Why do some infinities seem bigger than others, and how do mathematicians compare their sizes?
In: Mathematics
In terms of sets we compare sizes by whether or not we can draw a one-on-one correspondence between the elements of those sets. If we can do that then we consider those sets to be the same size. This is fairly simple when it comes to sets that have a finite number of elements but can get a little weird when we have sets with an infinite number of elements.
As it turns out, there are sets with infinite numbers of elements that we can’t draw one-to-one correspondences between, which means that one is bigger than the other, despite the fact that both are infinite in size. An example of this is the integers and the real numbers.
Because they are!
There is an infinite number of numbers between 1 and 2.
1.1, 1.2, 1.3, 1.4, 1.5. 1.6 1.7, 1.8, 1.9, 1.91 and so on
And there is an infinity twice as large between 1 and 3.
How do we compare them? Smart people do smart math that I can’t hope to know lol
The way I recall my mathematician friend explain it was like so:
Let’s look at the space between 0 and 1. What would be the middle of 0 and 1? That would be .5. What would be the middle of 0 and .5? .25. Then you can keep halving it forever, and you’ll find that between 0 and 1 there is an infinity.
Now, let’s look at whole numbers. We can count 1, 2, 3, and so on forever. It’s another infinity, except now we he have the inifities between 0 and 1 and 2 and 3 and so on. An infinity of infinities. The infinity of whole numbers is bigger than the infinity between 0 and 1.
At least that was my understanding!
One infinity being “larger” than the other is a mathematical concept. The formal name for the size of a set is called its cardinality. For finite sets, the cardinality is equal to the number of elements in the set. However for infinite sets, this definition is no longer useful. The basic infinity is the countable infinity (of size aleph-zero, another formal term). The simplest example is the cardinality of all natural numbers. The next larger infinity is the uncountable infinity an example is the cardinality of the set of all real numbers which would be aleph-one.
These concepts are not intuitive. So one would need a bit of learning about set theory to understand how and why these definitions “make sense”.
Larger infinities are also possible in mathematics like aleph-two etc etc but they become increasingly hard to describe and conceptualize and have no “real world” easy examples.
Some infinities seem bigger than others because, well, they are!
When we count sets of things, we create a one-to-one mapping between a that set of things and a set of objects whose size, or “cardinality”, is known. A one-to-one mapping is just that: each object in set A has exactly one corresponding object in set B, and vice-versa. A simple example is counting on our fingers: we know that we have ten fingers, so if we can match each object to one of our fingers, we know that we have ten objects in the set which we’re counting. In mathematics we attempt to find some mathematical function that creates this mapping for us.
The “smallest” infinity is the Natural Numbers: 1,2,3,4, … and so on. Consider the set of Real Numbers: things like -1, π, 0, 1,000,000, and so on. Can we find some mathematical function that gives us a one-to-one mapping between the Natural Numbers and the Real Numbers? Spoiler: no, we can’t, so those two infinite sets are not the same size.
If there’s any way that you can invent a “pairing rule” between two sets A and B, such that every element in set A has exactly 1 unique partner in set B, and vice versa, then you have proven that the sets are the same size.
If you can’t(*) do that, but you *can* invent a “pairing rule” such that every element in A has a partner in B, *but* not every element in B has a partner in A, then you can say that set B is strictly larger than set A.
So here’s an example of two sets:
Set A is all the real numbers from 0 to 1. Set B, is all the real numbers from 0 to 2.
You might be tempted to think that set B is bigger, since it’s a wider interval, right?
But here’s a “pairing rule.” For every number in A, you multiply it by 2 to find its partner in B. And for every number in B, you divide by 2 to find its partner in A. This means the sets are the same size!
edit:
*: where I said “if you can’t,” that should probably actually say: “if you can prove it’s impossible”. Because it’s conceivable that there *is* a possible pairing-rule which assigns unique partners to both sets, but you just couldn’t figure out what that rule would be.
Suppose you have two buckets, A and B, each with a bunch of doodads in them. There are (at least) two ways you could determine which bucket has more doodads in it:
1. Count up the number of doodads in A, then count up the number of doodads in B, and then see which number is bigger.
2. Take out a pair of doodads, one from each bucket, and then set them aside. Then take out another pair and set those aside, too. Keep doing this until you run out of doodads in one of the buckets. The bucket that runs out of doodads first must have had fewer doodads in it.
Now, Method #1 will work as long as there is a finite number of doodads in one or both of the buckets. But if there are an infinite number of doodads in both buckets, you run into a bit of a problem. Which, if either, of those infinities is larger? You can’t really say. You might say, well, infinity = infinity, so they must have the same number of doodads in that case. And indeed, this was the general attitude of mathematicians until around the 1870s.
Enter [Georg Cantor](https://en.wikipedia.org/wiki/Georg_Cantor), who said, hold up, if indeed it’s true that infinity = infinity, then we should also come to the same conclusion (i.e., that both buckets have the same number of doodads) if we apply Method #2. Cantor then showed that actually sometimes if we start pairing off doodads, we’ll end up running out of doodads in one bucket before the other, even when both have an infinite number of doodads in them. Thus was born the idea of differing “sizes” of infinity.
> how do mathematicians compare their sizes?
With slide rules in the men’s room.
Vsauce made a good video about the different types of infinities, I suggest you watch that as he does a very good job of describing the differences between certain infinities.
When scientists talk about different infinites, they are talking about the size of infinite sets. A set is a collection of items where each item in the set is unique. For finite sets it’s easy to compare their sizes, you just count how many items are in each set and compare the numbers. For infinite sets we have to get creative.
For infinite sets mathematicians use the buddy system to figure out if two sets are the same size. As an example, let’s use the set of all whole numbers (1, 2, 3…) and the set of all even numbers (2, 4, 6…). At first you might think there are more whole numbers than even numbers, but that is incorrect. If we create a rule where we multiply each whole number by two to find it’s buddy, we can see that each whole number has a “buddy” in the even number set, and each even number has a “buddy” in the whole number set.
It turns out that a lot of infinite sets are actually the same size, but there are some that are bigger. In 1891 a mathematician by the name of Georg Cantor came up with a clever proof to show that the set of Real Numbers (numbers with an infinitely long decimal part) is actually bigger than the set of whole numbers. He showed that no matter how you do it you can never set up a ‘”buddy system” between these two sets. There will always be real numbers that aren’t included in whatever buddy system you set up.
This proof is known as Cantor’s diagonal argument, and it’s actually pretty easy to understand. You can read about it here: https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
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