uncountable infinity

Richard Ferrara, Quora contributor, Software Engineer, Catholic, conservative Republican, Scrabble player, MS Mathematics & Physics, University of Copenhagen (1971), MSc Mathematics, University of Bristol (1990)
 
What is an uncountable infinity?
 
Let’s say you die and go to Hell.

You insist that there has been some mistake and file an appeal with Saint Peter. He convenes a panel of archangels to consider your case.
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Eventually, a compromise is reached: you can win your soul back, but to do so, you must beat the devil in a guessing game.

I.
Here’s how it works: the devil writes down a whole number on a piece of paper. This number never changes. It can be any positive whole number — there is no upper limit. It can be 52, a trillion, a googolplex, whatever. [Nine-year-old Milton Sirotta coined the term googol, 10 to the 100 power, and then proposed the term googolplex to be “one, followed by writing zeroes until you get tired.”]

Each day you get one guess. If you correctly pick the devil’s number, you go free. Otherwise, you’re stuck there for another day.

(We assume that you have some foolproof way of keeping track of what numbers you’ve previously chosen, the piece of paper is big enough to fit any number, the devil doesn’t cheat and change it, etc.)

What do you do?

Well, the bad news is that there’s no telling how long you’ll be in perdition. But the good news is that a simple strategy will get you free.

On the first day you guess one. If that’s the devil’s number, great, you’re free. If not, then on the second day you guess 2, then 3, 4, 5, and so on.

It may take eons, but eventually you will reach the right number and escape. That’s because even though there are an infinite number of positive integers, it’s a countable infinity — a set whose members can be arranged in order so that you can hit every one of them. The devil cannot pick a number so high or complex that you will never reach it.
 
II.
Now let’s change the rules of the game: this time the devil can pick any integer (positive, negative, or zero).

Obviously, you can’t follow the same strategy, because you will miss all the negative numbers. However, you can still win. Just start with zero, then plus 1, then minus 1, plus 2, minus 2, and so on.

Even though it looks like more choices have been added, the result is still a countable infinity. In math terminology, we say that the cardinality of the set hasn’t changed.
(It’s called “aleph-null” if you want to get technical.)

III.
Now let’s change the rules again: this time the devil can pick any rational number (i.e., any number that can be expressed as a fraction).

At first glance, it looks like the game is now unwinnable. What’s the fraction with the smallest value greater than zero? One half? No, one third is less. And one fourth is less than that. And what’s the first number greater than one fourth? 2/7? 3/11?

You are in trouble now.

Or are you?

See, there’s a trick: although you need to find some way to order your guesses, you don’t necessarily need to order them by value.

A fraction is just a pair of numbers: a numerator and a denominator.

Imagine those two numbers as X and Y coordinates on a plane. You can now win the game by starting from the origin and working outward in a spiral.

First you guess all the fractions whose numerator and denominator add up to 1. There’s only one of them: 0/1.

Next, the ones that add up to 2, which are 0/2 and 1/1.

Then 0/3, 1/2, 2/1. And so on.

It took some thinking this time, but once again you’ve escaped.

IV.
Okay, one more: this time the devil can pick any irrational number.

It can be pi, or e [Euler’s number, an irrational number with a non-recurring decimal that stretches to infinity], or the 20th root of a billion, or the square of the cosine of the cube root of the natural logarithm of the hyperbolic tangent of … you get the idea.

This time you’re screwed.

Going in value order won’t work. Going in a spiral won’t work. Going in alphabetical order by the description of the number won’t work. The number may have an infinitely long description or even none at all.

The set of irrationals is an uncountable infinity — there’s no way to order them so that you hit them all. A fellow named Georg Cantor proved it in 1874.

Whatever strategy you use, you’re going to miss some numbers.
 
Of course, you could still get lucky. The devil could pick something simple like the square root of 2, and you could just happen to guess it. But you’re no longer guaranteed to win the game.
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So … enjoy perdition. You’re going to be there for a while.