The theory of Zermelo-Fraenkel sets with Hilbert ɛ-terms

In this paper we study the role of functioning axioms on the deductive power of the system obtained from the Zermelo-Fraenkel ZF system by the introduction of -terms with the possibility of using them as a scheme for the substitution axiom. It is proved that if the system has a founding axiom the introduction of -terms does not extend the class of ZF theorems, while if the founding axiom is absent, there is an extension of the ZF theorems.Translated from Matematicheskie Zametki, Vol. 12, No. 5, pp. 569–575, November, 1972.