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Type: Posts; User: Marshmallow Marshall

## Post Author:Marshmallow Marshall

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### ►►Re: Infinities being bigger than others, "countable" and "non countable" infinities◄◄

Originally Posted by yzb25
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! If we could encode every real with a natural, we'd have a way of finitely expressing every real number at once. Can you imagine?! Well, we literally can't, but still!
Lol

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.

## Post Author:Marshmallow Marshall

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### ►►Re: Infinities being bigger than others, "countable" and "non countable" infinities◄◄

Originally Posted by Plotato
yes but is it countable?
Uncountable, but any real number between 0 and 1 is comprised anyway, so the real number you would "create" technically already is part of the infinity. What you're proving is that uncountable and countable infinities are different (and that the integers infinity is countable, while the [0, 1[ infinity isn't), but not that one is bigger than the other, since both can go on forever. There aren't "more numbers" in one set than in the other, even though it's impossible to pinpoint a rank in the real numbers list unlike in the integers list. You seemed to be debating the fact that the real numbers infinite set is uncountable... which nobody disputed, as far as I know.

I'm totally making you all waste your time on explaining this to me XD

## Post Author:Marshmallow Marshall

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1,757

### ►►Re: Infinities being bigger than others, "countable" and "non countable" infinities◄◄

Originally Posted by yzb25
Maybe the problem is how you're thinking about infinity, as you say. You have an infinitely long list, therefore you can include everything between 0 and 1. Because you have "endless ammunition" anything not on the list can be added to the list. Is that what you're thinking?
Anything between 0 and 1 not only can be added to the list, but is on it. That's the assumption I'm making. If it's just an arbitrary list of some stuff between 0 and 1, then sure, it works, but isn't that... not infinite?

## Post Author:Marshmallow Marshall

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1,757

### ►►Re: Infinities being bigger than others, "countable" and "non countable" infinities◄◄

Originally Posted by yzb25
oh also MM, if you have doubts about the veracity of modern mathematics, you might get a kick out of this --

https://en.wikipedia.org/wiki/Long_line_(topology)

Wikipedia doesn't have a picture for that one lmao ^^

"Intuitively, the usual real-number line consists of a countable number of line segments [0,1) laid end-to-end, whereas the long line is constructed from an uncountable number of such segments."
To be clear, I have more doubts about my understanding of maths than about maths themselves lol. But anyway, I misspoke and meant infinity comprises everything between 0 and 1 in this case. I thought it was clear at first, but now I see how it isn't. Whoops.
My issue is more that the infinitely long operation is forced to result in a number between 0 and 1 still (else you'd be out of the specific infinity set, which would be "cheating" and wouldn't prove anything). How is a real number between 0 and 1 not comprised in an infinity of real numbers between 0 and 1? Shouldn't everything between 0 and 1 be in there, thus including the result number?

As for your "do it yourself" suggestion, it doesn't solve my issue because I know it's going to give a different number, that's obvious. My issue is with the exact concept of infinity, I guess. It's what aamirus said here:
So, while yes we are constructing an infinite length element that is not in the enumeration, that element is still a valid member of the set we are trying to prove is uncountable.
Also thanks for the walls and fuck you plotato

## Post Author:Marshmallow Marshall

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1,757

### ►►Infinities being bigger than others, "countable" and "non countable" infinities◄◄

Some context:
https://en.wikipedia.org/wiki/Cantor...gonal_argument