This work was published in the international journal of mathematics:
and it is accessible on: GJARCMG
- In Euclidean geometry the primitive terms used for parallelism are: "straight line" and "point", which are introduced by the two following axioms:
given two points there is one and only one straight line (joining
them),
given a straight line and a point not on that line, there is one
and only one straight line passing through that point, and having
no point in common with it.
- The problem is that the second axiom is not self-evident because it requires to evaluate the possible presence of common points along an infinite space. So we cannot exclude that our space is described by the elliptical geometry, namely that it is true the following axiom:
given a straight line and a point not on that line, there is no straight line passing through that point, and having no point in common with it,
or that one concerning the hyperbolic geometry:
given a straight line and a point not on that line, there are more straight lines passing through that point, and having no point in common with it.
- My work seems to give a definitive answer to this question, allowing us to say that Euclidean geometry is the correct description of our space. In fact the parallelism developed by my work through the primitive terms: "straight line", "point" and "direction" is based on the following two axioms:
given two points there is one and only one direction (capable
to spatially connect them),
given a point and a direction there is one and only one straight
line (passing through that point and having that direction),
which, although are self-evident as the first axiom of Euclidean geometry, allow us to prove the second one.
