 The intuition that if one set is a proper subset of another, the former is smaller than the latter, traces back to Euclid—call this the part-whole intuition. Bolzano (1851) was sympathetic to it, and he tried to develop a theory of infinite sets that preserved it. He was not successful, but he warned his readers not to conflate one property of an infinite set—that it can be put in one-to-one correspondence with a proper subset of itself—with a criterion of ‘size’ (what he called the “multiplicity” of a collection). Cantor (see Hallett 1986), by contrast, later used one-to-one correspondence as the defining characteristic of cardinal numbers: the numbers that answer ‘how many?’ questions, and that generalize counting from the finite to the infinite in his set theory. He thus sided with the intuition that if there is a bijection between two sets, they have the same size—call this the bijection intuition. The intuition is clearly correct for finite sets. For example, the set of fingers on a normal human hand can be paired up with the set of toes on a normal human foot, and vice versa: there is a bijection between these two sets. And of course, the two sets have the same size (five). A central question is whether the intuition is correct also for infinite sets. We will discuss Cantor’s theory and, by contrast, some recent implementations of counting, known as theories of numerosities, that preserve the part-whole intuition also for infinite sets—see the