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Counting Sets with Surnatural Numbers

Peter Lynch, Michael Mackey

TL;DR

This paper develops the magnum function $m(A)$ that assigns a size to every countable set $A$ by mapping into the class of surnatural numbers $\boldsymbol{\mathsf{Nn}}$, thereby refining the usual cardinality to reflect intuitive size differences. It gives two complementary definitions: a genetic construction paralleling surreal numbers and a counting-function extension using the Axiom of Extension, with $m(A)=\widehat{\kappa_A}(\omega)$. The authors establish key principles such as the Euclidean property $B\subset A \Rightarrow m(B)<m(A)$, isobaric equivalence $m(A)=m(B)$ for partitions with equal window counts, and a General Isobary Theorem for fenestrations, plus extensions to relative magnums and Cartesian products. They compute magnums for fundamental sets ($\mathbb{N}$, $2\mathbb{N}$, $\mathbb{Z}$, $\mathbb{N}^2$, $\mathbb{Q}$) and analyze the primes and rationals under different orderings, revealing how magnums depend on reference sets and partitions. The framework provides a quantitative, intuition-aligned calculus of sizes for countable sets and opens questions about axioms, symmetries, and potential links to nonstandard approaches.

Abstract

How many odd numbers are there? How many even numbers? From Galileo to Cantor, the suggestion was that there are the same number of odd, even and natural numbers, because all three sets can be mapped in one-one fashion to each other. This jars with our intuition: cardinality fails to discriminate between sets that are intuitively of different sizes. The class of surreal numbers $\boldsymbol{\mathsf{No}}$ is the largest possible ordered field. In this work we define a function, the magnum, mapping a selection of countable sets to a subclass of the surreals, the surnatural numbers $\boldsymbol{\mathsf{Nn}}$. Set magnums are found to be consistent with our intuition about relative set sizes. The magnum of a proper subset of a set is strictly less than the magnum of the set itself, in harmony with Euclid's axiom, ``the whole is greater than the part''. Two approaches are taken to specify magnums. First, they are determined by following the genetic assignment of magnums in the way the surreal numbers themselves are defined. Second, the domain of the counting sequence, which is defined for every countable set, is extended, to evaluate it on the surnatural numbers. The two methods are shown to be consistent. For a subset $A$ of $\mathbb{N}$, the magnum is defined as the value at $ω$ of the extended counting function of $A$. Larger sets are partitioned into finite components and a more general definition of magnums is presented. Several theorems concerning the properties of magnums are proved, and are employed to evaluate the magnums of a range of interesting countable sets. The relativity of the magnum function is discussed and a number of examples illustrate how its value depends on the choice and ordering of the reference set. In particular, we show how the rational numbers may be ordered in such a way that all unit rational intervals have equal magnums.

Counting Sets with Surnatural Numbers

TL;DR

This paper develops the magnum function that assigns a size to every countable set by mapping into the class of surnatural numbers , thereby refining the usual cardinality to reflect intuitive size differences. It gives two complementary definitions: a genetic construction paralleling surreal numbers and a counting-function extension using the Axiom of Extension, with . The authors establish key principles such as the Euclidean property , isobaric equivalence for partitions with equal window counts, and a General Isobary Theorem for fenestrations, plus extensions to relative magnums and Cartesian products. They compute magnums for fundamental sets (, , , , ) and analyze the primes and rationals under different orderings, revealing how magnums depend on reference sets and partitions. The framework provides a quantitative, intuition-aligned calculus of sizes for countable sets and opens questions about axioms, symmetries, and potential links to nonstandard approaches.

Abstract

How many odd numbers are there? How many even numbers? From Galileo to Cantor, the suggestion was that there are the same number of odd, even and natural numbers, because all three sets can be mapped in one-one fashion to each other. This jars with our intuition: cardinality fails to discriminate between sets that are intuitively of different sizes. The class of surreal numbers is the largest possible ordered field. In this work we define a function, the magnum, mapping a selection of countable sets to a subclass of the surreals, the surnatural numbers . Set magnums are found to be consistent with our intuition about relative set sizes. The magnum of a proper subset of a set is strictly less than the magnum of the set itself, in harmony with Euclid's axiom, ``the whole is greater than the part''. Two approaches are taken to specify magnums. First, they are determined by following the genetic assignment of magnums in the way the surreal numbers themselves are defined. Second, the domain of the counting sequence, which is defined for every countable set, is extended, to evaluate it on the surnatural numbers. The two methods are shown to be consistent. For a subset of , the magnum is defined as the value at of the extended counting function of . Larger sets are partitioned into finite components and a more general definition of magnums is presented. Several theorems concerning the properties of magnums are proved, and are employed to evaluate the magnums of a range of interesting countable sets. The relativity of the magnum function is discussed and a number of examples illustrate how its value depends on the choice and ordering of the reference set. In particular, we show how the rational numbers may be ordered in such a way that all unit rational intervals have equal magnums.
Paper Structure (26 sections, 30 theorems, 124 equations, 5 figures, 2 tables)

This paper contains 26 sections, 30 theorems, 124 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Genetic Development of Magnums
  3. Pre-magnums and the Magnum Form
  4. Day-by-day Construction of Magnums
  5. Natural Density and Set Size
  6. Some Theorems Concerning Counting Sequences
  7. Some Theorems
  8. Isobaric Equivalence
  9. Extension of the Counting Function
  10. The Extension of the Domain of Definition of Functions
  11. The Extension of Subsets of $\mathbb{N}$ to Subsets of $\boldsymbol{\mathsf{Nn}}$
  12. Definition of the Magnum
  13. Some Theorems for Magnums of Sets in $\mathscr{P}(\mathbb{N})$
  14. Consistency of the Counting Function and Magnum Form
  15. The General Isobary Theorem
  16. ...and 11 more sections

Key Result

Theorem 1

Let $A = \{a_n:n\in\mathbb{N} \}$ be a subset of $\mathbb{N}$ with a monotone increasing defining sequence $a_A : k \mapsto a_k$. We interpolate $a_A:\mathbb{N}\to\mathbb{N}$ to a strictly increasing continuous function $\alpha:\mathbb{R}^{+}\to\mathbb{R}^{+}$ on the positive real numbers. Then, the

Figures (5)

  • Figure 1: The initial values of the defining sequence $a_A(n)$ (red circles), its inverse $a_A^{-1}(n)$ (blue triangles), and the counting sequence $\kappa_A(n)$ (black diamonds) for the set $A = 3\mathbb{N}\cup 4\mathbb{N}$.
  • Figure 2: The positive rational numbers (small black dots), rational numbers in $Q_1 = (0,1]_\mathbb{Q}$ with duplications (open circles, red online) and rational numbers in $Q_1$ in reduced form (large dots, blue online).
  • Figure 3: Ordering the set $\mathbb{N}^2 = \mathbb{N}\times\mathbb{N}$ with the usual square arrangement. Points represent ordered pairs $(m,n)$. Windows (shaded) are separated by dashed lines (red online).
  • Figure 4: Arranging $\mathbb{Q}$ in vertical bands of unit width. Large points (blue online) represent rationals (with $\text{gcd}(m,n)=1$), small points (black) are ordered pairs with $\text{gcd}(m,n)>1$, and $k$ is the integer part of $m/n$. Windows are separated by dashed (red) lines. The numbers in $Q_k$ are in the $k$-th vertical band.
  • Figure 5: Proportion of numbers less than $n$ that have an odd count of binary digits (open circles) and an even digit count (triangles, red online). Note that the $x$-axis is logarithmic.

Theorems & Definitions (67)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • proof
  • ...and 57 more