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.