Set Size and the Part-whole Principle

Recent work has defended “Euclidean” theories of set size, in which Cantor’s Principle (two sets have equally many elements if and only if there is a one-toone correspondence between them) is abandoned in favor of the Part-Whole Principle (if A is a proper subset of B then A is smaller than B). It has also been suggested that Gödel’s argument for the unique correctness of Cantor’s Principle is inadequate. Here we see from simple examples, not that Euclidean theories of set size are wrong, but that they must be either very weak and narrow or largely arbitrary and misleading. §