The Average Distance
Between Points

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o embody an infinite number of points in consecutive order requires, minimally, a ray.

But the distance between two successive countable points is unspecified.

It must be Unit Distance by definition. That is, the distance between two points on the ray will be defined as one unit length.

Clearly, if we select any finite distance on the ray there must be some definite probability of finding a countable point in that space. That probability cannot be 0 or 1 for every finite length else no sensible ray of the type required can be constructed. (If the probability is '0' then there are no countable points. If '1', there are an infinite number of countable points in the limit as the selected distance approaches zero relative to the selected 'unit' distace.)

The ray may be constructed only in the following way.

Divide the ray into equal unit segments. Into each segment let there be or not be one and only one countable point (the simplest conceivable, coin-toss layout). Now If we redefine the unit length on this 'prepared' ray, it can be seen to be equivalent to any other conceivable systematic layout scheme. Every logical objection is covered, i.e. "Why not two points per unit?" "OK, just double the size of the prepared unit length on the same ray, while leaving the points at their assigned positions, etc.".
We're simply examining the problem from another perspective.

And from this perspective it appears all right to assign exactly one countable point to each unit length.
There is, however, a problem with symmetry that makes this deterministic solution unviable. (To be taken up in a later section.)




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