Burali-Forti as a Purely Logical Paradox

Mathematics and Philosophy Seminar

• Speaker: Dr Graham Leach-Krouse (Kansas State University)

All welcome!

For more information please contact the convenor, Salvatore Florio.

Abstract

Russell's paradox is *purely logical* in the following sense: a contradiction can be formally deduced from the proposition that there is a set of all non-self-membered sets, in pure first-order logic—the first-order logical form of this proposition is inconsistent. This explains why Russell's paradox is *portable*—why versions of the paradox arise in contexts unrelated to set theory, from propositions with the same logical form as the claim that there is a set of all non-self-membered sets.

Burali-Forti's paradox, like Russell's paradox, is portable.  I offer the following explanation for this fact: Burali-Forti's paradox, like Russell's, is purely logical. Concretely, I show that if we enrich the language L of first-order logic with a well-foundedness quantifier W and adopt certain minimal inference rules for this quantifier, then a contradiction can be formally deduced from the proposition that there is a greatest ordinal.

Moreover, a proposition with the same logical form as the claim that there is a greatest ordinal can be found at the heart of several other paradoxes that resemble Burali-Forti's. The reductio of Burali-Forti can be repeated verbatim to establish the inconsistency of these other propositions. Hence, the portability of the Burali-Forti's paradox is explained in the same way as the portability of Russell's: both paradoxes involve an inconsistent logical form---Russell's involves an inconsistent form expressible in L and Burali-Forti's involves an inconsistent form expressible in L + W.