Resolving the trans-Planckian problem along the lines of a finite geometry

In black hole physics, inflationary cosmology, and quantum field theories, it is conjectured that the physical laws are subject to radical changes below the Planck length. Such changes are due to effects of quantum gravity believed to become significant at the Planck length. However, a complete and consistent quantum theory of gravity is still missing, and candidate models of quantum gravity have not yet overcome major formal and conceptual difficulties. Another problem is how to determine a geometry of physical space that features a minimal length scale such as the Planck length. In the present paper it is demonstrated that the said geometry can be any geometric system omitting continuity, i.e., a geometry that possesses only a finite number of points.