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Numerical Sets in Finitist Mathematics

Author:

Leon, Antonio

Category:

Research Papers

Sub-Category:

Mathematics and Applied Mathematics

Language:

English

Date Published:

November 19, 2025

File:

/Research Papers-Mathematics and Applied Mathematics/Download/10372

Downloads:

Keywords:

Actual infinity, potential infinity, numerical sets, densely ordered sets, natural numbers, integer numbers, rational numbers, irrational numbers, real numbers.

Abstract:

This article discusses the mathematical reality of numerical sets once the inconsistency of the actual infinity has been demonstrated (see below). Under these finitist conditions, only three numerical sets can consistently exist: the set of natural numbers, the set of integer numbers, and the set of exact rational numbers; the three of them potentially infinite. Since all irrational numbers have an actual infinite (and then inconsistent) number of decimal places, they are all, as one the meaning of its name indicates, irrational in the sense of inconsistent numbers. This formal "irrationality" of irrational numbers implies that the set of real numbers that contains them all cannot be a consistent set.