Cantor’s theory of cardinal numbers offers a way to generalize arithmetic from finite sets to infinite sets using the notion of one-to-one correspondence between two sets. As is well known, all countably infinite sets have the same ‘size’ in this account, namely that of the cardinality of the natural numbers. However, throughout the history of reflections on infinity another powerful intuition has played a major role: if a collection A is properly included in a collection B then the ‘size’ of A should be less than the ‘size’ of B (part-whole principle). This second intuition was not developed mathematically in a satisfactory way until quite recently. In this talk I begin by reviewing the contributions of some thinkers who argued in favor of the assignment of different sizes to infinite collections of natural numbers. Then, I present some recent mathematical developments that generalize the part-whole principle to infinite sets in a coherent fashion. Finally, I show how these new developments are important for a proper evaluation of a number of positions in philosophy of mathematics which argue either for the inevitability of the Cantorian notion of infinite number (Gödel) or for the rational nature of the Cantorian generalization as opposed to that, offered by Bolzano, based on the part-whole principle (Kitcher).