Functional Set Theory

A recent article in The Intelligencer described in some detail the theory of non-well-founded sets for the purpose of being able to solve freely equations of the sort X = A × X with A fixed ([Barwise & Moss, 1991]). Of course, the empty set is a solution, but what was wanted was a non-empty solution (in fact, the largest solution, if such exists). If X were a non-empty solution and x ∈ X, then we would have x = 〈a1, x1〉 for some a1 ∈ A and x1 ∈ X. Here I use 〈−,−〉 to denote ordered pairs. What happens next depends to some extent on the definition of ordered pair, but let us take the common definition that 〈u, v〉 = {{u}, {u, v}} which is a set with one or two elements, depending on whether or not u = v. It seems a little complicated, but it works and I believe that the same problem would occur no matter what definition of ordered pair is taken. Let us write u ∈2 U to indicate that there is a t with u ∈ t and t ∈ U . For example, u ∈2 〈u, v〉 and v ∈2 〈u, v〉. Now getting back to the equation x = 〈a1, x1〉 with a1 ∈ A and x1 ∈ X, we have that x1 ∈2 x. Since x1 ∈ X, it can be written as x1 = 〈a2, x2〉 with a2 ∈ A and x2 ∈ X. This implies that x2 ∈2 x1. Continuing in this way, we have a sequence of elements xn ∈ X such that · · ·xn ∈2 xn−1 ∈2 · · · ∈2 x1 ∈2 x This infinitely descending ∈-chain is banned (by a specific axiom of “well-foundedness”) from standard treatments of set theory and the main point of the Barwise and Moss paper is to describe a novel kind of non-wellfounded set theory in which such infinite chains are allowed. The purpose of this note is to suggest that the whole enterprise is wrong-headed. It is clear that the “intended solution” to the equation X = A × X is that X should be the set of all sequences