Cantor’s Abstractionism and Hume’s Principle

Although both Cantor and Frege broadly used the same abstractionist strategy to define
numbers (in particular, cardinal numbers), Cantor’s conception has generally been viewed
as less justifiable and, overall, more problematic than Frege’s. Frege himself sharply
criticised Cantor’s conception in his review of Cantor (1890) (Frege (1892)). In this paper, we
aim to reassess Cantor’s conception on new grounds, show its fundamental
indistinguishability from Frege’s conception, and, above all, suggest that Cantor’s
abstractionism might help establish the inevitability of Hume’s Principle as the logical basis
of the definition of number, in particular, of the extension of the notion of ‘cardinality’ to
the infinite. In order to do this, we take into account several alternative notions of cardinal
number, including that provided by the recently developed theory of numerosities (in,
among other works, Benci-Di Nasso (2003)), and show that ‘good company’ arguments, put
forward by Heck (1997) and further discussed by Mancosu (2016), based on all such
alternative notions of cardinal number, fail to challenge the inevitability of Hume’s principle,
if the latter is construed in a way which does justice to a refined version of Cantor’s
abstractionism.