  The study of non-wellfounded sets proposes to treat every graph as a picture of a unique set.  In order to make this work, some kind of change is needed in set theory.  The reason is that sets like Ω = {Ω} do not exist in the most commonly-used set theory, ZFC.  This is due to the Foundation Axiom (FA): we’ll discuss this issue further in Section 2 below.  For now, FA implies that the only graphs with decorations are those with no infinite sequence of points following the arrows.  The change in set theory that we make is simply to replace this axiom FA with a different one called AFA.  The content of AFA is that every graph has a unique decoration (alternatively, every system of set equations has a unique solution).