Counting Sets with Surnatural Numbers

Abstract

How many odd numbers are there? How many even numbers? From Galileo to Cantor, the suggestion was that there are the same number of odd, even and natural numbers, because all three sets can be mapped in one-one fashion to each other. This jars with our intuition: cardinality fails to discriminate between sets that are intuitively of different sizes. The class of surreal numbers $\boldsymbol{\mathsf{No}}$ is the largest possible ordered field. In this work we define a function, the magnum, mapping a selection of countable sets to a subclass of the surreals, the surnatural numbers $\boldsymbol{\mathsf{Nn}}$. Set magnums are found to be consistent with our intuition about relative set sizes. The magnum of a proper subset of a set is strictly less than the magnum of the set itself, in harmony with Euclid's axiom, ``the whole is greater than the part''. Two approaches are taken to specify magnums. First, they are determined by following the genetic assignment of magnums in the way the surreal numbers themselves are defined. Second, the domain of the counting sequence, which is defined for every countable set, is extended, to evaluate it on the surnatural numbers. The two methods are shown to be consistent. For a subset $A$ of $\mathbb{N}$, the magnum is defined as the value at $\omega$ of the extended counting function of $A$. Larger sets are partitioned into finite components and a more general definition of magnums is presented. Several theorems concerning the properties of magnums are proved, and are employed to evaluate the magnums of a range of interesting countable sets. The relativity of the magnum function is discussed and a number of examples illustrate how its value depends on the choice and ordering of the reference set. In particular, we show how the rational numbers may be ordered in such a way that all unit rational intervals have equal magnums.