Inconsistency of Infinity in a geometric context

Inconsistency of Infinity in a geometric context Enrico P. G. Cadeddu * 28 September 2025 Abstract Representation of Nand then its finite sub-chains along a line-segment (or a line) leads to a contradiction concerning actual infinity; the longest linesegment, corresponding to N, contains some natural numbers not contained in any shorter line-segments corresponding to all sub-chains. Inconsistency proof This proof of inconsistency, regarding the actual infinity then the set of all natural numbers, has already been treated [1], even if in a less detailed way on the geometric aspect. I1⊂I2⊂I3⊂I4⊂I5⊂.... ⊂In⊂.... ⊂N(1) Each finite set (or chain) Ii, with the form {0,1,2,3,4, ....i −1}, has one more number than the previous one. It is a proper subset of other following sets and obviously of Nwhich is infinite. It must be: Si∈NIi=N. We take a finite line-segment in which the chain N≡ {0,1,2,3,4, ....}of all natural numbers is represented; a number is identified with a point and the distance between two points is dx, an infinitesimal distance (Figure 1). Figure 1 In this line-segment we have the infinite chain of natural numbers and then all its finite proper sub-chains Ii≡ {0,1,2,3,4, ...i −1}.ω /∈Nand delimits the infinite * Email Address [email protected] 1 chain N. Every finite Iicontains less numbers than N, which contains all natural numbers, see relation (1). Then infinite chain N, corresponding to the entire line-segment, corresponds to the longest segment and finite sub-chains Iicorrespond to shorter segments. Clearly Ncannot coincide with any proper sub-chain Ii. So this entire segment, corresponding to N, contains some points, then some natural numbers, not contained in any other shorter segment corresponding to a finite sub-chain Ii. But this isn’t possible by definition, because any natural number (of N) has to belong to a finite sub-chain Iiand then contained in a corresponding segment shorter than the entire segment. There is a contradiction. Taking a finite line-segment isn’t a necessary condition for the proof. We can take a line (an infinite segment) with a finite distance ∆xbetween two points corresponding to two natural numbers. The result is the same: the line contains some points, then some natural numbers, not contained in any other segment corresponding to a finite sub-chain, which gives a contradiction. But in this case we also see that the line, corresponding to the infinite chain N, has some points, then some natural numbers, at infinity. This implies a natural number (which is finite in any case) defining an infinite distance, an impossible statement, n·∆xbeing a finite line-segment ∀n, then a contradiction. This proof was achieved because all segments lie along the same segment (or line) and a comparison can be made, in particular we can see that the longest segment has to contain some points, then some numbers, not contained in any other segment. From this we see the importance of a numerical-geometric approach. A purely numerical approach or a purely geometric one appears not sufficient to obtain an inconsistency proof. It should also be emphasized that inconsistency of actual infinity implies inconsistency of infinitesimals (a line-segment would contain an infinite amount of infinitesimals, then a contradiction). Geometry is directly connected to space, which in turn is connected to physical reality, then this proof seems to affirm the finiteness of physical reality. References [1] Enrico P G Cadeddu. Inconsistency of N with the set union operation. Zenodo https://doi.org/10.5281/zenodo.10530599 2024-2025. 2