At my fourth birthday party my friend and I were playing a game. One of us would say a number and the other would have to produce a larger number. I didn’t quite understand what came after a million so I started shouting out nonsensical numbers like one-thousand one-million and three. My friend went for googol, which made even less sense to me, so I responded with googol plus one. Then my friend went for the trump card with infinity. I quickly tried infinity plus one, but he told me there is no such thing and obviously I thought he was cheating and got very upset. This started my quest to better understand infinity.
In case this seems outlandish for a 5 year old, my friend graduated from MIT last year and is currently enrolled in a PhD program.
A common use for numbers is counting. So what does infinity count?
Let’s start with a simple example. How many sheep are in this picture?
It’s pretty obvious that there are 3 sheep. So if I wanted to produce positive whole numbers I can just assemble that many objects. If I want seven sheep I just go to a farm, round them up, and I get seven.
What about producing fractions?
Looks like this pizza is broken into fourths and 3/4 of the pizza is left because 1/4 was someone’s lunch. Notice that 3/4 and 1/4 consist of an integer divided into an integer. In the pizza example we would make the number of slices our denominator and then consume slices to make the value left in the dish less than 1, producing the desired fraction.
The question I’m hinting at is how do you mathematically construct infinity? What does infinity look like?
So far we have been counting using examples such as sheep and pizzas. The sheep represent integer values (whole numbers) and the pizza represented rational numbers which includes integers and fractions. I introduced you to real world counting examples using integers and fractions, but now we have to answer the question what does infinity count?
Turns out there are two concepts of infinity when counting. Namely countably infinite and uncountably infinite. Countably infinite means that I can map the set of integers to whatever I am counting. What do I mean by that? Let’s go back to the sheep. Imagine a never ending line of sheep. You could start at the first one and go down the list forever and ever going “1, 2, 3, 4,…”, and you would be able to count the sheep endlessly.
However uncountably infinite is a little more tricky. Let us try to count every number between 1 and 2. So we would start at 1 and go “1, 1.1, 1.2,…”, but wait what about 1.11 and 1.11111 and 1.111113? The complete list of all decimal type numbers is the real numbers and we cannot list the real numbers between 1 and 2. Even given a starting point like 1, it’s impossible to tell when we’re done listing numbers starting with 1.1 and then move onto numbers starting with 1.2.
As far as infinity goes, the list of objects we are trying to count is always endless. The difference between countably infinite and uncountably infinite lies in the counting process. Countably infinite means you will never finish counting, but you will be able to continue counting forever. Uncountably infinite means you will hit a roadblock in your counting process that inhibits your ability to continue. In the sheep example we kept counting and all was good. With all real numbers between 1 and 2 we had trouble shifting from 1.1 to 1.2 because there were an infinite amount of numbers starting with 1.1 (It doesn’t matter if its 1.1 or 1.11 or 1.111 etc, same principle applies; you don’t know when to stop and go to the next increment).
Infinity has many applications as an abstract concept, but in the counting world this is the closest thing we have to a tangible way to understand infinity.
Small Disclaimer: Yeah this was not a rigorous proof or anything. If you’re in the group that knows how to give a rigorous proof of this concept, I was trying to explain this to a general audience and make the concepts as accessible as possible.