Atla wrote: ↑Thu Oct 17, 2019 7:52 pm
I didn't object to the union operation. To be honest I don't even see what sets have to do with the question of 0d points vs 1d lines.

Oh ok. Good point. Perhaps I didn't make the connection clear.

In modern math -- again, this is a historically contingent theory made up by humans, and has arguably nothing (at least for sake of this conversation) to do with the actual universe -- but in modern math, the real numbers form a

*set*. A set is informally thought of as a collection of objects. So for example the set of natural numbers {0, 1, 2, 3, ...} is a collection containing each of the numbers 0, 1, 2, 3, ...

Because of the way the union operation is defined we can write

{0, 1, 2, ...} = {0} ∪ {1} ∪ {2} ∪ ...

where the little '∪' is the union operation.

In face any set whatsoever is equal to the union of its singleton subsets. So the unit interval, the set of real numbers between 0 and 1, is the union of the singletons of its elements. This is incontrovertible unless you are prepared to throw out the whole of set theory on which modern math is based. [Again you are free to do so, but you must then rebuild math and physics etc. according to your new theory. The burden is on you].

So the unit interval, which has length 1, is equal to the union of infinitely many singleton sets {x} where x is some real number.

@surreptitious57 claimed that therefore the unit interval must have length 0. This of course contradicts high school math. One can hardly build science on a foundation that claims that the distance from point 0 to the point 1 is zero.

What exactly is your view on how to handle the fact that the unit interval has length 1, and it's the union of all these singleton sets each containing a point of length 0?

There is by the way a mathematical way of handling this. We don't require lengths to add up for arbitrary infinite sets; only for

*countably infinite sets*. In probability and measure theory this is known as countable additivity.

https://en.wikipedia.org/wiki/Sigma_additivity
The reason we reject arbitrary additivity is exactly because of this paradox: that the unit interval has length 1, and each individual point has length 0.

I do in fact regard this as a mystery. Math doesn't explain it; math only finesses the issue by only requiring countable additivity for lengths and probabilities.

We do have a formalism in calculus that the integral from 0 to 1 of dx is 1. That expresses the fact that informally "adding up all the 0-length points gives you 1." We have a formalism but not truly a philosophical explanation, in my opinion.

What's your take?