of natural numbers, the element n such that D={m : mRn}.

REPLY [15 votes]: This can also be done using Godel-Roesser instead of Tennebaum.  Suppose M is a  model of ZFC.  There is an element $a$ of that M believes is the set of Godel codes for the sentences true of the integers of M. Of course $a$ may contain nonstandard Godel codes, but if we let T be the set of standard sentences with Godel codes in $a$, then T will be a complete consistent extension of Peano Arithmetic.  If M is computable, then T would be computable, but there are no computable complete consistent extensions of Peano Arithmetic.
Of course the appeal to Godel-Roesser just hides Joel's argument about recursive inseparable r.e. sets.