Side Note - You brain might hurt if you think for too long for infinity. We'll make sure, it doesn't.
The idea of infinity is so crazy that it hurts your brain if thought about for too long and too hard. It is a concept that we are all taught in just middle school, but so complex that we do not actually understand the idea of a number or something being infinite.
But, what is infinity and does it exist?
Aristotle was the first scientist to go into this concept of something being infinite and discovered the two types of infinity: potential infinities and actual infinities. Potential infinities he classified as a type of infinity that is unending, for example the natural number system, pi, or even the universe. These infinites could actually exist in specific scenarios and focuses. Actual infinities are a type of infinity that Aristotle was convinced could never exist. These are relating to an infinite set of “things” in a finite space with a beginning and ending. Aristotle then went on to prove that actual infinities can never and will never actual exist because they are a paradox.
Existence of a Potential Infinity
In the world of mathematics, there is a more of a possibility for the infinity to exist. The idea that "there's always a next one" is encapsulated by the principle of induction, which says that "if n exists, so does n + 1". Georg Cantor, a German mathematician from the 1800s, developed the theory of a countable infinity, which is what we think of when looking at the real number number system. He discovered that in mathematics, infinity could exist and that infinity could be added and subtracted by other infinities and that some infinities are bigger than other infinities. He wanted to prove that the potential infinity existed.
He showed these theories multiple different ways. For example, Cantor wanted to prove that there is an equal set of square numbers while compared to natural numbers. While thinking about it for a quick second, you would think that there must be many more natural numbers that square numbers in the real number system. But, as Cantor looked at it, if you have an infinite set of natural numbers, that means that each number has a square number, therefore an infinite number of square numbers. And if you match each square up with a natural number (example: (1,1) (2,4) (3,9) (4,16)...) and so on matching each number with its paired square number, you will see that there is the same infinite square numbers then there is natural numbers. Infinity, while maybe being confusing at times, makes perfect sense when you draw it out and actually think about each individual number in the infinite set that you have.
Infinity is endless and it is incredibly complicated to wrap your head around it. Just thinking about whether or not infinity exists depends in which context you are trying to prove it in. Understanding the idea that “∞+1=∞” is completely plausible because when something is already endless, endless plus one is still endless. The rules of infinity are completely justified to only an infinite set and no other set of numbers, in mathematical terms. But still, no one can ever truly know if infinity exists. If you ask different mathematicians, they will give you different answers with different reasonings. So, the next time you have spare time and want to twist your brain a little, think about the idea of infinity and what it means for something to be infinite.
Written by: Ellie Humphreys
Ellie Humphreys is the Board Member for Voyager Space Outreach for writing Blog posts on STEM/Space Topics.