[QUOTE=yzb25;942889]It might help to bare in mind when we talk about the "size" of these infinite sets, that is a very informal way of referring to something called their "cardinality". In the popular culture, we've gotten very used to talking about "cardinality" as a measure of "size", but it may be slightly more accurate to think about cardinality in terms of "information".

For example, if you consider the set of positive whole numbers (1,2,3,4,5...) vs. the set of even numbers (2,4,6,8,10...) the first set seems strictly larger than the second set (in some sense, it has literally double the stuff). However, from the point of view of "cardinality", they both have the same amount of information. I can label every positive whole number with a unique even number like so, in a well-defined manner:

2->1

4->2

6->3

...

And when we say the real numbers have a higher cardinality, we are somehow making a statement that the real numbers are simply too complicated to be encoded in terms of positive whole numbers. There is no way of labelling every real number with a unique positive whole number.

If we could label every real number with a unique positive whole number, that would be kind of revolutionary for our notation. We use these garish "infinite decimals" to encode real numbers... but no matter how many decimal places you write down, there's still so many possible numbers you could be referring to when you write the next digits! [COLOR="#008080"]If we could encode every real with a natural, we'd have a way of [I]finitely[/I] expressing every real number at once. Can you imagine?! Well, we literally can't, but still![/COLOR][/QUOTE]

[COLOR="#008080"]Lol :D[/COLOR]

I think I understand a little better now (it basically comes down to plotato's thing about mapping integers and real numbers one to one, right?), and well... I guess that's why I didn't go too far into maths xD.

LolIt might help to bare in mind when we talk about the "size" of these infinite sets, that is a very informal way of referring to something called their "cardinality". In the popular culture, we've gotten very used to talking about "cardinality" as a measure of "size", but it may be slightly more accurate to think about cardinality in terms of "information".

For example, if you consider the set of positive whole numbers (1,2,3,4,5...) vs. the set of even numbers (2,4,6,8,10...) the first set seems strictly larger than the second set (in some sense, it has literally double the stuff). However, from the point of view of "cardinality", they both have the same amount of information. I can label every positive whole number with a unique even number like so, in a well-defined manner:

2->1

4->2

6->3

...

And when we say the real numbers have a higher cardinality, we are somehow making a statement that the real numbers are simply too complicated to be encoded in terms of positive whole numbers. There is no way of labelling every real number with a unique positive whole number.

If we could label every real number with a unique positive whole number, that would be kind of revolutionary for our notation. We use these garish "infinite decimals" to encode real numbers... but no matter how many decimal places you write down, there's still so many possible numbers you could be referring to when you write the next digits! If we could encode every real with a natural, we'd have a way offinitelyexpressing every real number at once. Can you imagine?! Well, we literally can't, but still!

I think I understand a little better now (it basically comes down to plotato's thing about mapping integers and real numbers one to one, right?), and well... I guess that's why I didn't go too far into maths xD.