The status of Cantorian numbers

Abstract

A critical evaluation of Cantor's number conception is undertaken against which the interpretations by Wang and Hallett of Cantoran set theory are measured. Wang takes Cantor's theory to tend to be a theory of numbers rather than a theory of sets, while Hallett takes Cantor as proposing an ordinal theory of cardinal numbers which however permits Cantor to accept ordinal numbers as given without defining them. The evidence presented, however, shows that Cantor conceived numbers, both cardinals and ordinals, as extensional objects, and while either Wang's or Hallett's interpretations eliminate certain difficulties of Cantoran set theory, neither one of them is an accurate depiction of Cantor's theory.