Conceptualizing and axiomatizing Mike Barr's experience with teaching membership-based set theory is shared by many mathematicians, and quite a few share his conclusions. One conclusion is that clarification is needed on even more basic questions than just the large/small issue (which concerned Goedel, Mac Lane, and Perez), in order to arrive at conceptions and axiomatizations compatible with the practice of mathematics. For example, I was aiming at such a clarification in pp. 118-128 of my 1976 paper in honor of Professor Eilenberg's 60th birthday, where I advocated some rational connection between conceptualizing and axiomatizing. The complete lack of such a connection in a recent article in the journal "Mathematical Structures in Computer Science" could have been avoided by the editors, if not by the authors. In a section labeled "Basic Set Theory" (p.510) they quote from my above paper a description of the notion of abstract set: 1. ...each element of X has no structure whatsoever. 2. X itself has no internal structure except for equality and inequality of pairs of elements.... immediately followed by their absurd conclusion: "axiomatically this corresponds to taking the membership relation epsilon as the only primitive notion of set theory and to postulating .." some axioms typical to Zermelo-style membership-based theory! Of course those axioms are NOT compatible with the conception quoted: they violate (1) because according to the Zermelo primitives and axioms, an element usually has elements, which would be structure; and they violate (2) since according to those primitives and axioms, a pair of elements of X may stand itself in the membership relation, which would be an internal structure other than equality. The authors neglected to quote the third clause which (as in the example that Mike mentions) their axioms also violate. The notion of abstract set (Kardinalen in Cantor's sense) is basic among the many other notions of cohesive and/or variable sets (Mengen) to the extent that we can model the Mengen via diagrams of maps between abstract sets. Abstract sets may be "abstracted from" less abstract sets, as Cantor did, or used, as most modern mathematics in practice does, as all-purpose memory cells or parameterizers or nodes in such diagrams. In addition to the papers by Colin McLarty mentioned in his message of November 11, 1998, the following papers should help to clarify this notion and its role in mathematics. J. Isbell Adequate sub-categories Illinois J. Math. vol. 4, pp 541-552, 1960 F. W. Lawvere An elementary theory of the category of sets Proc. Nat. Acad. Sc. USA, vol. 52, 1964, pp 1506-1511 F. W. Lawvere Variable quantities and variable structures in topoi, (see especially pp 118-128) in Algebra, Topology, and Category Theory, ed. Heller & Tierney, Academic Press, 1976 F. W. Lawvere Cohesive toposes and Cantor's lauter Einsen (concerning Cantor's neglected Kardinalen) Philosophia Matematica, vol. 2, 1994, pp 5 - 15 W. Mitchell Boolean topoi and the theory of sets (the membership-free content of Goedels constructible sets still needs to be clarified further) Journal of Pure and Applied Algebra, vol. 2, 1972, pp 261-274 ******************************************************************** F. William Lawvere Mathematics Dept. SUNY wlawvere@acsu.buffalo.edu 106 Diefendorf Hall 716-829-2144 ext. 117 Buffalo, N.Y. 14214, USA *********************************************************************