Is Infinity Even or Odd?

The question is naturally taken in a variety of ways—About the ordinals? Specifically ω? But are we dividing into pairs or cutting in half? Standard or natural arithmetic? Omnific integers? Cardinals?

A curious child asks:

Is infinity even or odd?

How shall we answer?

Think about it...

The child’s question may seem innocent, but actually there are diverse ways to take it, leading to a corresponding spectrum of answers. Perhaps it is natural initially to answer from the point of view of the least infinite ordinal ω with the standard ordinal arithmetic, for example, but even in this case there are two distinct concepts of even, depending on whether we imagine dividing into a sequence of pairs or cutting in half; we shall get still another answer if we should use the natural arithmetic; and we shall find further different answers when using different infinite ordinals; or when considering the question in the natural ring of ordinals; or in the omnific integers; we can answer with the cardinal numbers of set theory, either with the axiom of choice, or without—and it matters. Thus we shall discover a great diversity of answers for the child’s question, senses in which infinity is even, senses in which it is not, senses in which the answer depends on the particular infinity we have in mind, and senses in which the answer depends on our mathematical axiomatic foundations.

Let us explore!

Even and Odd in the Ordinals

Let us begin by considering the concept of even and odd in the transfinite ordinal numbers with the standard ordinal arithmetic.