The other day VRBones commented that the discussion about ever larger sizes of infinities sounds like a playground argument:
1: You’re an idiot!
2: Am not!
1: Are too!
2: Am not, not!
1: Are too, too, too, too, too!
2: Am not infinity!
1: Are too infinity times infinity!
Far be it for me to argue with the wisdom of children, but it turns out that in additon to describing how infinities can be different, another of Cantor’s insights was how to say when two infinite sets are the same size. Basically, they are the same when you can match up their elements one to one.
This leads to some pretty crazy sounding but absolutely correct conclusions. For example, the number of even numbers (2,4,6,8,…) is the same as the number of counting numbers (1,2,3,4…), because you can match them up one to one:
| 1 | 2 | |
| 2 | 4 | |
| 3 | 6 | |
| 4 | 8 | |
| … | … |
Now we get to where VRBones quotes his schoolyard mathematician friend: “Are too infinity times infinity!”. It turns out that even if you multiply an infinity by itself, you just get the same sized infinity. Here’s how Cantor showed this to be true for the counting numbers:
Making a list
If you multiply the set of numbers 1,2,3,…∞ by itself, you get a two dimensional block of row/column pairs:
| 1,1 | 1,2 | 1,3 | 1,4 | … |
| 2,1 | 2,2 | 2,3 | 2,4 | … |
| 3,1 | 3,2 | 3,3 | 3,4 | … |
| 4,1 | 4,2 | 4,3 | 4,4 | … |
| … | … | … | … |
Clearly this forms an infinite set. But here’s the question: Can we number the things in this set, using the numbers 1,2,3,…? If we can, then the set is no bigger than the set of counting numbers.
Cantor showed just how to do that:
| 1 | 2 | 4 | 7 | … |
| 3 | 5 | 8 | 12 | … |
| 6 | 9 | 13 | 18 | … |
| 10 | 14 | 19 | 25 | … |
| … | … | … | … |
I’ve used colors to highlight the way he went through the list along the diagonals, column by column. In this way you can assign a number 1,2,3,… to every pair of numbers in ∞×∞. In other words, infinity times infinity is just infinity!
Checking it twice
Cantor also showed that the real numbers times the real numbers is no bigger than just the real numbers, as follows:
Consider the real numbers from zero to one. This is the same as considering all points along a line segment from zero to one:
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If you multiply the infinite set of real numbers between zero and one by itself, you get all possible pairs of such real numbers, like (0.73245.. , 0.26347…). But this is just another way of talking about all points in a square:
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The first number describes left-to-right position of a point in the square, and the second number represents top-to-bottom position.
I know, you’re probably going to say it’s crazy to talk about a line segment and a square having the same number of points. But Cantor came up with a very easy way to prove that it’s true.
Just take any point in the square, say (0.73245.. , 0.26347…), and weave together their decimal coordinates into a single number, like so:
0.7236234457
I’ve alternated every other digit color so you can see how the two original numbers each contributes its digits. You can do the sme thing for any pair of real numbers to get a single number.
Furthermore, you can also always go the other way: For any single number you can think of, like 0.3141592653…, you can make a pair numbers out of it by separating out its even and odd numbered digits – in this case: (0.34525… , 0.11963…).
Since you can match up their points one to one, you know that there are just as many points in a line segment as there are in a square.
In other words (yet again) infinity times infinity is the same as infinity.
VRBones, you might want to try telling that to those kids in the playground the next time you see them. But beware – they just might surprise you: If you explain it clearly, you might find that they understand exactly what you are talking about.
Actually I was kind of pleased when infinity came up in the schoolyard. Infinity times infinity would equal infinity, which isn’t bigger but the same, therefore the progression ends with the first person to mention it. Introducing Aleph into the schoolyard would mean there’s no end in sight!
Actually I’ve enjoyed the infinity posts, but once you start talking about the ‘size’ of a set I can’t see why it doesn’t undo the initial infinity logic? I’m happy to accept that there’s an infinite amount of primes, and that there’s an infinite amount of counting numbers, but by saying that some numbers are uncountable (reals) is like saying some countable numbers aren’t prime. Why is the countable set so different to any other arbitrary set (primes, evens, fibonacci)?