文档介绍：Reprints in Theory and Applications of Categories, No. 11, 2005, pp. 1–35.
AN ELEMENTARY THEORY
OF THE CATEGORY OF SETS (LONG VERSION)
MENTARY
F. WILLIAM LAWVERE
Received by the editors 2005-04-01.
Transmitted by M. Hyland, A. Kock, R. Rosebrugh. Reprint published on 2005-05-23.
2000 Mathematics Subject Classification: 18B05, 00A30, 03A05.
Key words and phrases: Category of sets, Axiom of choice, Mathematical logic and foundations.
This article is an expanded version of ‘An elementary theory of the category of sets’, Proceedings of
the National Academy of Science of the 52, 1506–1511. Article mentary c F. William
Lawvere and Colin McLarty. Permission to copy for private use granted.
1
2 F. WILLIAM LAWVERE
ETCS and the Philosophy of Mathematics
Colin McLarty
Philosophers and logicians to this day often contrast “categorical” foundations for math-
ematics with “set-theoretic” foundations as if the two were opposites. Yet the second
categorical foundation ever worked out, and the first in print, was a set theory—Lawvere’s
axioms for the category of sets, called ETCS, (Lawvere 1964). These axioms were writ-
ten soon after Lawvere’s dissertation sketched the category of categories as a foundation,
CCAF, (Lawvere 1963). They appeared in the PNAS two years before axioms AF
were published (Lawvere 1966). The present longer version was available since April 1965
in the Lecture Notes Series of the University of Chicago Department of It
gives the same definitions and theorems, with the same numbering as the 5 page PNAS
version, but with fuller proofs and explications.
Lawvere argued that set theory should not be based on membership (as in Zermelo
Frankel set theory, ZF), but on “isomorphism-invariant structure, as defined, for exam-
ple, by universal mapping properties”(p. 1). He later noticed that Cantor and Zermelo
differed over this very issue. Cantor gave an isomorphism-invariant account of sets, where