I am writing this post because I am fascinated by the idea of infinity and I think it is something that we do not talk enough about.

Moreover off late I am reading about Theoretical Computer Science and I find encountering this concept again and again. I am just beginning to understand the connection between Computer Science and Mathematics.(note that I used Computer Science as not Computer Engineering)

To be honest infinity is a very weird and un-intuitive concept to get your head around and I don't think I understand it fully at this point and I don't I ever will. However, here goes nothing.

Let's begin.

What is Infinity ?

Infinity is a concept that describes the of never-ending-ness of something. If I were to say that something is infinite it means that that something is never ending. It is never-ending no matter what we take away from it or add to it.

This concept can be illustrated in the famous hotel called the Hilbert's Hotel.

This hotel has an interesting property - it has an infinite number of rooms which are all occupied.

So if they are all occupied then how do they accommodate new customers ?

In Hilbert's hotel, whenever a new customer comes into the hotel management shifts all the other customers to the next room and the new customer is allotted the first room.

Thus this hotel can accommodate an infinite number of people.

Infinities in Mathematics

There are a whole lot of things to talk about infinity, however neither am I qualified enough nor do we want a post of infinite length. So we will focus on two infinite sets in mathematics.

The set of whole numbers in Mathematics. This set is a set of all positive integers beginning from 0 and this is an infinite set. This is the famous set \(\mathbb{N}\) and the set of all Real numbers. This set consists of any number that can be represented on a number line. This is the famous set \(\mathbb{R}\).

Are all infinities equal ?

That is a loaded question!! It sounds simple doesn't it? If infinity means that something is never ending then all infinities must be equal; right ?

WRONG !! Turns out there are different kinds of infinities. Countable Infinities and Un-countable infinities.

Let's unpack this argument from the beginning and in the end we will compare the two sets we talked about before (\(\mathbb{N}\) and \(\mathbb{R}\))

Two sets P and Q are said to be equal sets if they have the same number of elements (or they sets have the same cardinality). Now the question arises that if two sets are infinite then how do we compare if they have the same number of elements or not.

This is the brilliance of it all. To establish the equality between \(P\) and \(Q\) we do not need to and we can not count the number of elements in each set and show that they are equal, we just need to show that if counting the number of elements were possible then the sets would be equal. We do this by showing a one-to-one correspondence (Bijection) between the elements of the two sets. This, is a fancy way to say that we have to show that we can pair every element of the first set with an element of the second set.

Comparing the two sets (\(\mathbb{N}\) and \(\mathbb{R}\)) will make this idea more clear. Set \(\mathbb{N}\) is the set of Natural numbers and set \(\mathbb{R}\) is the set of Real numbers. For this experiment we will only consider the numbers between 0 and 1 from the set \(\mathbb{R}\). Thus this small interval will include numbers like \(0.01, 0.0001, 0.00235421, ...)\). Now, let us select such numbers from this interval in ascending order and pair them up with each element from the set \(\mathbb{N}\).

Soon, we will run out of numbers from \(\mathbb{N}\). This happens because no matter how you choose, you can not choose numbers in ascending order (or any order for that matter) from set \(\mathbb{R}\). Say you choose \(0.01\) and the next number you choose is 0.02, however there are still numbers like \(0.001\) and \(0.000001\) and so on. In fact, there are infinitely many numbers between any two numbers from the set \(\mathbb{R}\).

You can image this as having numbers on a number line and you can just zoom into the line indefinitely and all you would do is just increase the granularity of the numbers. The numbers would never end.

Thus the set \(\mathbb{N}\) is a countably-infinite set while the set \(\mathbb{R}\) is uncountably-infinite.

Another WILD idea ...

Take the set of all Natural numbers. This set contains all numbers from 1 to infinity. Take another set containing all the even numbers from \(2\) to infinity.

So now we have two different sets. What is the size of each of these sets ?

Basic intuition suggests that the set of all natural numbers will be twice as large as the set of all even numbers. But, that is not the case. Both the sets are of infinite length. In fact both of these sets have exactly the same number of elements.

This suggests some weird arithmetic. If you add infinity with a number; you get back infinity. If you remove a number from a infinity; you get back infinity. If you divide infinity by a number; you get back infinity. If you multiply infinity by a number; you get back infinity.

Crazy isn't it ?? :-P

Stretch your brains a bit more

If the last section didn't rattle your brains enough, then let us look at something more bizarre.

This is a thought experiment. Let us imagine a circle with radius \(r\). Thus the circumference of this circle is \(2 \pi r\). Thus the circumference of a circle is not rational. It is however a Real Number.

Now, if you start from point \(A\) on the circumference of this circle and you take \(\pi\) steps every time then you will never reach the same point \(A\) ever again.

This happens because of the irrationality of the circumference. You can imagine the circumference of the circle as two points marked on the Real number line at \(0\) and at \(2 \pi r\). As we have seen previously there can be infinitely many numbers between any two numbers on the real number line. Now if we were to remove the original point \(A\) the circle would remain as it is and the "gap" in its position will be replaced by some other point since we have an infinite supply of points.

There is also an incredible paradox called the Banach-Tarski Paradox.

The non-mathematical version of this paradox can be paraphrased that there are certain methods in which a sphere can be decomposed (broken down or taken apart) into a finite number of subsets such that these subsets can be recombined to form two exact copies of the original sphere

If this were true is the physical world then it would mean that we could take an orange and turn that into two oranges. However, the physical world is different from the mathematical one in a way that the physical world is bound by the laws of physics, while the mathematical world is not. Hence the fun sphere in the mathematical world does not follow the law of conservation of mass while the poor sphere in the physical world needs to follow it. This is why we can not create infinite oranges and we can not have a circle form which you can remove a point and the size of the circle will still remain the same.

Bummer right!! :-P (I KNOW!!!)

Conclusion

I first read about the Banach-Tarski paradox in a video and I was blown away by it. The idea was so revolting, but at the same time it made so much sense.

If this blog post scratched that part of your brain that says "wait-a-minute-but-how" then please go ahead and read about infinity!

It's amazing.

And finally a huge thanks to Vasauce for this video.

This post was primarily inspired by that video!! :-)