COMPARISON OF THE AXIOMS OF LOCAL AND GLOBAL UNIVERSALITY

1. In 1965 D. SCOTT introduced in an unpublished paper (see [S]) an axiom strongly contradicting the Axiom of Foundation.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].