Numerosities

For the philosophically, or foundationally, inclined, the below post argues for something called "numerosities" that give "part-whole" set relationships priority over "one-to-one" relationships, and in so doing, counter the usual Cantorian orthodoxy, which permits a partial set (say odd numbers), to be deemed equal in size to an entire set (all integers):

http://www.newappsblog.com/2014/03/counting-infinities.html

Excerpt: "The philosophical implications of the theory of numerosities for the philosophy of mathematics are far-reaching... Philosophically, the mere fact that there is a coherent, theoretically robust alternative to Cantorian orthodoxy raises all kinds of questions pertaining to our ability to ascertain what numbers ‘really’ are (that is, if there are such things indeed)."