# How to count

### Shall we count together in the ordinals? Let us venture into that transfinite realm beyond infinity. Is infinity even or odd? Can we count to an uncountable ordinal?

### The ordinal numbers

The ordinal numbers are the natural continuation of the natural numbers into the transfinite realm beyond infinity. Anyone can learn to count into the ordinals, even a child, and so let us explore this realm together.

We start with zero at the bottom and count through the familiar natural numbers.

We have hardly begun. Let us imagine that we have finished counting through *all* the finite numbers. Afterwards we shall encounter the first infinite ordinal number, the first ordinal after all finite numbers—this is the ordinal known as ω, pronounced as the Greek letter *omega*.

What? How can there be any numbers *after* all the finite numbers? After all, if we were counting them, there wouldn't be any time left to pronounce ω, since we would use up all our time counting the finite numbers, right?

Some people find infinite ordinals easier to grasp if we imagine counting through the finite numbers faster and ever faster in the supertask manner we discussed in earlier chapters—we pronounce the number 0 in the first half hour, the number 1 in the next quarter hour, and 2 in an eighth hour, and so on. After one hour, we will have pronounced every finite number, with plenty of time remaining for more.

In my view, however, that supertask process is an unnecessary extravagance. To understand the ordinals, it is simply not necessary to embed them into what we perceive as the structure of our temporal reality. Rather, the ordinals have their own order structure, their own integrity. Let us understand the nature of the ordinals directly in terms of their own order structure, such as it is. Let us count in *ordinal time*, rather than physical time, giving each ordinal its own pace and dignity.

The ordinal ω, the first infinite ordinal, is a *limit* ordinal, which means it is approached by smaller ordinals, the finite numbers, but it has no immediately preceding number. After all, the successor of a finite number *n* is the number *n* + 1, which is still finite. There is no ordinal number immediately prior to ω.

#### Counting to ω^{2}

Let us count now in earnest.

## Keep reading with a 7-day free trial

Subscribe to Infinitely More to keep reading this post and get 7 days of free access to the full post archives.