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.

**Answer**

In the context of transfinite ordinals, the usual

definition is that an ordinal number α is *even* if

it is a multiple of 2, specifically: if there is another

ordinal β such that 2⋅β=α. In other

words, the order type α can be viewed as β

many pairs in sequence, or in other words, α 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 ω, which is even on

this definition, since having ω many pairs in

sequence is order-isomorphic to ω, and so

2⋅ω=ω. Meanwhile, the next infinite ordinal

is ω+1, which is odd. The ordinal ω+2 is

even, since it is equal to 2⋅(ω+1), even though

it is not β+β for any β.

(Please note that α=2⋅β is not at all the

same as saying α=β+β, since β copies

of 2 is not the same order type as 2 copies of β,

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:

1,2,3,⋯,ω,ω+1,ω+2,⋯,ω+ω=ω⋅2,ω⋅2+1,⋯,ω⋅3,⋯,ω2,ω2+1,⋯,ω2+ω,⋯⋯

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 ω is

even, ω+1 is odd, ω+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

more information about the ordinals.

**Attribution***Source : Link , Question Author : Kevin , Answer Author : JDH*