 Dedekind showed how to fill the gaps between rational numbers; Cantor showed how to extend (ordinal and cardinal) numbers beyond the existing finite numbers. John Conway (1976) integrated both ideas. He developed a very different system that generalizes von Neumann’s representation of Cantor’s ordinals, as well as Dedekind’s representation of the real numbers, to generate a much larger field that has become known as the “surreal numbers”. It contains a copy of each ordinal and cardinal number, while defining operations that work just like addition, subtraction, multiplication, division, exponentiation, and the taking of roots, on the standard reals. In particular, even the infinite and infinitesimal surreal numbers are amenable to these operations. Thus, as well as familiar numbers, we now have numbers such as \(\sqrt\omega, \omega/2, -\omega, 1/\omega, -\omega^\omega,\) and so on. Indeed, as Ehrlich (2001, 2012) observes, the surreal numbers may plausibly be regarded as including “all numbers great and small”! The surreal numbers can apparently be applied in cases where there is no straightforward way to use hyperreals, as for example in the treatment of Pascal’s Wager, discussed in section   7.3—see Hájek (2003a).