$ST$ 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.

completewith respect to a language, if for every sentence $\sigma$ in that language, either $T\vdash\sigma$ or $T\vdash\neg\sigma$ (some people also insist that $T$ should be consistent). In this terminology, although Presburger arithmetic is complete in the language of addition, it is not complete in the language of arithmetic, because it neither proves nor refutes the axioms of PA. In your case, are you proposing to change the language of set theory? If not, then your weak theory will be incomplete, whether or not it supports coding. $\endgroup$ – Joel David Hamkins Oct 29 '15 at 13:18