Comment on the article “Dispute over Infinity Divides Mathematicians” by Natalie Wolchover in Scientific American / December 3, 2013.
Kantinomus Verlag, Tübingen, 2023. ISBN: 978-3-911041-04-1. 5 pages. 2,99 Euro.
”Let’s discuss the problem in (slightly more intuitive) geometric terms.
Let’s say infinity is the vanishing point at the edge (horizon) of the geometric plane. The question of whether infinity is something or nothing will actually be: Where is the vanishing point, inside or outside the geometric plane (on the earth’s surface or in the sky)?
In Euclidean geometry, the vanishing point lies outside the terrestrial plane, it lies in the sky and can never be reached (from this terrestrial point of view infinity is nothing) and therefore the postulate of parallelism remains intact. Euclidean geometry is a ZFC; consistent but incomplete, Gödel would say; an orthodox system, one might say.
In non-Euclidean geometries, the vanishing point is something on the ground (infinity lies inside the geometric plane) and therefore Euclid’s fifth postulate is violated. Consequently, in non-Euclidean geometries, all points in the plane are, in fact, cardinal points (poles of geometric space) towards which an infinity of parallel lines (such as meridians) converge.
Non-Euclidean geometries are ZFC+ systems (with infinity within them), complete but not inconsistent (as they appear from the point of view of Euclidean geometry) − they are paraconsistent (in da Costa’s terms); supercompact (in Koellner’s terms); or simply paradoxical.
Logically speaking, we are dealing with one and the same problem when we ask whether or not the cardinal of a set is an element of the set.”