 Foundational programs such as neo-logicism also start from a notion of “equinumerosity” based on the Cantorian bijection intuition. It was not until the early 2000s that a group of mathematicians working on non-standard analysis (Benci, di Nasso, and Forti) developed a theory of “numerosities” that agrees on finite sets with Cantorian cardinality but that also upholds the part-whole intuition for infinite sets, and thus diverges from Cantorian cardinalities. (See Benci and Di Nasso 2003, 2019; Benci, Di Nasso and Forti 2006, 2007). One can consider numerosities as a refinement of Cantorian cardinalities. Every two sets that have the same numerosity have the same cardinality, but the converse does not hold. For instance, in this approach the set of squares has numerosity strictly less than the set of natural numbers. See the