$ST$ 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][1].


  [1]: https://en.wikipedia.org/wiki/General_set_theory