Georg Cantor at the Dawn of Point-Set Topology - Number Line vs. Numberness

Cantor creates a bijection between the number line and number values. He calls an element on the number line a “point” and an element of \(B\) a “number value” or “value,” and seeks a bijection between values and points. Again, the idea here is that after we have established the bijection, we will then have a rigorous foundation to justify arithmetic operations on the real numbers.

Fix a point \(o\) on the number line (think of this as the origin). To show that the number line injects into the collection of number values, Cantor first considers the case when the distance from the given point to \(o\) has a rational relationship. If point \(a\) on the number line is rational, then we may associate the constant sequence \(a,a,\dots,a\dots,\) which possesses numberness, to the rational number \(a\) on the number line. This seemingly obvious fact is not without its criticisms. As Russell [17, p. 285] points out

There is absolutely nothing in the above definition of the real numbers [number values] to show that \(a\) is the real number defined by a fundamental series [sequence] whose terms are all equal to \(a.\) The only reason why this seems self-evident is, that the definition by limits is unconsciously present, making us think that, since \(a\) is plainly the limit of a series whose terms are all equal to \(a,\) therefore \(a\) must be the real number defined by such a series.

However, this may be problematic only when dealing with the arithmetic of number values, not when defining a bijection with the reals. So if point \(a\) is rational, we associate \(a\) with the sequence \(a,a,\dots,a,\dots\); that is, with whatever symbol the limit symbol of the sequence may be. Now suppose that a point \(b\) on the number line is irrational. Cantor asserts that

It is always possible to give a sequence \[a_1,a_2,\dots,a_n,\dots\quad\quad\quad\quad(1)\] [so that the] distance ... from the point \(o\) is equal to \(b,\) where \(b\) is the corresponding numerical quantity of sequence \((1)\).

In other words, Cantor associates the equivalence class [\(a_n\)] \(\in B\) to \(b.\)

Cantor took the converse, that "the geometry of the straight line is complete," as an axiom. He writes,

[T]o make the geometry of the straight line complete is only to add an axiom, which simply consists in [declaring that] any numerical quantity belongs to a certain point of the straight line ... I call this theorem an axiom because it is in its nature to not generally be provable. (emphasis original).

Before diving into what we now call topology, Cantor points out that this bijection is helpful only for conceptual purposes.

Given the previous, we now assign number values to the points on the line. For clarity (not that it is essential), we use this notion in the following and have, when we speak of points, values in mind by which they are given.

Notice that once again, Cantor stresses that thinking of points on the real line as number values is "for clarity" and not essential to what he is doing.

Though it may seem like the work we have done in Sections 4 and 5 is not at all related to topology, the building of such a rigorous foundation is necessary to make the definitions that appear in Section 6. We will see that these definitions abstract away the distance between real numbers, but keep the nearness between them, precisely by making the distinction between "points" and "values."