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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 [I]finitely[/I] expressing every real number at once. Can you imagine?! Well, we literally can't, but still!

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!

[QUOTE=Marshmallow Marshall;942863]Uncountable, but [I]any[/I] 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[/QUOTE]

As a matter of fact, you have given our hollow lives purpose!

Originally Posted by Marshmallow Marshall

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

As a matter of fact, you have given our hollow lives purpose!

[QUOTE=Plotato;942845]this is misleading

the set of integers will always have a bigger number, this is an approach but never reach argument which implies the set of integers is uncountable, but it is. cantor's diagonalization demonstrates by contradiction that you can generate more real numbers than a prescribed infinite number of integers, therefore making the set of real numbers, [I]not[/I] countable by mathematical definition.

personally, analogies that try to "list" or "count" infinities will always end up being confusing because the verb implicitly ascribes a mathematical countability (i can count +1 everytime) or uncountability (i can go on forever, can't count forever) to it, depending on whoever's interpretation. some infinities will be smaller or larger [B]in size, or the number of elements[/B] contained in them, but the only thing defining countability infinite is whether there is mathematical indication that you can map one-to-one from the infinite set of integers to whatever.[/QUOTE]

I didn't mean to suggest in the forever-list you're missing a number because you haven't reached it yet. I meant even "after" completing the forever-list, you still wouldn't have every number. You could list every natural number given "forever", by writing the number n at second n, or every integer by alternating between writing a negative and a positive each second. Understand? This is a legitimate way to think about it. If the thought is clearly conveyed, the forever-list is no different from a function from the naturals.

I agree it might be better to let go of the forever-list stuff if it's derailing things and just work with a function from the naturals. I just like it because you can explain it without giving explicit formulae. >.>

Originally Posted by Plotato

this is misleading

the set of integers will always have a bigger number, this is an approach but never reach argument which implies the set of integers is uncountable, but it is. cantor's diagonalization demonstrates by contradiction that you can generate more real numbers than a prescribed infinite number of integers, therefore making the set of real numbers, not countable by mathematical definition.

personally, analogies that try to "list" or "count" infinities will always end up being confusing because the verb implicitly ascribes a mathematical countability (i can count +1 everytime) or uncountability (i can go on forever, can't count forever) to it, depending on whoever's interpretation. some infinities will be smaller or larger in size, or the number of elements contained in them, but the only thing defining countability infinite is whether there is mathematical indication that you can map one-to-one from the infinite set of integers to whatever.

I didn't mean to suggest in the forever-list you're missing a number because you haven't reached it yet. I meant even "after" completing the forever-list, you still wouldn't have every number. You could list every natural number given "forever", by writing the number n at second n, or every integer by alternating between writing a negative and a positive each second. Understand? This is a legitimate way to think about it. If the thought is clearly conveyed, the forever-list is no different from a function from the naturals.

I agree it might be better to let go of the forever-list stuff if it's derailing things and just work with a function from the naturals. I just like it because you can explain it without giving explicit formulae. >.>

If you do not necessitate that the numbers are being written down, one at a time, once per second, forever, then perverse things can happen. For example, if you allow god to take the pen, maybe god is capable of instantaneously producing a vast list that includes every such number. Maybe he uses his long line for space to do it (lol) but that is neither here nor there.

As Plotato is repeating, the list needs to be [i]countably infinite[/i], which I suppose for our purposes is the mathematically formal way of ruling out the possibility god took the pen. It restricts you to either using a [I]numbered well-defined list[/I], or some kind of scenario where you only get to write down one entry per second forever.

If you do not necessitate that the numbers are being written down, one at a time, once per second, forever, then perverse things can happen. For example, if you allow god to take the pen, maybe god is capable of instantaneously producing a vast list that includes every such number. Maybe he uses his long line for space to do it (lol) but that is neither here nor there.

As Plotato is repeating, the list needs to be countably infinite, which I suppose for our purposes is the mathematically formal way of ruling out the possibility god took the pen. It restricts you to either using a numbered well-defined list, or some kind of scenario where you only get to write down one entry per second forever.

[QUOTE=Marshmallow Marshall;942838]Anything between 0 and 1 not only [I]can[/I] be added to the list, but [I]is[/I] on it. That's the assumption I'm making. If it's just an arbitrary list of [I]some[/I] stuff between 0 and 1, then sure, it works, but isn't that... not infinite?[/QUOTE]

Hm... Maybe it would benefit both of us for you to be more cautious with how you phrase these things. An arbitrary list of [i]some stuff[/i] may very well be unending (which is exactly what the word [I]infinite[/I] means). What I assume you mean is, "clearly that list isn't complete".

What I am asserting is there is no [i]complete[/i] list of all the numbers between 0 and 1, whether it's unending or not. By virtue of merely demanding that you have to write each number down, one by one, you have already imposed a subtle restriction on your ability to account for every number. Even if you write down one entry per second from now until the end of time, you will still be missing a number. That is what the proof illustrates, and that is what makes it profound.

Originally Posted by Marshmallow Marshall

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?

Hm... Maybe it would benefit both of us for you to be more cautious with how you phrase these things. An arbitrary list of some stuff may very well be unending (which is exactly what the word infinite means). What I assume you mean is, "clearly that list isn't complete".

What I am asserting is there is no complete list of all the numbers between 0 and 1, whether it's unending or not. By virtue of merely demanding that you have to write each number down, one by one, you have already imposed a subtle restriction on your ability to account for every number. Even if you write down one entry per second from now until the end of time, you will still be missing a number. That is what the proof illustrates, and that is what makes it profound.

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?

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?

[QUOTE=Marshmallow Marshall;942833]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 [I]between 0 and 1[/I] 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 [I]everything between 0 and 1[/I] 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:


Also thanks for the walls and fuck you plotato :D[/QUOTE]

hmm, maybe there is a misunderstanding about what this infinite list represents exactly. This list is just some arbitrary list of numbers between 0 and 1. The list does [i]not[/i] necessarily have the infinity of numbers between 0 and 1. Indeed, we are about to prove it doesn't. The list could be something dumb like this:

1) 0.1
2) 0.01
3) 0.001
4) 0.0001
...

Or it could be a more earnest attempt to hit every number between 0 and 1:

1) 0.110100110001...
2) 0.110001111111...
3) 0.101010110101....
...

It doesn't matter. The point of the proof is that, no matter what infinite list is given, we can apply the construction to get a number between 0 and 1 not on the list.

Originally Posted by Marshmallow Marshall

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:


Also thanks for the walls and fuck you plotato

hmm, maybe there is a misunderstanding about what this infinite list represents exactly. This list is just some arbitrary list of numbers between 0 and 1. The list does not necessarily have the infinity of numbers between 0 and 1. Indeed, we are about to prove it doesn't. The list could be something dumb like this:

1) 0.1
2) 0.01
3) 0.001
4) 0.0001
...

Or it could be a more earnest attempt to hit every number between 0 and 1:

1) 0.110100110001...
2) 0.110001111111...
3) 0.101010110101....
...

It doesn't matter. The point of the proof is that, no matter what infinite list is given, we can apply the construction to get a number between 0 and 1 not on the list.

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

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

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

[I]"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."[/I]

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."

[QUOTE=Marshmallow Marshall;942814]Some context:
[url]https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument[/url]
[url]https://www.youtube.com/watch?v=HeQX2HjkcNo[/url]

I've been randomly looking into this and... I don't believe Cantor's argument! I'm not great at maths, though, so it's very possible I'm just being dumb, but I think that the argument is flawed (and that the conclusion is too). Here's why: it's impossible to add one to [I]x[/I] rank of each number on an infinite list without getting to a number of the list, because... the list is infinite! The result obtained through this cannot be unique, as infinity necessarily comprises [I]everything[/I]. Thus, isn't the proof based on a misunderstanding of the concept of infinity?

And yes, it is very ballsy of me to attack something that apparently was proven a century ago, and I dare hope you're gonna prove me wrong or tell me I misunderstood something, O you knowledgeable people :P[/QUOTE]

Well, it might be better for [i]you[/i] to flesh out your understanding of this and us to then reply to you. Because this reasoning:

[I]"because... the list is infinite! The result obtained through this cannot be unique, as infinity necessarily comprises everything."[/I]

Sounds like an argument for why the statement itself must be wrong rather than any issue with the proof itself. As lag said, you can certainly have infinitely many things without having [i]everything[/i]. You just need to have a never-ending amount.

Nonetheless, this discussion is very fun so I'll say a few things --

The proof simply constructs a new real number from a list of other real numbers which is not anywhere on the aforementioned list. Do you have an issue with some aspect of that process? Do you have an issue with the idea of applying an "infinitely long operation" to construct the new real number? Or are you unconvinced that the new real number will necessarily not appear anywhere on the list?

The following advice may help you to think about it:

Rather than trying to grasp the entire infinite process at once, try to think about what's happening at each entry. You can't possibly grasp the new real number this construction generates, because changing the n'th entry on every n'th row will take infinitely long. However, you can ask me about any of this new real number's digits - ask me what the 50th digit is, ask me what the 100th digit is - and I can run 100 rows down my list and 100 entries along and find the digit and tell you "ah, the new real number's 100th digit is a 1!". This is no different from how you understand any other real number. You do not know every single digit of pi, but given enough time (and enough motivation ^^) you can find the 100th digit or the 200th digit. And that's enough to satisfy you that the number exists and makes sense.

The same is true for the process of [i]checking[/i] whether this new real number is different from every entry on the list. You can't possibly check [i]every[/i] number on the list to see if it's different from your new real number. Attempting such a thing would be foolish. But, give me any row. Ask for the 1000th row even. I can go to that row, and verify for you that our new real number is indeed different from the number on that row, by plodding along to the 1000th entry and checking that our new real number does indeed have a different digit.

It may sound daft, but maybe try literally carrying out this construction for just a few numbers. Write out 5 or 6 real numbers to 10 digits, then reverse the n'th digit on the n'th row, and stare at the resulting construction until it makes sense how it would work exactly the same in an endless list ^^.

Originally Posted by Marshmallow Marshall

Some context:
https://en.wikipedia.org/wiki/Cantor...gonal_argument
https://www.youtube.com/watch?v=HeQX2HjkcNo

I've been randomly looking into this and... I don't believe Cantor's argument! I'm not great at maths, though, so it's very possible I'm just being dumb, but I think that the argument is flawed (and that the conclusion is too). Here's why: it's impossible to add one to x rank of each number on an infinite list without getting to a number of the list, because... the list is infinite! The result obtained through this cannot be unique, as infinity necessarily comprises everything. Thus, isn't the proof based on a misunderstanding of the concept of infinity?

And yes, it is very ballsy of me to attack something that apparently was proven a century ago, and I dare hope you're gonna prove me wrong or tell me I misunderstood something, O you knowledgeable people :P

Well, it might be better for you to flesh out your understanding of this and us to then reply to you. Because this reasoning:

"because... the list is infinite! The result obtained through this cannot be unique, as infinity necessarily comprises everything."

Sounds like an argument for why the statement itself must be wrong rather than any issue with the proof itself. As lag said, you can certainly have infinitely many things without having everything. You just need to have a never-ending amount.

Nonetheless, this discussion is very fun so I'll say a few things --

The proof simply constructs a new real number from a list of other real numbers which is not anywhere on the aforementioned list. Do you have an issue with some aspect of that process? Do you have an issue with the idea of applying an "infinitely long operation" to construct the new real number? Or are you unconvinced that the new real number will necessarily not appear anywhere on the list?

The following advice may help you to think about it:

Rather than trying to grasp the entire infinite process at once, try to think about what's happening at each entry. You can't possibly grasp the new real number this construction generates, because changing the n'th entry on every n'th row will take infinitely long. However, you can ask me about any of this new real number's digits - ask me what the 50th digit is, ask me what the 100th digit is - and I can run 100 rows down my list and 100 entries along and find the digit and tell you "ah, the new real number's 100th digit is a 1!". This is no different from how you understand any other real number. You do not know every single digit of pi, but given enough time (and enough motivation ^^) you can find the 100th digit or the 200th digit. And that's enough to satisfy you that the number exists and makes sense.

The same is true for the process of checking whether this new real number is different from every entry on the list. You can't possibly check every number on the list to see if it's different from your new real number. Attempting such a thing would be foolish. But, give me any row. Ask for the 1000th row even. I can go to that row, and verify for you that our new real number is indeed different from the number on that row, by plodding along to the 1000th entry and checking that our new real number does indeed have a different digit.

It may sound daft, but maybe try literally carrying out this construction for just a few numbers. Write out 5 or 6 real numbers to 10 digits, then reverse the n'th digit on the n'th row, and stare at the resulting construction until it makes sense how it would work exactly the same in an endless list ^^.