[cubefox]

The existence of a bijection is 7indeed just a possibility to define "size". But since it leads to counterintuitive results (some sets have bijections to proper subsets of themselves) this is hardly the definition of our intuitive concept of size. It leads also to many mathematical paradoxes, in fact most mathematical paradoxes involve infinities and this definition of size, e.g. the Ross–Littlewood paradox and the Banach–Tarski paradox.