As discussed his week by Oswald Wyler, Vaughan Pratt, and Ernie Manes, the opposite of the category of small sets is equivalent to the algebras over the monad T obtained by dualizing into Z for Z>1 ; there is really no difference which Z is used. But that changes if we consider a truncation of T, i.e., the part supported by arity I for some I (the truncated monad has value at X equal to the union of the images of T(I)--> T(X) over all I-->X). There is still the possibility that some subcategory of sets will be full in the opposite of the category of algebras. For example, as Vaughan points out, if Z=3, I=1, at least the finite sets can be so captured. An important way to capture all small sets is to take Z=2 and I= countable (Ulam) or Z=countable and I=1 (Isbell in 1960 showed that these are equivalent). Taking unary operations (I=1) is always possible by enlarging Z. (For some reason Ulam spoke of measures, which is really quite misleading because measures are additive but here the condition that they be multiplicative is at least as important; likewise the set -theorists' terminology "measurable" (for sets too big to be captured) is misleading because "to be capturable" is intuitively to be measurable by procedures valued in Z.) I advocate that an additional axiom on small sets is that all such monads (Z infinite, I=1) should be the identity, because all known constructions of geometry and analysis preserve capturability and moreover all mathematical situations where one can reasonably expect a space/quantity duality are spoiled without this axiom on the underlying discrete sets. It is well known that the negation of the axiom, as applied to all sets, implies its consistency. Recognizing a category of small sets, so defined, as an ordinary object in the cartesian-closed category of categories of course in no way prevents the consideration of possible much larger sets or categories in the same system, no more than does recognizing the category of finite sets.