Irregular Webcomic! #3727

If you remember back to Han's dianogalisation argument, we've established that there are (at least) two different types of infinity. The easiest one to get to grips with is: How many natural numbers are there?

The natural numbers are the standard counting numbers that you would use to count whole numbers of things, like how many apples are in the fruit bowl, or how many times you've had a birthday. They go: 0, 1, 2, 3, 4, 5, 6, ... and so on.^[1]

So, how many of these numbers are there? Well, no matter what natural number you care to name, there is always a bigger one. You can just add 1, for example, to create a bigger natural number. So "the number of natural numbers" cannot be a natural number itself. There are in fact infinitely many natural numbers, because no matter how many of them you care to name, you can always name more of them. That's pretty much a definition of infinity.

"The number of natural numbers" is not a natural number, but we can still give it a specific name. As we shall see, the name "infinity" is actually not specific enough to describe this number. Mathematicians have given this number, "the number of natural numbers", the name ℵ₀. This is the Hebrew letter aleph, with a subscript of zero, and is pronounced either as aleph-zero, aleph-naught, or (as I learnt it) aleph-null.

Now, recall everything we said back in the annotation to comic #2292 (Han's dianogalisation argument). Using a clever way of mapping the positive and negative integers to positive numbers, we can assign a natural number label to every integer, both positive and negative (and zero). So the number of integers is the same as the number of natural numbers. In other words, the number of integers is also ℵ₀.

Remember also that we managed to construct a way to assign a natural number to every single rational number that exists - that is, every number that can be written as a fraction. So the number of rational numbers is also the same as the number of natural numbers. In other words, the number of rational numbers is also ℵ₀.

And recall also that we showed that, no matter how we try to assign a natural number to all of the real numbers (i.e. every number that can be written out as a decimal, possibly requiring an infinite number of decimal places), there is always demonstrably some real number that we missed out on. So there are more real numbers than the number of natural numbers. In other words, the number of real numbers must be greater than ℵ₀.

If the number of real numbers is (an infinite number) greater than ℵ₀, then mathematicians would like to know "how big" this (infinite) number is as well. In fact, using some mathematical proofs which I won't go into here, you can show that the number of real numbers is meaningfully equal to 2^ℵ₀. This is very different from the countable infinity ℵ₀.

Now, if you count the natural numbers, forever, you will eventually count up to any given natural number. Let's say you count an average of one number per second (allowing for rest breaks). How long will it take you to count to:

A thousand? Easy, 16 minutes and 40 seconds.
A million? As it turns out, a bit over 11 and a half days.
A billion? 31 years and 8.5 months. If you started when you were born, some of you could have done this by now.
A trillion? 31,709 years. Difficult, but if you lived long enough, you could eventually get there.

The point is that any natural number I name, doesn't matter how big it is, you could theoretically count up to it, given enough time. We say that the number of natural numbers is infinite, but that it's countably infinite. Any given natural number, no matter how big, you can theoretically count up to. So ℵ₀ is a countable infinity.

Because of the way we've mapped the integers and the rationals to natural numbers, you can do the same with them. You can assign them in a fixed, particular order, and say the name of one integer, or one rational number, every second, and eventually you will reach any given one that anyone cares to name. Again, it could take many years, but eventually you'll get to it. So the number of integers, and the number of rationals, is also countably infinite.

But the real numbers cannot be ordered in this way. If you arrange the real numbers in a fixed, ordered sequence and then you say the name of one real number per second, following the defined order, then I can always say the name of a real number that you will never reach, no matter how long you keep reciting numbers. And for this reason we say that 2^ℵ₀, the number of real numbers, is an uncountable infinity, and that the number of real numbers is uncountably infinite.

Crikey!!

There is another method of defining infinite numbers larger than ℵ₀ as well. You could define a number ℵ₁ to be the next biggest infinite number after ℵ₀. We already know there is an infinite number bigger than ℵ₀ - the number of real numbers 2^ℵ₀ is demonstrably a bigger infinity. But is it the next biggest infinite number after 2^ℵ₀, or is there some other infinite number in between? This question is the subject of the so-called continuum hypothesis, which basically hypothesises that there is no sensible infinite number between ℵ₀ and 2^ℵ₀. Or in other words that 2^ℵ₀ = ℵ₁. If you make some assumptions about how set theory and large numbers like this work, you can prove this statement, but those assumptions are not necessarily true. Basically, whether the number of real numbers is equal to ℵ₁ or not depends on what axioms you choose to establish your set theory.

Crikey²!!

And, yep, you guessed it. Mathematicians would hardly define two new numbers as ℵ₀ and ℵ₁ if they didn't want to keep going and also have even bigger infinite numbers: ℵ₂, ℵ₃, etc. Personally, I don't have a particularly clear idea in my head what these bigger infinite numbers correspond to, but if you're interested you can read more on Wikipedia's page on aleph numbers.

ℵ_Crikey!!

[1] Some definitions exclude 0 from the natural numbers. For the purposes of today's discussion, this makes no difference.