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My infinity is bigger than yours!

April 20, 2013

Warning: Attempt at explaining some real mathematics here! 🙂

A common exchange between young children:

Child 1: My blank is better than yours!
Child 2: Nuh uh! Mine’s twice as good!!
Child 1: Mine’s 100 times as good as yours!
Child 2: Well mine’s infinity times better!
Child 1: Oh yeah?! Well mine’s infinity plus one better than yours!!

Ad infinitum.

Now, I hope the average reader is already aware that “infinity plus one” is not a number. Infinity is not a number either; not in any usual sense of numbers anyway. Let’s think of infinity as simply a concept, of sorts. Mathematicians refer to the regular counting numbers, 1, 2, 3, 4… as natural numbers and we use the symbol \mathbb{N} to denote them. So if I were to ask you how many natural numbers there were, then you’d obviously say there are infinitely many; however steer clear of saying you have infinity of them.

For argument’s sake, let’s see what happens if we do take infinity to be a number; specifically, we’ll take it to be the total number of natural numbers that there are. If there are “infinity” natural numbers, then there we’ll just exclude the number one and then we’ll have one less number: 2, 3, 4… etc. But surely this isn’t satisfying to say this new set of numbers starting at two, has any less than infinity numbers! Equivalently, we could pair up every single number in the new set with a unique natural number simply by subtracting one from it. So two in the new set gets paired with one in the old; three in the new set get paired with two in the old one, and so on.

There is exactly one number in the top set for every number in the bottom one.

There is exactly one number in the top set for every number in the bottom one.

This means that infinity minus one would still be equal to infinity – hardly makes sense as a number! After all, if \infty -1=\infty then we could subtract infinity from both sides and be left with -1=0. Congratulations, you broke math.
I hope I’ve convinced you that infinity can’t be a number (in the usual sense of the word), and that the collection of counting numbers starting from 2, 3, 4… has just as many numbers in it as if we had started from 1, 2, 3…

Now I want to tell you about “bigger numbers” than this notion of infinity; this is where it may start messing with your head. A mathematician will call a collection of things, a set; think of a set as being any old bag of… stuff. If I have a set, or a bag containing three things, I’ll say the cardinality of this set is three. Cardinality is just a mathematician’s notion of the size of a collection – just the number of things in the collection (at least if there are only finitely many things in it). So what if our “bag” has all of the natural numbers (1, 2, 3…), so there are infinitely many things in the set? I’ve already said we want to talk about bigger infinities than this one, so let’s not use the symbol \infty for the size (or cardinality) of the natural numbers. Instead, mathematicians say the cardinality of the natural numbers is \aleph_0, pronounced “aleph naught” or “aleph null”. So to recap: the cardinality (size) of a finite set (collection) is just the number of elements (things) in it, and the cardinality (size) of the natural numbers (1, 2, 3… etc.) is something we call \aleph_0, “the smallest infinity”.

Now obviously there are far more numbers than just the natural numbers that I keep talking about. What about -2, 0.5, \pi and \sqrt{2}?  Well it turns out that if we include all of the things that you would regularly consider numbers in one big set (collection), then the cardinality (size) of this set is actually bigger! This larger set of numbers is called the real numbers and is usually denoted by \mathbb{R}. I won’t bore you with the proof  here but any good real analysis book will have one; even Wiki’s page for Cantor’s diagonalisation argument proves it, although reading it makes me want to edit the page to something more understandable (warning, link contains math). Let’s call the cardinality(size) of the real numbers C, for now.

But you may have expected the set of real numbers to be larger than the set of natural numbers; it does intuitively seem like there are more real numbers. The real kicker, is thinking about something in between! What about the set of numbers that can be written as fractions? Like a half, or negative three quarters or \frac{23234268}{423768923}? We call these numbers rational numbers (they’re written as a ratio of two integers) and mathematicians use the symbol \mathbb{Q} for them. How big is the set of rational numbers?

We can actually list the rational numbers in some order, 0,1,-1,\frac{1}{2},2,-\frac{1}{2},-2,\frac{1}{3},\frac{2}{3},3,\frac{3}{2},-\frac{1}{3},-\frac{2}{3}..., so there is a first, second, third etc. Since we can write it like this, then we can pair up each of the natural numbers with this list of rational numbers; the first in our list gets paired with the number one, the second is paired with the number two and so forth. This is exactly like the picture above. Since we can do this, there must be just as many rational numbers as there are natural numbers, so we say the cardinality (size) of the rational numbers is also \aleph_0 – a little bizarre, hey?! Just to kick your intuition square in the nuts, between any two real numbers there are infinitely many rational numbers! Even though there are way more real numbers than rational ones…

Now for the really amazing part!
We use the notation \aleph_0, because it’s the “smallest infinity”, and we can say \aleph_1 is the next smallest, \aleph_2 is the next smallest after that, and so on. It is unknown if the cardinality of the real numbers (which we’ve denoted by as C) is equal to \aleph_1. But worse than that! It is impossible to ever prove or disprove this using the standard axioms of mathematics! (See Continuum hypothesis)

For the mathematically curious, check out the following Wikipedia articles: Natural number, rational number, irrational number, real number, ordinal number, cardinal number.

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From → Math

7 Comments
  1. I really enjoyed learning about the concept of cardinality. 🙂
    Thanks for the revision!

  2. Also, the links at the bottom of the page to next and previous post are screwing with me. I know it probably wasn’t you that put ‘next’ on the left and ‘previous’ on the right, but maybe there’s an option somewhere… 🙂

    • Thanks for the comments!
      I had no idea those links were even there; unfortunately, almost nothing is customisable without subscribing to a premium kind of thing though.

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