$ST$ is the weak set theory built upon identity theory and containing (1) the axiom for \textit{empty set}, (2) the axiom for \textit{adjunction} and (3) the axiom for \textit{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.