$ST$ is the weak set theory built upon identity theory and containing
- the axiom for empty set,
- the axiom for adjunction and
- 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.