$ST$ 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 some notions, cfr. General Set Theory in Wikipedia.