Why real numbers and points can't model continuum

A loss-less cut is defined to be a split in a continuum which ensures
that if the two parts are joined back we get the whole again without
any loss of continuity. Let P be a point at which a loss-less cut is
made in a continuum. Three possibilities arise after the cut:

P belongs to only one of the segements.

If this is true, then the cut was made between the points but not at
the point. But this is not possible because there are no adjacent
points. So this possibility does not arise.

P is lost in the cut and doesn't belong to any of the new segments.

This means if we join the parts back, we lose a point, and the
continuity is lost implying the cut was not loss-less. This
possibility is ruled out by the requirements of the cut.

P is split into two points of same location.

This means we have produced a duplicate point and a duplicate real
number with same value as some other point or number. This is ruled
out trivially by the properties of real numbers and as well as
positional uniqueness of points.

The above implies that a cut can not be made at point on a line
segment, which is a contradiction. This contradiction arises due to
modeling the continuum with points as well as with real numbers. Both
the concepts are incorrect.

- venkat
.

Relevant Pages

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