Decidability of model of set theory. I am wondering if there exists a model of set theory (KPU or ZFC) that has decidable theory?
Thanks.
 A: Any model of set theory includes the full true theory of arithmetic. Even very weak theories of arithmetic, let alone the full True Arithmetic, are undecidable.
A: The viewpoint that is usually taken in the literature is to look at extensions of the original theory rather than models of it. A theory is said to be essentially incomplete if it has no decidable complete consistent extension.  Essential incompleteness implies, in particular, that the theory cannot have a model whose elementary diagram is decidable. 
A corollary of the incompleteness theorems is that many effective theories are essentially incomplete. For example, Peano arithmetic and ZFC are both essentially incomplete, so they cannot have models whose elementary diagrams are decidable. 
On the other hand, the theory of groups is incomplete, because it cannot prove or disprove $(\exists x)(\exists y)(x \not = y)$, but it can be made complete by adding the axiom $(\forall x)(\forall y)(x = y)$, and the resulting theory (which is categorical) has a model whose elementary diagram is decidable. 
