WEEKLY WHINE

Let’s read a finite number of words

If you exist, you are probably a finite being. You have an extent that can be measured in some way. If you are a physical being, you have a height and mass. If you are an algorithm, you have a size in lines of code or in bytes. If you are a social construct like marriage or saying gesheunteit after a sneeze, you have a temporal duration.

Other things that are finite can still be very large. The list of all valid words in the English version of Scrabble according to Collins Scrabble Words, for instance, is 267,751.

But there are many things that are infinite, such as, perhaps, our Universe. Infinity is a popular concept and has inspired such objects as towers, jewelry, and pools. Everyone has an idea about the infinite, what it means, and why it exists.

So let’s now explore your ideas of the infinite in the GoobNet Mailbox. As always, please remember to keep the GoobNet Mailbox away from infinite mirror rooms, as the total number of simultaneous GoobNet Mailboxes that the human visual cortex can process at one time is not infinite. Not in the least.

How big is infinity?

– Charlie Aphrodine
Fimbert, QC, Canada

Bigger than the biggest thing ever and then some. One way to think of the infinite is as the total number of natural numbers. If you have a finite set of natural numbers, you can always find one more to put into the set. Therefore, the total number of natural numbers is not finite.

But there are more real numbers than natural numbers, right? So the number of real numbers is even more infinte than the number of natural numbers.

– Charles Baker-Pratt
Constantine, NY, USA

You are correct. The total number of real numbers is uncountably infinite, as opposed to the total number of natural numbers, which is countably infinite.

How can something be countably infinite? If it’s infinite, you can’t count it.

– Adam Jeffers
Ottawa, ON, Canada

That’s where you’re wrong. Some things that are infinite can be counted. For example, the set of natural numbers, N = {1, 2, 3, 4, ...}. There are infinitely many of them, but you can uniquely assign a natural number to each one. On the other hand, there are also infinitely many real numbers, but you can’t assign a natural number to each one of them.

How many atoms are there in the entire Universe?

– Kelly Page Wright
Foxhen, MA, USA

A lot. Most estimates place the total number of atoms in the observable Universe at somewhere in the neighbourhood of 10⁸⁰, give or take a few powers of ten.

What about the unobservable Universe?

– Ryan Algaard
Bath, England, UK

We don’t know. We can’t observe it.

You’ve talked about natural numbers. Is there such a thing as an unnatural number?

– Vickie Smefferty
Blantyre, Malawi

Sure. Any number that is not a natural number, like -5 or 7.7.

I can assign a natural number to every real number. Watch: I’ll assign 1 to 1. Then I’ll assign 2 to 1/2, 3 to 1/3, 4 to 1/4, 5 to 1/5, and so on. There! Done!

– Caleth Gardenstein
Globe, AZ, USA

That’s not all the real numbers. What about 3 or 4? Or 2/3? Or 6/7? Or -77.12? Or п? All you have succeeded in is assigning natural numbers to their reciprocals.

Oh. Then should I go away?

– Caleth Gardenstein
Globe, AZ, USA

That would probably be best.