Subject: categories: re: Sets

From: F W Lawvere <[email-masked]>
Date: Mon, 16 Nov 1998 15:07:52 -0500 (EST)

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

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F. William Lawvere			Mathematics Dept. SUNY 
[email-masked]               106 Diefendorf Hall
716-829-2144  ext. 117		        Buffalo, N.Y. 14214, USA

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