If you had two flock2 of sheep and didn’t know how to count, there’s still a way by which you can tell if a flock is bigger than the other one: you just pair them up and see if there are sheep left over from one set.
Now, when you have infinite sets, counting doesn’t really make much sense, for numbers usually mean 0,1,2,3… BUT you can still tell if two sets have the same amount of numbers. You can use the sheep-comparing method.
Suppose you were standing before a room full of sitting people: you could tell that there’s at least as many chairs as there are people if you see that everyone has a seat, although there may be too many people to count.
This is the better way to tackle the infinity subject: if you want to know if two sets are “equal”, meaning they have the same amount of elements, you just have to relate their elements. This way, you can tell that there’s the same amount of natural numbers and even numbers, because you know you can go like this:
1 —> 2
2 —> 4
3 —> 6
…
And there’s no need to go on, because you know how it goes on and you can tell that there’s no natural number left out on the left side nor there is an even number left out on the right side.
You can do the same thing for the real numbers between 0 and 1 and the ones between 0 and 2. You just multiply by two:
0 —> 0
0.02 —> 0.02
0.35 —> 0.7
0.8 —> 1.6
…
And of course I’m leaving out many, many numbers, but you know how it goes, and you know no number is being left out.
“Infinity” is a real tough concept, and I find astonishing how it means many different things to a mathematician while it seems to have just one meaning for the rest of people. Infinity can mean an infinite amount of things in a set, a queue which is infinitely long (what mathematicians call “ordinals”), analysis objects like lim 1/x when x approaches 0, and a couple more things which have totally different meanings from one to the other.
