In response to John Baez and Steve Vickers, let me put my question more directly. What categories of "finite" (in any sense of finite you like) objects can be characterized up to equivalence as the finite objects (again feel free to define this notion yourself) common to all categories of some elementary class? I showed that finite sets and finite Boolean algebras could be so characterized. What about finite Abelian groups, or finitely presented (by generators and relators) Abelian groups, or finite dimensional vector spaces over some field? Once an appropriate notion of finiteness is settled on, these become straightforward yes-no questions. A related question is, what categories can be characterized up to equivalence by purely elementary means? I hope it's clear why I asked what I did and not this. (Loewenheim-Skolem and all that.) Like John Baez, I like free algebras and cofree coalgebras as methods of characterization. (Why else would Dusko Pavlovic and I bother to characterize the continuum as a cofree coalgebra?) I would immediately withdraw whatever I said that conveyed the opposite if I knew what it was. In asking about definability in a given framework (here first order logic plus cardinality restrictions) I had not intended to imply even endorsement of that framework, let alone rejection of other frameworks. In defense of sets, I very much like them as a foundational concept, in considerable part because one can reach a larger audience by starting from sets than from sheaves. I was impressed that anyone would dislike sets so much as to compare starting from them to standing in wet concrete until it sets (or were you just making a pun about sets, Steve?). Sheaves as a starting point is fine for category theorists, who are equipped to benefit from its greater generality, but they are singularly inappropriate for most other mathematical audiences, for most of whom experience with adding up the restaurant bill has made natural numbers much more familiar than sheaves. I'm not objecting to crash courses on sheaves here, just to talks for a general mathematical audience that start out "Ladies and gentlemen, let S be a sheaf." Mentioning categories on Steve Simpson's FOM mailing list typically brings on a diatribe from someone railing against categories. It would be nice if one could mention sets on this mailing list without the analogous response. John Baez makes exactly the right connection between sets and the free monoid on one generator (Peter Selinger made a related remark to me privately about the free cocomplete category on one generator satisfying FinC ~ FinSet). The categorification of the semiring of natural numbers yielding the distributive category of sets is natural, simple, beautiful, and easily understood. However I disagree that the assumption of finiteness constitutes smuggling in FinSet. A tiny part of it, fine, but that's a long way from smuggling in the whole notion of function. The notion of linear order with endpoints can be characterized elementarily up to isomorphism if one restricts attention to countable linear orders, but how much of the notion of linear order does countability smuggle in here? Cardinality restrictions as an axiomatization strategy convey no substantial structural information, and are one of the mildest possible excursions outside first order logic when such is unavoidable. They have a long and distinguished history in logic. Vaughan Pratt