[tobiasSoftware]

There are, and it turns out that this is a significant mathematical concept.

The integers between 0 and infinity are defined as "countably infinite". Other infinities are considered countably infinite, or the "same" infinity, if and only if you can arrange it in a list such that each item in the list pairs to an integer in our 0 to infinity list. So the set of even numbers is countably infinite because for every i that is an even number, it pairs with the number i/2.

To demonstrate: 0 -> 0, 2 -> 1, 4 -> 2, 6 -> 3, ...

The decimal (real) numbers between 0 and 1 are not countably infinite, and we know this from a concept called Cantor diagonalization. What Cantor did was a proof by contradiction: assume that the numbers are countably infinite, then you can arrange them in a list. However, he then builds a number by altering the first decimal place of the first number, the second decimal place of the second number, and so on. Finally, he shows that this built number is both a real number and is not on the list. Therefore, the real numbers between 0 and 1 cannot be ordered into a list, therefore they are not countably infinite, and there are more decimal numbers between 0 and 1 than integers between 0 and infinity.

[denton-scratch]

> The decimal (real) numbers between 0 and 1

The way I parse "decimal number" in this context is a number expressible as a (finite?) string of decimal numerals. Those numbers are not reals, they are rationals.

[dbtc]

"Countably infinite" makes zero sense to me.

Whatever method you use to generate your decimals, you can just slap an integer on each step of the way. You'll never run out of integers.

I'll put Cantor and his proof in a box, tell him to give me his fancy decimals quick as he can, and I can match each one with an integer no problem.

And pairing one infinite list with another infinite list doesn't make either one any more countable, because however high you count, they keep on going.

[amanwithnoplan]

I think "countably infinite" makes no sense to you because you have a different idea of countable than a mathematician (disclaimer: not a mathematician).

A mathematician compares the size of two sets of stuff by pairing off items from each set, but this is not a mechanical process taking a finite or even unbounded amount of time: they just need to show such a mapping exists or that nonexistence would lead to a contradiction; they don't need to actually carry out the process mechanically. By definition (according to mathematicians), something is countable if it is the same size as the set of natural numbers {0, 1, 2, ...} or smaller, and "countably infinite" just means it is the same size as the naturals (and not smaller, which would make it finite).

A small minority of mathematicians hold the position that proof-by-contradiction is not good enough, and that you really do need to positively prove something. They are called intuitionists.

Presumably, an even smaller minority of mathematicians hold the position that this proof must (theoretically) be able to be carried out in a mechanical manner. They're some flavor of constructivists, but maybe they're better called programmers. <- This is where you are.

[feoren]

> Whatever method you use to generate your decimals, you can just slap an integer on each step of the way. You'll never run out of integers.

Exactly correct! This holds true of everything you can generate stepwise, even infinite sets. Cantor proved that you cannot "generate" (stepwise) all Reals between 0 and 1. Any infinite set you can generate stepwise is Countably Infinite.

> I'll put Cantor and his proof in a box, tell him to give me his fancy decimals quick as he can, and I can match each one with an integer no problem.

Exactly correct! And then infinitely later, when you're "done", having generated every Real between 0 and 1, he will then generate a new Real not on your list. Oops! You have not generated all Reals between 0 and 1, even with infinite time.

> And pairing one infinite list with another infinite list doesn't make either one any more countable, because however high you count, they keep on going.

Exactly correct! Any two sets you can pair together (via a bijection) have the exact same cardinality. Neither is more infinite nor countable than the other. Cantor proved you cannot "pair" the Reals with the Natural Numbers.

You and Cantor agree completely. You're very close to understanding why the Reals are bigger.

[quantified]

Cantor's proof is a single attempt. Suppose you could construct a space-filling curve that did indeed map all numbers between 0 and 1 to all integers? Has there been a proof that such a curve does not exist? The fact that his proof leans on a specific set of decimal places at every juncture has always seemed a weakness of his proof, because you can always map any set of numbers from 0 to 1 with any set of decimal places to a set of integers.

[bombolo]

does decimal numbers mean fractions or real numbers?

If it means fractions only, they are countable.

[shidoshi]

This account discrete maths. Bravo!