wtf wrote: ↑Thu Oct 17, 2019 3:58 am
surreptitious57 wrote: ↑Thu Oct 17, 2019 12:24 am
wtf wrote:

How do you account for the length of the unit interval then

The unit interval is composed of zero length points and so it cannot ever be longer than zero

That does not magically change just because there happens to be an infinite number of them

The unit interval is the interval on the real line between 0 (inclusive or not doesn't matter for us) and 1 (likewise). If you took analytic geometry in high school you know that its length is 1. If not, you can take my word for it. For example the Euclidean distance between the point 0 and the point 1 in the real numbers is |1 - 0| = 1.

But then again don't they teach the real line in first year high school algebra (Algebra I in the US I think) to ninth graders, 14 year olds? Did you miss all this? Genuinely curious as to how someone could deny that the length of the unit interval is 1. Did you just miss high school algebra? No shame in it I suppose but that's the only explanation for your erroneous belief. I know this forum's a bit of an alternate math universe, but still.

It's a bit of a mystery out a one-dimensional line segment of length 1 can be made out of zero-dimensional points each of length zero. But there it is.

That is your conclusion.."it is a bit of a mystery"? What are you a f"""ing mystic now? That is just saying "our measurements our made up and we only believe them because we built so much on these assumptions". It is a gravity fallacy....mass of information does not necessitate the information is correct.

There is no point 1 on the unit interval as you begin quantifying 0's and equating them to 1. It is a contradiction unless 1=0, which it does, but standard math does not accept it. So according to standard math the 1 is the line itself.

The distance between 0's necessitates any two points as having a length, thus infinite points is to say infinite 1d lines.

The unit interval is thus composed of further number lines as lines in themselves. It is composed of fractals that are lines in themselves and fractions that when viewed individually are whole numbers.

Thus each unit interval shows a progression of positive numbers approaching zero, where each unit interval can be observed as a number line in itself...thus each unit interval is a proof all number lines are approaching zero.