Measuring Infinities: a Short Journey into Number Theory
Jakub Dranczewski
Could you say that one infinity is somehow bigger than another one? It seems completely senseless. How can one claim that one thing of infinite size is greater than another thing of infinite size if, by definition, they both have no proper end? Let’s just stop here for a second though. What does it actually mean that two sets of things have the same number of elements? Your first answer would probably be ‘if you count the elements in them, you end up with the same number’. But what if we cannot count the number of elements in the set? This definition fails in that case. We have to look for something more general, and that something is a simple action: matching the elements of the two compared sets into pairs. If we manage to find a method of matching that leaves no elements of either set without a pair, we have proven that the number of elements in both sets is the same! Is this in any way useful for you? Well, that surely depends on your definition of usefulness, but one can make some fairly decent proofs with the above method. For example, we can quickly prove that the number of even numbers is exactly the same as the number of positive natural numbers! ‘But wait!’, you scream, ‘shouldn’t there be twice as many natural numbers as even numbers? Surely there is an additional odd number for every even number’. Intuitively that is correct, but let’s use the pairing method and see what happens. Take every natural positive number and multiply it by two, then assign the result as a pair to that number. If you write that down you’ll notice that all the even numbers were assigned to a natural positive number. … This means that the number of positive integers and the number of even integers are exactly the same. We just compared two infinite sets! It can be proven in a similar way that the number of integers is the same as the number of negative integers (I for one was initially rather surprised by that fact) or that there are exactly as many points on a line segment as on the entire line. The proofs go on and on! But hey, one could argue that all infinities are of the same size – as they are all infinite this seems to be a fair assumption. So now the real fun begins: let us prove that some infinities are bigger than others! 1 2 3 4 5 6 7 … ▼ ▼ ▼ ▼ ▼ ▼ 10 ▼ 12 ▼ 14 2 4 6 8
Not all infinities are equal
The proof above involved the size of a set of all integers. Let’s call this amount ℵ 0 (aleph-zero, the term is actually used by mathematicians, and it originates from a Hebrew letter). Now to help us find an infinity bigger than that I’ll quickly go through the proof that the size of the set of all rational numbers is also ℵ 0 (bear with me, it’s simple). First, make a table as below:
0 1
1
2
3
4
… … … … … …
1/1
2/1
3/1
4/1
-1
-1/1
-2/1
-3/1
-4/1
2
1/2
2/2
3/2
4/2
-2
-1/2
-2/2
-3/2
-4/2
…
…
…
…
…
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