# Is infinity an odd or even number?

My 6 year old wants to know if infinity is an odd or even number. His 38 year old father is keen to know too.

In the context of transfinite ordinals, the usual
definition is that an ordinal number $\alpha$ is even if
it is a multiple of $2$, specifically: if there is another
ordinal $\beta$ such that $2\cdot\beta=\alpha$. In other
words, the order type $\alpha$ can be viewed as $\beta$
many pairs in sequence, or in other words, $\alpha$ is
left-divisible by $2$. Otherwise, it is odd.

It is easy to prove from this definition by transfinite
recursion that the ordinals come in an alternating even/odd
pattern, and that every limit ordinal (and hence every
infinite cardinal) is even. Many transfinite constructions
proceed by doing something different on the even as opposed
to the odd stages, just as with finite constructions.

The smallest infinite ordinal is $\omega$, which is even on
this definition, since having $\omega$ many pairs in
sequence is order-isomorphic to $\omega$, and so
$2\cdot\omega=\omega$. Meanwhile, the next infinite ordinal
is $\omega+1$, which is odd. The ordinal $\omega+2$ is
even, since it is equal to $2\cdot(\omega+1)$, even though
it is not $\beta+\beta$ for any $\beta$.

(Please note that $\alpha=2\cdot\beta$ is not at all the
same as saying $\alpha=\beta+\beta$, since $\beta$ copies
of $2$ is not the same order type as $2$ copies of $\beta$,
a phenomenon at the heart of the non-commutativity of
ordinal multiplication. )

To explain the idea to a child, I would focus on the
principal idea: whether finite or infinite, a number is
even when it can be divided into pairs. For finite sets,
this is the same as the ability to divide the set into two
sets of equal size, since one may consider the first
element of each pair and the second element of each pair.
In the infinite context, as others have noted, there are
numerous concepts of infinity, each with its own concept of
even and odd. In my experience with children, one of the
easiest-to-grasp concepts of infinity is provided by the
transfinite ordinals, since it can be viewed as a
continuation of the usual counting manner of children, but proceeding into
the transfinite:

This concept of infinity is attractive to children, because
they can learn to count into the infinite this way. Also,
this concept of infinity has one of the most successful
parity concepts, since one maintains the even/odd pattern
into the transfinite. The smallest infinity $\omega$ is
even, $\omega+1$ is odd, $\omega+2$ is even and so on.
Every limit ordinal is even, and then it repeats even/odd
up to the next limit ordinal.

See the Wikipedia entries on transfinite
ordinals
and ordinal arithmetic for