This is an interesting remark.bahman wrote: ↑Fri Mar 13, 2020 10:45 pm

This is my understanding of the subject matter: Any mathematical category, like real number, is internally consistent if you can reach from one entity to another one. There are two geometrical entities in real number, point and segment. But you cannot reach a point to a segment, by this I mean that you cannot construct a segment from an assembly of points since point size is zero.

First, in modern set theory, the real numbers are a "set" of points. That means you can express the entire set of real numbers as a set of individual points.

This echoes Euclid's idea that a line is made up of points. Although when I looked for a reference, I could not find anyone claiming that Euclid actually said that. But it's commonly understood in math that a line is made of points, and that in modern math, the set of real numbers consists of all the individual real numbers; and we may, by a leap of imagination, take the set if real numbers as modeled by a straight line; and the individual real numbers represent the addresses, if you will, of locations on the line.

Now it turns out that there's a philosopher Charles Sanders Peirce, with exactly that spelling, who has pointed out that a continuum must have the property that its parts are the same as the whole; and that the set-theoretic model most definitely does NOT satisfy that property, for exactly the reason you mentioned. Points aren't anything like a continuum! So to Peirce, and to you, and also to some or many others these days, the standard set of real numbers is not entirely satisfactory as the conceptual model of a continuum.

If that's what you mean, I understand your point. As a "math guy" I am aware of these philosophical issues but nevertheless I will persist in calling the real line "the continuum" unless pressed by a Peircean.

I see what you're getting at. You want to construct space out of smaller parts, and points simply won't do; because they have zero size. Yes I agree that's a philosophical mystery. I'm not convinced fractals are the answer but I do see what you're getting at.

Yes good. I was about to say that so I'm glad we agree.

Are you looking for something like the space-filling curve?

Measurability is a subtle problem in math. There's a subject called measure theory in which we try to assign a number, or measure, to various sets of points in a way that generalizes length, area, volume, and so forth.

It can be proved that there are non-measurable sets; that is, sets of real numbers that can not possibly be assigned a sensible measure that's consistent with how we think of length/area/volume.

There is also a famous fractal called the Cantor set, which serves as an example of an uncountable set of measure zero.

Is any of this on point to what you're trying to say?

ps -- Also the infinitesimals of nonstandard analysis will not help you. They're non-Archimedean. That means that no matter how many infinitesimals you put next to each other you'll never get enough length to cover the real line. They don't give you the fractal properties you need.