Tom Leinster awarded 2019 Chauvenet Prize
Tom Leinster has been awarded the 2019 Chauvenet Prize. This prize is awarded to the author of an outstanding expository article on a mathematical topic. First awarded in 1925, the Prize is named for William Chauvenet, a professor of mathematics at the United States Naval Academy. It was established through a gift in 1925 from J. L. Coolidge, then President of the Mathematical Association of America. Winners of the Chauvenet Prize are among the most distinguished of mathematical expositors.
"Rethinking Set Theory," The American Mathematical Monthly, 121 (2014), no. 5, 403-415. DOI: 10.4169/amer.math.monthly.121.05.403
Every mathematician knows that modern mathematics is an axiomatic system based on a theory of sets defined by the Zermelo-Fraenkel axioms plus the Axiom of Choice (ZFC). But how many of us can recite these axioms? Even after looking them up, are they in accord with our working understanding of sets? Or is the ZFC conception of sets necessarily nonintuitive as a result of having to rectify the difficulties of naive set theory discovered by Russell? In this paper, Tom Leinster tackles this issue with clarity and finesse. In 1964, F. William Lawvere proposed a revolutionary alternative axiomatization. Presented in the language of topos theory, an esoteric branch of category theory, Lawvere's axioms languished unrecognized by the majority of mathematicians of the era. Leinster reformulates Lawvere's axioms with striking simplicity. By expanding the primitive terms of the axiomatization to include functions and the composition of functions, Leinster replaces the ZFC axioms with ten axioms that can be easily understood—and accepted—by any undergraduate math major. Leinster issues a challenge: how many of us would be troubled to discover one day that ZFC is inconsistent? He speculates that most of us would be "unlikely to feel threatened by the inconsistency of axioms to which we never referred anyway." "Discovering an inconsistency in Lawvere's much more natural and intuitive axioms," Leinster conjectures, would be "devastating." Read the paper and judge for yourself!