What is Infinity?

To read more by Shunsuke: https://msurjblog.com/author/shunsukekatayama/

“I love you to infinity”, “I love you to infinity plus one”, “I love you to infinity plus infinity”, …

Let’s look at this mathematically, together, as you were just about to do on your own.

Mathematicians are to mathematics as geologists are to rocks.

Not many people gain pleasure from observing an ordinary rock, just like nobody ever gets excited when seeing an integral. Rocks have been in our hands as tools for a very long time, and we have developed many great things from them. Similarly, mathematics has been the flashlight we’ve used to discover new things: from the Pythagorean theorem to Einstein’s theory of general relativity. But it’s not mathematics that’s fascinating, but the results that we find beauty in. While having said that, geologists look very carefully at rocks and listen to the stories that they have to tell.

So let’s listen carefully to the sweet sounds of whisper of mathematics – to infinity and beyond.

Yes, infinity – the idea of infinitude is as beautiful as it is vast. Let’s put aside the “you’re infinitely beautiful,” “no, you’re infinity plus infinity times more beautiful,” “no no, you’re infinity multiplied by infinity times more beautiful” he-said-she-said – neither is complimenting the other any more than what they just said. As heartbreaking as it was when Andy had to leave for college in Toy Story 3, there are so far only two (sadly not infinite) types of infinities; namely, countable and uncountable infinities.

Countable infinity is a set of numbers that if you lived to be infinity years old, you can count all the numbers in the set. For example, the set of odd numbers {1, 3, 5, …}. If you count an odd number everyday for infinite number of days, you can count all odd the numbers. Let’s take a closer look at this after we define “uncountable infinity”.
Uncountable infinity is a set of numbers that even if you lived to be infinity years old, there’s no way that you can count all the numbers in the set. An example is the set of real numbers which is all the numbers that can be represented by decimals. You can spend an infinite amount of time counting all the numbers between 0 and 1 but you’ll still have the numbers between 1 and 2, 2 and 3, and infinitely many remaining intervals.

So, back to countable infinity. Here is a question: which set of numbers do you think is bigger, the set of prime numbers {1, 2, 3, 5, 7, 11, …} or natural numbers {1, 2, 3, 4, 5, …)? Euclid proved that there are an infinite number of prime numbers, which is a cool little fact, but would you believe me if I told you that there are exactly the same number of prime numbers as natural numbers?

Let’s say that there is a set of boys each representing a natural number and, likewise, a set of girls each representing a prime number.  We are going to form couples to have them go on a blind date. If there are more boys than girls – in other words, more natural numbers than prime numbers – we’ll know that by seeing that at one point we’ve run out of all the girls and we’ll have a (fairly sad) situation with only boys. But that’s not the case, since even when 20 couples or 100 couples or a million “couples” are matched, there will still be an infinite number of boys and girls that can be happily matched.
This is called the one-to-one correspondence: we will never find a boy or a girl – a natural number or a prime number – who cannot “find a date”. That may sound unrealistically perfect for the real world, of course,  but the world of mathematics is as perfect as it gets.
So, to recap: infinity = infinity + 1 = infinity + infinity = infinity × infinity. Those lovebirds were just repeating the same thing to each other [1].

Another fun fact is that the number of rational numbers – the set of all the numbers which can be written as fraction (e.g. 1/2, 2/3, 11/4) – is countable infinity as well. This means that there is the same number of rational numbers as prime numbers or even numbers.

Now that we have a little taste of what infinity is, do you think there is another infinity that we don’t know of? Maybe something that’s in between uncountable and countable infinity? I have no idea what that infinity might look like, but if you have an idea, perhaps this branch of mathematics – called number theory – might be for you.

NOTE:

[1] Assuming that these infinities being referred to are countable infinity.

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