So, supposedly a line is actually a set of an infinite amount

>>8895458
>wouldn't this imply that you could start with a dimensionless point with length 0, and keep adding more of these until you get some non-zero length

Nope

A dimensionless point is a point, any more than that and you have a line

>>8895476
so a line is a set of points, but a set of points won't ever make a line? How does that make sense?

Because a line isn't a set of points. A line contains an infinite amount of points, but that's solely because a line has length, while a point is dimensionless

>>8895458
>If this is the case, wouldn't this imply that you could start with a dimensionless point with length 0, and keep adding more of these until you get some non-zero length?
No because you'd have to add infinitely many you silly goose.

>>8896027
So 0 times infinity = a finite length? As in, an infinite number of points with length 0 can make up a line of definite length?

>>8896063
It seems so

>>8896011
because a line is not a set of points

>>8895458
>If this is the case, wouldn't this imply that you could start with a dimensionless point with length 0, and keep adding more of these until you get some non-zero length?
Yes. But in R^n you have to add unaccountably many points to get a non zero length.

>How the fuck does that work?
I think it is pretty intuitive.

File: scan0011.jpg (39KB, 400x322px) Image search: [Google]

39KB, 400x322px

>>8896063
0 is technically a finite length and we have [math]0 \times \infty = 0[/math] measure-theoretically, but that's probably not what you meant.

The technically correct answer is that it depends on how you define "infinite" as well as the ambient set theory you're working in, because the standard axiom of infinity doesn't actually give you any way to control the size of the set of natural numbers N, and hence no way to control what it means for your indexing set to be countable.
In order to do this you have to distinguish between the syntax of your set theory (usually ZFC) and the semantics, this is a bit like how you want to distinguish the (countably infinitely many) bitstreams 0.xxxx... with the (uncountably many) reals in [0,1]. Then you can define methods to translate between different models of ZFC while keeping them separate enough to avoid things like Skolem's paradox. One of the most common methods is called "forcing" and this was most famously used to prove the independence of the continuum hypothesis, which (intuitively) means that there can be no consistent, absolute answer to your question, so long as ZFC remains the foundational set theory of mathematics.

>>8896299
*Correction: ignore the part about bitstreams, I don't know what I was thinking when I wrote that.

>>8896282
>I think it is pretty intuitive.

Not really. If you have a line of length 5, and you assert that it is entirely comprised of an infinite amount of points of length 0, then you are saying that 0*infinity=5. That's completely nonsensical.

>>8896480
Do you know what an "Integral" is?

There you basically add up infinitely many 0s and get something which is not 0.

I also think that you are confusing infinities here. If there are countably many of these points, then you are absolutely right. It would be impossible to construct a line from that, but if you take unaccountably many you already have the same "amount" of points as there are real numbers.

Then it should become obvious that a line made up out of uncountably many points can have a non-zero length.

You can look into measure theory (more specific the Lebesgue (or Borell) measure) where this Idea is explored further.

>>8896511
thanks, that was really informative. I hadn't thought about how closely related this idea is to an integral.

>>8895458
https://youtu.be/3bacYDSy19Q?t=324
>a real number is uncomputable with probability 1 and a real number is computable with probability 0
>in other words if you pick a real number between 0 and 1 at random it is possible that you will get a computable real but it is infinitely unlikely.
>this unit interval, the commutable reals are an infinitesimally small part of this.
>this shows you how insignificant the computable reals are. the only problem is that every real number that we've ever seen or used is a computable real. the uncomputable reals are in a way a mathematical fantasy.
>and to make it worse, because most real numbers we cannot really refer to so they exists only a sense of a fantasy.

https://youtu.be/xaAhPo5KKUI?t=303
>how many possible names or ways are there are to refer to real numbers? its only a countable infinity of possibilities so not just the computable reals have probability 0 but all the nameable real numbers have probability 0 most real numbers will never be nameable in any way at all so they are completely inaccessible.
>Chaitin is trying to argue that real numbers don't exist.