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[QUOTE=yzb25;942830]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][/QUOTE]

This is great.

"It's like, the number line... but [I]longer[/I]"

Also a perfect example of why I stuck to applied mathematics xP

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."

This is great.

"It's like, the number line... but longer"

Also a perfect example of why I stuck to applied mathematics xP

[QUOTE=aamirus;942818]Let's try a proof that some infinite set is countable then. In general, to prove this we need to show that you can map a natural number (these are the numbers 0, 1, 2, 3 4, 5, .... to infinity, sometimes we include 0 sometimes we don't depending on textbook) to every element in the set.

The natural numbers are trivially countable since you can just assign 0 to 0, 1 to 1, 2 to 2, 3 to 3, and so on.

Integers (.... -3, -2, -1, 0, 1, 2, 3 ...) countable proof:
Map 0 to 0, 1 to 1, 2 to -1, 3 to 2, 4 to -2, 5 to 3, 6 to -3....

Now you may say that "HEY! wait a minute, then you're going to run out of natural numbers long before you finish the integers!" But the natural numbers is an infinite set. Pick any integer and I can give you the natural number that would produce it. We just needed to show that we can assign a unique natural number to every integer. Hence the integers are countable.

Now to show that a set is UNcountable you'd need to prove that there is no formula out there which can map a unique natural number to every single element in the set. Of course trying every possible formula would be impossible, hence we look for a more clever way to prove that any formula will fail.
For cantor's diagonalization theorem, in the video they are proving the set R(real numbers) is uncountable. In the wiki and generally the way the proof is shown, it's used on the set of all infinite length binary sequences. In either case, the idea is we first assume that there DOES exist a formula out there which manages to enumerate all the elements of the target set. Then we show that this enumeration must be missing an element of the set and therefore fails to map a natural number to every element in the set. In both cases, the elements here are all infinite in length. 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.[/QUOTE]

This is a great explanation

[QUOTE=Marshmallow Marshall;942814]The result obtained through this cannot be unique, as infinity necessarily comprises [I]everything[/I].[/QUOTE]

The one thing I would add is to address the underlying source of the misunderstanding.

If infinity [I]did[/I] comprise everything by definition, then your reasoning would be correct - but something can be infinite, and [I]not[/I] contain everything.

For example, there are infinite counting numbers (1, 2, 3, 4, ...)
But the counting numbers will never contain 0.5, no matter how high you count. Nor will they ever contain 1.5, or 2.5, or 3.5. In fact despite being there being infinite counting numbers, there are still infinite numbers that are [I]aren't[/I] counting numbers.

The important takeaway stems from a pretty common misunderstanding of what infinity is. Infinity doesn't mean it contains everything, it just means that you can continually list things forever.

===

On a sort of a tangent, what you're describing as infinity (a set that contains [I]everything[/I]) can actually be shown to be impossible due to Russell's paradox that they talk about in the Veritasium video you linked.

Originally Posted by aamirus

Let's try a proof that some infinite set is countable then. In general, to prove this we need to show that you can map a natural number (these are the numbers 0, 1, 2, 3 4, 5, .... to infinity, sometimes we include 0 sometimes we don't depending on textbook) to every element in the set.

The natural numbers are trivially countable since you can just assign 0 to 0, 1 to 1, 2 to 2, 3 to 3, and so on.

Integers (.... -3, -2, -1, 0, 1, 2, 3 ...) countable proof:
Map 0 to 0, 1 to 1, 2 to -1, 3 to 2, 4 to -2, 5 to 3, 6 to -3....

Now you may say that "HEY! wait a minute, then you're going to run out of natural numbers long before you finish the integers!" But the natural numbers is an infinite set. Pick any integer and I can give you the natural number that would produce it. We just needed to show that we can assign a unique natural number to every integer. Hence the integers are countable.

Now to show that a set is UNcountable you'd need to prove that there is no formula out there which can map a unique natural number to every single element in the set. Of course trying every possible formula would be impossible, hence we look for a more clever way to prove that any formula will fail.
For cantor's diagonalization theorem, in the video they are proving the set R(real numbers) is uncountable. In the wiki and generally the way the proof is shown, it's used on the set of all infinite length binary sequences. In either case, the idea is we first assume that there DOES exist a formula out there which manages to enumerate all the elements of the target set. Then we show that this enumeration must be missing an element of the set and therefore fails to map a natural number to every element in the set. In both cases, the elements here are all infinite in length. 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.

This is a great explanation

Originally Posted by Marshmallow Marshall

The result obtained through this cannot be unique, as infinity necessarily comprises everything.

The one thing I would add is to address the underlying source of the misunderstanding.

If infinity did comprise everything by definition, then your reasoning would be correct - but something can be infinite, and not contain everything.

For example, there are infinite counting numbers (1, 2, 3, 4, ...)
But the counting numbers will never contain 0.5, no matter how high you count. Nor will they ever contain 1.5, or 2.5, or 3.5. In fact despite being there being infinite counting numbers, there are still infinite numbers that are aren't counting numbers.

The important takeaway stems from a pretty common misunderstanding of what infinity is. Infinity doesn't mean it contains everything, it just means that you can continually list things forever.

===

On a sort of a tangent, what you're describing as infinity (a set that contains everything) can actually be shown to be impossible due to Russell's paradox that they talk about in the Veritasium video you linked.