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Numbers and numerosities

Vieri Benci

TL;DR

The paper develops a comprehensive framework for numerosity theory that extends classical counting to infinite sets while preserving Euclid's principle. It systematically builds and analyzes counting systems, introduces label-trees and the Λ-limit to unify discrete counting with continuum-scale magnitudes, and connects numerosities to ordinal, cardinal, hyperreal, and surreal numbers. A central achievement is constructing a numerosity counting system that yields a real-closed Euclidean line $ extbf{E}$, embedding numerosities into hyperreal and surreal settings and enabling a rich algebraic and measure-theoretic interplay. The work provides both a formal model and explicit computations for numerosities of key sets (e.g., $ extbf{N}$, $ extbf{Q}$, $ extbf{R}$), clarifying how different exponentiations and continuum-sized objects behave under the numerosity perspective, with potential implications for nonstandard analysis and the foundations of counting infinite structures.

Abstract

We develop new aspects of the the of numerosity theory; more exactly, we emphasize its relation with the ordinal numbers, cardinal numbers, hyperreal numbers and surreal numbers. In particular, we combine the notion of numerosity with the idea of continuum and we get a definition of Euclidean line which includes all the sets of infinite numbers mentioned above.

Numbers and numerosities

TL;DR

The paper develops a comprehensive framework for numerosity theory that extends classical counting to infinite sets while preserving Euclid's principle. It systematically builds and analyzes counting systems, introduces label-trees and the Λ-limit to unify discrete counting with continuum-scale magnitudes, and connects numerosities to ordinal, cardinal, hyperreal, and surreal numbers. A central achievement is constructing a numerosity counting system that yields a real-closed Euclidean line , embedding numerosities into hyperreal and surreal settings and enabling a rich algebraic and measure-theoretic interplay. The work provides both a formal model and explicit computations for numerosities of key sets (e.g., , , ), clarifying how different exponentiations and continuum-sized objects behave under the numerosity perspective, with potential implications for nonstandard analysis and the foundations of counting infinite structures.

Abstract

We develop new aspects of the the of numerosity theory; more exactly, we emphasize its relation with the ordinal numbers, cardinal numbers, hyperreal numbers and surreal numbers. In particular, we combine the notion of numerosity with the idea of continuum and we get a definition of Euclidean line which includes all the sets of infinite numbers mentioned above.

Paper Structure

This paper contains 38 sections, 45 theorems, 264 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries ad notation
  3. Counting systems
  4. The numbers
  5. Operations with numbers
  6. The main counting systems
  7. The finite counting systems
  8. The Euclid's Principle and the Natural numbers
  9. The Euler's infinite
  10. The Hume's Principle and the cardinal numbers
  11. The ordinal numbers
  12. The numerosities
  13. Numerosities and transfinite numbers
  14. Numerosities and ordinal numbers
  15. Operations with ordinal numerosities
  16. ...and 23 more sections

Key Result

Proposition 1

The numbers is a linearly ordered set with respect to the following order relation: given ${ \if@compatibility \mathchar"010B {} \mathchar"010B } =\mathfrak{n}(A)$ and ${ \if@compatibility \mathchar"010C {} \mathchar"010C } =\mathfrak{n}(B)$

Figures (1)

  • Figure 1: Schematic representation of a pivotal tree. Horizontal arrows connect points such that $a\equiv b$.

Theorems & Definitions (71)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Proposition 1
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Proposition 2
  • ...and 61 more