Is there an effective set theory $T$ such that $T + $$TA$ is consistient and complete. It should at least prove all theorems of $ZF$ true, so that it is a "standard" set theory. In particular, the axiom of infinity is required, since otherwise finite set theory would work, making the answer trivial. By complete, I mean every statement in the language of set theory is either provable or disprovable.
Another way of phrasing this question is what are a set of axioms of $T$ (which is either finite or such that its contents can be listed by a Turing machine), such that for any statement in the language of set theory, it can be decided by $T$ and statements in arithmetic (which are true of the standard integers).
If it fails, perhaps the requirement that $T$ be effective could be replaced by a requirement that $T$ is an Arithmetical set. Also, perhaps $TA$ could be replaced with some other consistent and complete theory of arithmetic (although this wouldn't be as useful as with $TA$).
The nice thing about such a theory is that, once $T$ is finalized, selecting new axioms would be metaphysically simpler, you could say. Instead of arguing over whether a given axiom is actually true of sets or not, the problem would be reduced to arguing about whether statements in arithmetic are true of integers or not (after $T$ has been selected).
tl;dr. Can you have a complete set theory, modulo arithmetic.