I was watching Veritasium’s video about math having a fatal flaw. He explained that if we make a set of all the numbers in between 0 and 1, then added one to the first digit of the first number and added one to the second digit of the second number etc, we would always have a new number. He said this proved that there were more numbers in between 0 and 1 than natural numbers.
I was confused as to why you can’t do this with natural numbers, and how that proved one infinity was smaller than another.
In: Mathematics
There are two ways to compare sizes of two sets A and B:
1. count how many elements are in A, count how many elements are in B, compare the result
2. Pair up the elements of A with the elements of B, see in which set you run out of elements first
While they seem similar, if you think about it you come to realize there is a principal difference between them: the first method assigns a “size” to A and then to B and then compares them. It does not only check if A and B have the same amount of elements, it also calculates what that amount is.
Both methods are fine (and equivalent) if A and B are finite, but when we try to extend them to the infinite we run into problem with the first method, namely, you would never finish counting the elements of A.
The second method does not require you to do so. Granted, if the sets are infinite then there are infinitely many pairs, but you are not required to present the pairing pair by pair, just to produce a rule.
We say that two sets are of the same size, denoted |A|=|B|, if there is a rule which matches every element of A with every element of B such that every element of B is paired exactly once. For example the set of natural numbers (1,2,3,…) can be paired with the set of even numbers (2,4,6,…) via the rule “x pairs with 2x”.
(To be more detailed: if there is a pairing such that every element of B is paired with at most 1 element of A [but not all of them must pair] we say that B is at least as large as A and denote |A| <= |B|. If there is a pairing with each element of B is paired with at least one element of A [but it could be more than 1] we say that A is at least as large as B denoted |A| >= |B|. It is not trivial at all to show that |A| >= |B| and |A| <= |B| implies |A|=|B| [or even, under my notations, that |A| <= |B| implies |B| >= |A|], but it does indeed hold. This result is called the Schröder–Bernstein theorem)
Now that we have a working definition of when one infinity is larger than other, one can try to argue that not all infinities are the same by showing there are two infinite sets A and B for which |A| <= |B| holds, but |A| >= |B| does not. The textbook example is to show that there are more natural numbers than real numbers. There are many videos of the proof you can look up, but the general idea is to:
1. note that it is easy to show that |Naturals| <= |Reals|: every natural number already is areal number, so matching x with x suffices.
2. Prove that if you try to do the opposite, that is, pair with each natural number a real number hoping that you’ve covered all real numbers, then there must be a number you have missed. This is done via a simple* argument called Cantor diagonalizations which has since become a staple of modern mathematics.
*by simple I don’t mean trivial. It is natural for people with no background to struggle with it. Don’t be disparaged if you don’t get it immediately, just keep revisiting it. It is a common saying that the main difference between mathematicians and non mathematicians is that mathematicians are so used to not understanding things they don’t get phased by it.
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