Counting systems and the First Hilbert problem

TL;DR

This paper explores the First Hilbert problem using a novel methodology that differentiates between mathematical objects and languages, and introduces a new numeral system to measure infinite sets with varying accuracies.

Contribution

It presents a new approach combining a methodology and a numeral system to analyze infinite sets and their cardinalities, offering a fresh perspective on the problem.

Findings

Different mathematical languages describe objects with varying accuracies.

The new numeral system enables expressing and measuring different infinite numbers.

Comparison of traditional and new approaches highlights their differences.

Abstract

The First Hilbert problem is studied in this paper by applying two instruments: a new methodology distinguishing between mathematical objects and mathematical languages used to describe these objects; and a new numeral system allowing one to express different infinite numbers and to use these numbers for measuring infinite sets. Several counting systems are taken into consideration. It is emphasized in the paper that different mathematical languages can describe mathematical objects (in particular, sets and the number of their elements) with different accuracies. The traditional and the new approaches are compared and discussed.