I remember as a kid we’d settle arguments by whoever could say the highest number (or multiply by it; such as “I hate you more times 100, or I’m 50 times better than you). At some point we all learned of the existence of infinity, and that would settle it. Whenever someone said “infinity plus one”, the other would object by saying “but infinity plus one is still infinity!” implying that infinity was as big as you could get, and that all infinities were the same size. Turns out they’re not.

This may sound counter-intuitive to some of you. After all, infinity alludes to an endless size right? How can two things that are endless be bigger or smaller than the other?

Well the properties of infinite sets have been studied for quite some time. I’m not sure if he was the first, but Galileo Galilei identified a conceptual problem with infinite sets in his final scientific work Two New Sciences. I’m not interested in giving a history lesson so I’ll keep this post short and let you do you own research if you don’t believe me (and by now, you should have much more faith in me than that!).

His work identified a property of infinite sets that is now known as Galileo’s Paradox. The gist of it is this:

  1. Some numbers are squares (2^2 = 4; 3^2=9; etc.) whereas some numbers are not (prime numbers and other non-square integers, etc.). Therefore all numbers must consist of both squares and non-squares, and thus be more numerous than just the squares alone.
  2. However, every square has exactly one positive number that is its square root (basically just reversing the square operation). Therefore, there are as many squares as there are square roots.
  3. For every number, there is exactly one squared value (you can square any number by multiplying it by itself). Therefore, there are as many square roots as there are numbers (because every number can be squared so every number has a square root). Since there are as many squares as there are square roots, and there are as many square roots as there are numbers, then there must be as many squares as there are numbers.
  4. However, our premise was that there are more numbers than there are squares. Therefore it is a “paradox”, the solution to which is that some infinities are larger than others.

Galileo simplified this in his work with a very apt analogy that I think will help people visualise this concept better. Imagine two lines, one longer than the other. You can say that both lines are made up of an infinite amount of infinitesimally small points, however, you can also clearly see that one is longer than the other.

Later on, the mathematician Georg Cantor provided mathematical proof and definitions for sets and trans-infinite numbers, to which a great deal of resistance was put up by the other mathematicians of the time. The existence of larger infinities was and is of great philosophical importance. You can Google more of Georg Cantor if you wish; this post is merely for me to show you that not all infinities are the same.

Also, here’s an 8 minute video explaining this concept with some different examples by Numberphile. The guy presenting has a creepy smile and is way too excited about maths, but it’s still quite informative: http://www.youtube.com/watch?v=elvOZm0d4H0

And a much faster explanation by MinutePhysics (about 2 minutes): http://www.youtube.com/watch?v=A-QoutHCu4o

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