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The Bigger Infinity

Let’s challenge our notion of infinity by stating a downright crazy idea… and then proving it beyond any doubt.

Welcome to another part of the “Maths For Normal People” series. You will find all the previous parts in this article.

So far we have been examining various sets of numbers: natural numbers (<$1,2,3,\ldots$>), whole numbers (<$\ldots,-3,-2,-2,0,1,2,3,\ldots$>), and fractions (<$\frac{p}{q}$>).

We found out that they are all infinite, but still somehow of the same size. Their cardinality (size) is always the same infinity, <$\aleph_0$> (Aleph-0).

The case of the set of real numbers (denoted <$\mathbb{R}$>) is surprisingly different. Of course, there are infinitely many real numbers, but…

The Real Is More Bizarre Than Your Imagination

The set of real numbers <$\mathbb{R}$> is bigger than the set of natural numbers, or <$|\mathbb{R}|>\aleph_0.$> In other words, the cardinality of real numbers is a number bigger than <$\aleph_0$>, which is infinite. In still other words, there exists a kind of infinity that is bigger than our “usual” infinity!

This extraordinary claim demands proof. First, let’s reduce the problem only to real numbers between 0 and 1. If we prove that these alone are more numerous than natural numbers, it will follow that the whole set <$\mathbb{R}$> is bigger than <$\mathbb{N}$>.

Proof By Contradiction

Proof by contradiction is a popular method in mathematics. Instead of directly proving that something is true, it shows that it is impossible for it not to be true.

We pretend for a while that the proposition we want to prove is false, and deduce that this would result in some logical impossibility (contradiction), like <$1=2$> or <$x \neq x.$>

Cantor’s Diagonal Method

We shall use the so-called Cantor’s Diagonal Method, which is a type of proof by contradiction. We want to prove that there are more real numbers between 0 and 1 than there are all natural numbers. So let’s pretend the opposite proposition is true instead:

There are as many real numbers between 0 and 1 as there are natural numbers.

If that is true, it must be possible to list the real numbers in an endless table and number them by natural numbers (that is, establish a bijection between the reals and the naturals). For example:

natural numberreal number
10.16461652…
20.76249817…
30.76299017…
40.12941439…
29,836,527,465,7820.6739239…

Under our assumption, all real numbers between 0 and 1 are in this infinite table and have natural numbers assigned to them.

We are going to arrive at a contradiction of this assumption by constructing a real number between 0 and 1 which is not in the table.

Creating such a treacherous number is not difficult. Let’s call it <$X$>. We take the table and look at the first decimal digit of the first number. Let’s call it <$a_1$>. Then we set the first decimal digit of our <$X$> to be anything else than this decimal digit <$a_1$>. For instance, if the first number is 0.16461652… then <$a_1~=~1$>, and we set our <$X$> to be 0.2. In other words, <$X$> differs from the first number at the first decimal place.

We then do the same with the second decimal digit, say, <$a_2$> of the second number. We set the second decimal digit of <$X$> to be different from <$a_2$>. <$X$> now differs from the first number (in the first decimal place) and from the second number (in the second decimal place). For instance, if the second number is 0.76249817…, we set <$X$> to be 0.27.

We continue this process with the third, fourth, fifth,… number. For instance, suppose the beginning of the table looks like this:

natural numberreal number
10.16461652…
20.76249817…
30.76299017…
40.12941439…
50.62132848…

Our number <$X$> then could start with 0.27353…. Note that it differs from each of the first five numbers in at least one decimal place.

If we go all the way through the infinite table in this manner, we end up with <$X$> that cannot be in the table!

Thus our original assumption (that a complete table of reals between 0 and 1 can be constructed and numbered by natural numbers) must be false.

That means there are more real numbers between 0 and 1 than there are all natural numbers. That implies that there are more real numbers in total than all natural numbers.

<$$ |\mathbb{R}| > |\mathbb{N}|. $$>

The infinity of real numbers is strictly greater than the infinity of natural numbers.

Implications

It may be hard to see behind all the mathematical symbolics, but we have just proved a staggering fact: once we accept the existence of infinity as such, our system of logic makes it necessary for more than one infinity to exist. And as we shall soon see, it doesn’t stop with just two infinities! Once we open the door for infinity ever so slightly, it bursts into our logic like an untamable Hydra with, er, infinitely many heads.

The existence of several infinities is a blow to the implicit idea of infinity that we usually have. It leads one to reconsider just what infinity is, whether it actually exists, and what is precisely meant by that existence.

In mathematics, the difference between <$|\mathbb{R}|$> and <$|\mathbb{N}|$> is crucial. Many theorems or whole theories work only for sets with cardinality <$|\mathbb{N}|$> (these are called countable sets) and break down with sets of size <$|\mathbb{R}|$> (these are called uncountable sets). There are sets that can be either finite or uncountable, but can never be countable.

We are nearing the end of our exploration of infinity in mathematics. Through obscure logical reasoning, we’ll soon discover facts about infinity that will lead us from mathematics to the realms of philosophy.

Want to read more? You will find the outline of all parts of the series in this article.

September 29, MMXI — Mathematics, Maths For Normal People, Infinity.