  Adopting AFA not only helps with circularly defined sets, but it also helps with streams and trees.  As we have mentioned, if one uses FA, there are no streams or trees according to our definitions.  That is, N∞ is literally the empty set with FA, as is Tr.  But with AFA these sets are non-empty.  Moreover, one can prove that under AFA, N∞ and Tr have the properties that one would want them to have.  (For example, one can prove that N∞ corresponds to the function space NN in the way we have discussed.) Finally, working out the resulting theory gives tools that are useful in studying collections of circularly-defined objects such as streams and trees.  The point is that this one axiom AFA gives us all of this, and more.