Is the one-dimensional point always the same?

But what exactly is the number of points? At first glance, this question can only answer "infinity".

However, Dedekin's friend, the Russian-born mathematician Georg. Cantor, mathematically, examines the number of points that make up a one-dimensional space. The result is that this number is much larger than the number of all integers, that is, infinity, and indicates that the number of points that make up any one-dimensional space is the same. This shocked his other mathematicians.

From a common sense point of view—perhaps for Euclidean, who thinks that the points on the line are evenly arranged—because the long line also includes the long line formed by the combination of short lines, the number of points must be long and straight. But Cantor said that whether it is a straight line that is 1 millimeter long, a curve of ten thousand kilometers in length in the Pacific submarine cable, or a straight line that leads to the end of the universe that is longer than human imagination, or an infinite length of line that we cannot even imagine, the number of points that make up these lines is the same.

Georg Cantor

Euclid, in his previous writings, proposed the axiom that the whole is greater than the part, which is a matter of course that anyone can understand. Cantor's answer, however, was different. He proposed that the number of points constituting any one-dimensional space is equal, further to say that there is a situation in space where "parts and wholes are equal".

The key difference between zero and one dimensions is that one dimensions have size (length). Because zero-dimensional space is a point, the number of points is only one, but it is a point without a quantity (physical quantity). In contrast, once there are physical quantities like one-dimensional space, the number of points that make up zero-dimensional space is infinite.

According to Cantor, if the number of integers mentioned earlier is set to א₀ (Aleph0) - א (Aleph) is the first letter of Hebrew, the number of points in one-dimensional space is "2 to the א₀ power". I didn't expect that infinity actually had a size difference. Thus, the number of infinite points in space is represented as the "infinite power of 2", which cannot be calculated by humans and can only be represented by mathematical symbols.

Other questions arise here. In mathematics , a one-dimensional line or curve (two-dimensional space or three-dimensional space as well) is a collection of an infinite number of points. The size of each point is 0, but when they are grouped together, why produce quantities of one dimension or higher? Aren't several zeros together also 0?

It seems that no mathematician in the world has found a convincing answer to this question. In addition, a larger question arises as to whether mathematical space can be applied to actual physical space.