transfinite arithmetic, cardinal arithmetic, ordinal arithmetic prime field, p-adic integer, p-adic rational number, p-adic complex number arithmetic geometry, function field analogy The cardinal numbers (or just cardinals) constitute a generalisation of natural numbers to numbers of possibly infinite magnitudes. Specifically, cardinal numbers generalise the concept of ‘the number of …’. In particular, the number of natural numbers is the first infinite cardinal number. Naïvely, a cardinal...