Forcing and antifoundation

It is proved that the forcing apparatus can be built and set to work in ZFCA (=ZFC minus foundation plus the antifoundation axiom AFA). The key tools for this construction are greatest fixed points of continuous operators (a method sometimes referred to as "corecursion"). As an application it is shown that the generic extensions of standard models of ZFCA are models of ZFCA again. 1 Preliminaries It is well known that the constituents of forcing machinery, including the forcing relation - itself, are defined by -recursion. So it is natural to ask what happens when foundation is missing. Let ZFCbe ZFC without the foundation axiom. We shall show that in ZFCplus an antifoundation axiom, we can restore the forcing machinery by the help of greatest fixed points of continuous operators. ZFCwill be our basic system, although there are other interesting unfounded set theories, like New Foundations (NF). The reason is that after [1], the standard approach to non-well-foundedness has ...

• Type: Journal
• Year: 2005
• Where: AML
• Authors: Athanassios Tzouvaras
• Keywords: AML 2005, Antifoundation Axiom, Antifoundation Axiom Afa, Continuous Operators