$ST$ 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.