  First, the standard modeling of pairs in set  theory[2]  would have us believe that from the beginning of this section onwards, we have been talking about things which do not exist: as we have literally defined them, there are no streams of numbers whatsoever!  We discuss this at length in Section 2.2.1, when we talk about the Foundation Axiom of set theory.  The point is is that this axiom forbids object-level circularity in a way that precludes streams in the exact form that we have them.  Thus if one wants to model the intuitive notion of a stream as we have introduced it, one would need to say something like: “By a stream, we mean a function on numbers.  We adopt special notation to make it look like streams are pairs of a certain sort, but deep down they are just functions on numbers.”