COMPARISON OF THE AXIOMS OF LOCAL AND GLOBAL UNIVERSALITY by MARCO FORTI and FKJRIO HONSELL in Pisa (Italy)') 1. I n 1965 D. SCOTT introduced in an unpublished paper (see [S]) a n axiom strongly contradicting the Axiom of Foundation. After him several authors, e.g. BOFFA ([I], [2]), HAJEK ([S]), considered the possibility of consistently replacing Foundation by axioms which postulate a strong non-well-foundedness of the membership relation. The authors examined this problem ([3], [ S ] ) , through a different approach, i.e. giving a "Free Construction Principle" for sets, while researching at the seminar on Foundations of Mathematics held by Prof. E. DE GIORGI a t the Scuola Normale Superiore in Pisa. The axioms derivable from this principle are studied in detail in [4].
In the absence of Regularity several problems, such as the classification of settheoretical formulas, the models of I-calculus, have an easier end more natural expresaion.
All these kinds of axioms share the idea that the membership structure is "universal" in some sense, that is it can model all of a wide class of extensional structures. Strange enough these axioms are far from being all equivalent ; the complexity of the E-relation can in fact be expanded to several degrees. M. BOFFA, in particular, proved the consistency of the Axiom of Universality which states that every binary extensional relation is isomorphic to the E-relation over a tranaitive set. As it happens for the Axiom of Choice or the Axiom of Regulaxity, in Gael-Bernays set theory both a class form (U) and a set form (U,) of the Axiom of Universality can be given. In [2], where BOFXA proved also the consistency of U, the problem of the equivalence between U and U, is left open.