Using math of course!
Infinity is an incredibly important concept in mathematics, but it isn’t always easy to wrap your mind around. You may have heard someone use the terms “countably infinite” or “uncountably infinite” before, but do you know what the difference is? The name sort of gives it away, but not in the way you might think. Countably infinite doesn’t mean that you can actually count all of the numbers, it just means that technically you could find a method that could enumerate them.
Before we go any further, we’re going to need a few definitions:
Set: A set is a group of objects — numbers in the cases we’re talking about here. They are denoted by curly brackets. A few examples are {blue, red, green}, {26, 87, 1987, 20394}, all the multiples of 3.
Subset: A subset is a subgroup of a set. A subset of the multiples of 3 could be: {9, 12, 27}.
Natural Numbers: These are the numbers that we use for counting: {1, 2, 3, 4, 5, 6, …}. Some people include 0 in this set.
Integers: Integers include the natural numbers, but also include 0 and negative numbers: {…, -3, -2, -1, 0, 1, 2, 3, …}
Real Numbers: Real numbers express values along a continuous line. They include all integers, fractions (rational numbers), and irrational numbers like π that can’t be written as a fraction of two integers. Almost any number you can think of is a real number.
Now that we’ve got our definitions, let’s get on with proving what we set out to: that there is more than one type of infinity.
SEE ALSO: Is it Possible to Imagine Infinity in Our Minds?
What do I mean by that? I mean that there are two sets of infinite numbers that are of different sizes. We call two sets the same size if you can make a one to one relationship between all of their values. Now you might be thinking that I just defined three sets of numbers above (naturals, integers and reals) and that are obviously a different size, but let’s take a closer look.
If you had the sets {Anna, Bob and Claire} and {1, 2, 3} it would be easy to see that you can find a one to one relationship: Match Anna with 1, Bob with 2 and Claire with 3. We can actually do the same thing with the natural numbers and the integers.
| Naturals | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | … |
| Integers | 0 | 1 | -1 | 2 | -2 | 3 | -3 | 4 | -4 | 5 | … |
Yes, the natural number that each integer is matched with will always be bigger, but since both sets are infinite, we can go on forever and show that the sets are the same size. We call them both countably infinite.
If we look at the real numbers, you might think that we could use a trick similar to the one we just used to match them up to integers or natural numbers, but there are just too many real numbers.
To show this, we’re going to go through a thought experiment or what mathematicians call proof by contradiction. We are going to pretend that our claim is true — that you can find a one to one relationship between real and natural numbers — and then show that it is impossible.
SEE ALSO: Mathematician Solved a Nearly Impossible Maths Problem
Start by writing down all real numbers (not a feat that we can accomplish here or anywhere except sort of in your mind so I’m going to write down just a few) and match them with natural numbers:
1 ↔ 0.87293…
2 ↔ 3.14987…
3 ↔ 0.83912…
4 ↔ 1.16734…
5 ↔ 2.09765…
You can see as I’m going that I could have picked any numbers to match in each case. We’re saying that continuing in this manner we could make a list of all the real numbers. Now here comes the tricky part. We’re going to create a new number by taking a digit from each of these and adding one to it (or changing a 9 to 0).
We get: 0.95046… It isn’t the same number as any of the ones that we have on our small list, but it won’t be the same number as any of the ones that we have on our infinite list either because it will be at least one digit off from each of the numbers. It is a real number, but it doesn’t show up on our list which contradicts our first assumption that we had a list containing all real numbers.
That completes our proof by contradiction, and we can say that the real numbers and integers are sets of different sizes. We call the real numbers an unaccountably infinite set.
Even if we had a countably infinite number of seconds to count them all, we could never do so. Just let that sink in for a minute.
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