— 3 min read

On infinity

Proving that some infinities are bigger than other infinities.

In this post we will look closely at the notion of infinity. Particularly we will try proof that infinity comes in different sizes.

First of all we need to define a couple of notions, and we start we the notion of countable. Any set that is equipotent to , i.e. there’s a one-to-one correspondence with is called countable, personally I find this term a bit extreme, they probably should have called it listable. In other terms, given such set, one can count the elements of the set and eventually can reach any element.

Some example of countable sets are:

For we can consider the application

Which makes equipotent to a subset of

Now we shall prove that is not countable, for that reason we will show that the set of reals in interval is not countable.

Let’s consider a function , and let be a sequence of closed real intervals such that .

We know that one of these intervals does not contain .

Let be such interval.

Again one of these intervals does not contain .

Let be such interval.

We suppose that we construct for a certain such that

We will denote and again one of these intervals does not contain

Let be such interval.

Hence by induction, we just construct a sequence of intervals .

Let

We have the following

It follows then that

Hence is not surjective, and so it’s not bijective.

In other terms we proved that there’s no bijection from to .

Therefor is not countable and so is .