H. P. Grice, "Cantorianum implicatum"

CANTORIANUM IMPLICATUM -- Cantor, Georg (1845–1918) German mathematician, born in St Petersburg, Russia, Professor, University of Halle. Cantor’s account of set theory and transfinite arithmetic established the basis of the logicist program of deriving mathematics from set theory and the mathematics of infinity. His treatment of the ordering of infinite sets, continuity and discontinuity, and the paradoxes of set theory have all had major consequences for mathematics and philosophy. His works are contained in Gesammelte Abhandlungen (1932). Cantor’s paradox Logic A paradox showing that we cannot treat the set of all sets as a set-theoretical entity. It was discovered by Georg Cantor through comparing the number of sets contained in the set of all sets S and the number of sets contained in PS (the power set of S), where the power set of a set is the set of all the subsets of that set. Cantor’s theorem shows that for any set A, its power set PA contains more sets than A. The paradox arises because no set can contain more sets than the set of all sets S, yet the power set of S does contain more sets than S. Cantor’s paradox and Burali-Forti’s paradox together are called the paradoxes of size. “In Cantor’s paradox it is argued that there can be no greatest cardinal number and yet that the cardinal number of the class of cardinal number . . . must be the greatest.” Quine, Selected Logical Papers