$ST$ is the weak set theory built upon identity theory and containing 


1. the axiom for *empty set*,
2. the axiom for *adjunction* and
3. the axiom for *extensionality*.


It is known that $ST$ interprets Robinson Arithmetic, and so $ST$ is incomplete.


Is there a very weak set theory $ST^\*$ which is like $ST$ minus the axiom for extensionality, though possibly with some other very weak principles, so that $ST^\*$ is incomplete for Gödelian reasons by supporting arithmetization and the definition of a Gödelian provability predicate and the Gödel-Carnap diagonal lemma?


For some notions, cfr. [General Set Theory in Wikipedia](https://en.wikipedia.org/wiki/General_set_theory).



Is Extensionality needed for the incompleteness of very weak set theories?
$\newcommand\ST{\text{ST}}$Here is an example of a set theory $\ST^\*$ with your properties. Let $\ST^\*$ be the theory that is just like $\ST$, but it asserts extensionality only for nonempty sets. So $\ST^\*$ is consistent with the existence of multiple empty sets, and therefore it does not prove extensionality. But we can interpret $\ST$ in $\ST^\*$, simply by using only one of the empty sets (and restricting to those sets that do not use the other one in their transitive closures), or alternatively, by identifying the two empty sets hereditarily. 


So this is a weak set theory, a proper subtheory of $\ST$, which does not prove extensionality, but which still supports coding.


There are many other similar such theories, by allowing extensionality to be violated in a controlled manner, in such a way that you can still interpret $\ST$.


