Counting Sets with Surreals.
Part I: Sets of Natural Numbers

Peter<EMAIL_ADDRESS>& Michael<EMAIL_ADDRESS>
School of Mathematics & Statistics, University College Dublin.

16 November, 2023

###### Contents

  1. 1 Introduction
  2. 2 Desiderata
  3. 3 Magnum Spaces
  4. 4 Genetic Definition of Magnum
  5. 5 Using Natural Density to Define Magnums
  6. 6 Extending the Domain: the Transfer Axiom
  7. 7 Theorems Following from the Counting Function
  8. 8 Application of Theorems to Sample Sets
  9. 9 Equivalence between Magnum Forms
  10. 10 Concluding Remarks
  11. References

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 different
in size. The class of _surreal numbers_ $\boldsymbol{\mathsf{No}}$ is the
largest possible ordered field. In this work we define a function, the
_magnum_ , mapping the power set of the natural numbers to a subclass of the
surreals, the surnatural numbers $\boldsymbol{\mathsf{Nn}}$ (the non-negative
omnific integers). 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 counting sequence, which is defined for
every set of natural numbers, is extended using a Transfer Axiom, to evaluate
it on the surnatural numbers. The two methods are shown to be consistent.
Several theorems are proved, establishing that the family of sets with magnums
form what we call a magnum space. Finally, these theorems are employed to
evaluate the magnums of a number of subsets of $\mathbb{N}$.

##  1 Introduction

Our objective is to define a number $m(A)$ for each subset $A$ of $\mathbb{N}$
that corresponds to our intuition about the size or magnitude of $A$. We will
call $m(A)$ the _magnum_ of $A$.333As ‘magnum’ is a contraction of ‘magnitude
number’, we write the plural as magnums, not magna. For this, we require
quantities that are beyond the real numbers and more refined than the ordinal
numbers. The surnatural numbers, a subclass of the class of surreal numbers,
are suitable for our purposes.

The surreal numbers, denoted by $\boldsymbol{\mathsf{No}}$, were discovered by
John H. Conway around 1970. They form the largest possible ordered field, with
all the basic arithmetical operations, and sensible arithmetic can be carried
out over $\boldsymbol{\mathsf{No}}$. Moreover, quantities like $\omega/2$ and
$\sqrt{\omega}$ are meaningful (where $\omega$, the smallest infinite ordinal,
is the “first-born” infinite surreal number).

A subclass of $\boldsymbol{\mathsf{No}}$ called the omnific integers
$\boldsymbol{\mathsf{Oz}}$, plays the role of integers in the surreal context.
We define the _surnatural numbers_ $\boldsymbol{\mathsf{Nn}}$ to be the class
of non-negative omnific integers. The _magnum_ for any subset $A$ of the
natural numbers $\mathbb{N}$ is defined to be a surnatural number. If $A$ is a
finite set, then $m(A)$ is just the cardinality of $A$. The magnum of
$\mathbb{N}$ is $\omega$; the sets of even and odd natural numbers both have
magnums $\omega/2$. We find the magnums of a wide range of other sets which
are consistent with our intuition about the relative sizes of these sets.

#### Background

Galileo observed that the natural numbers and their squares could be matched
one-to-one, with each $n$ paired with its square $n^{2}$ and each perfect
square with its root. This caused him to doubt the meaning of size for
infinite sets. Bolzano recognized the crucial property of an infinite set,
that it may be in one-to-one correspondence with a proper part of itself.
Dedekind later identified this as the defining characteristic of infinite
sets.

Cantor defined transfinite numbers and showed how every set has a cardinal
number. However, cardinality is a _blunt instrument_ : the natural numbers,
rationals and algebraic numbers all have the same cardinality, so $\aleph_{0}$
fails to discriminate between them. Moreover, the arithmetic of the cardinals
is awkward at best: for example, $\aleph_{0}+1=\aleph_{0}$ and
$\aleph_{0}^{2}=\aleph_{0}$.

Cantor also showed that every well-ordered set has an ordinal number. For
infinite sets, there are many possible orderings, so that a given set may have
more than one ordinal assigned to it. The ordinals have the unattractive
property of being non-commutative for both addition and multiplication. Thus,
neither cardinals nor ordinals provide a good basis for a calculus of
transfinite numbers.

All infinite subsets of the natural numbers $\mathbb{N}$ have the same
cardinality. However, intuition strongly indicates that a proper subset of a
set should be “smaller than” the set itself. This idea is generally called
Euclid’s Principle,444The Euclidean Principle is also known as Aristotle’s
Principle. as it accords with the fifth Common Notion as usually listed in
Euclid’s _Elements_ [He08, pg. 232] [Risi20].

In a series of publications starting with [BeNa03] and culminating with the
book [BeNa19], Vieri Benci, Mauro Di Nasso and others defined the concept of
_numerosity_ as a generalisation of finite cardinality that also applies to
infinite sets. Like magnums, numerosity respects the Euclidean Principle, that
“the whole is greater than the part”: the numerosity of a proper subset is
strictly smaller than that of the whole set. There are similarities, but also
striking differences between the numerosities defined by Benci et al. and
magnum spaces defined here. In particular, the codomains are different:
numerosities take values in the hyperreal numbers whereas magnums are
surnatural numbers. However, where comparisons are possible, the set sizes we
have found are consistent with results presented in [BeNa19].

#### Outline of Contents

A preliminary study of magnums, which appeared in [Lyn19], is developed and
greatly extended in this paper. In §2, we list the desired properties of the
magnum of a set. In §3, we introduce the concept of isobaric equivalence and
define the structure of a magnum space. The genetic development of set
magnums, following the evolution of surreals themselves, is presented in §4.
In §5, we explore the value of natural density in defining the sizes of sets
and discover its limitations. In §6, we introduce the Transfer Axiom, which
allows us to extend properties of sequences of natural numbers to the
surnatural domain. This provides us with a general definition of the magnum of
a set. Several theorems are stated and proved in §7 and, using them, we show
that our definition of set magnums results in a magnum space. The theorems are
applied in §8 to deduce the magnums of a selection of sample subsets of
$\mathbb{N}$. In §9 we show that the definition of magnums using the counting
sequence is consistent with the genetic approach. Concluding remarks follow in
§10.

##  2 Desiderata

We should like to assign a “size” to any subset $A$ of $\mathbb{N}$ that
accords with our intuition about its magnitude. We denote the size of a set
$A$ by $m(A)$, called the “magnitude number” or _magnum_ of $A$. We list below
some desirable properties of the magnum function $m(A)$.

  1. 1.

For a finite subset $A$, we require $m(A)=\\#A$, the number of elements in
$A$.

  2. 2.

The magnum of a subset of $A$ should not exceed that of $A$:

$A\subseteq B\implies m(A)\leq m(B)\,.$

  3. 3.

A _proper_ subset555Throught this paper we denote subsets by $\subseteq$ and
proper subsets by $\subset$. $A$ of set $B$ should have a _smaller_ magnum
than $B$:

$A\subset B\implies m(A)<m(B)\,.$

  4. 4.

Addition of a new element $b\notin A$ should increase the magnum by one:

$m(A\uplus\\{b\\})=m(A)+1$

  5. 5.

Removal of an element $a\in A$ from a (non-void) set should reduce the magnum
by one:

$m(A\setminus\\{a\\})=m(A)-1$

  6. 6.

The magnums of the union of two disjoint sets should add: if $A\cap
B=\varnothing$, then

$m(A\uplus B)=m(A)+m(B)$

  7. 7.

For the complete set of natural numbers, $m(\mathbb{N})=\omega$.

  8. 8.

The odd and even numbers should behave _sensibly_ :

$\displaystyle 2\mathbb{N}=\\{2,4,6,\dots\\}$ $\displaystyle\implies$
$\displaystyle m(2\mathbb{N})=\textstyle{\frac{1}{2}}\omega$ $\displaystyle
2\mathbb{N}-1=\\{1,3,5,\dots\\}$ $\displaystyle\implies$ $\displaystyle
m(2\mathbb{N}-1)=\textstyle{\frac{1}{2}}\omega\,,$

ensuring that
$m((2\mathbb{N})\uplus(2\mathbb{N}-1))=m(2\mathbb{N})+m(2\mathbb{N}-1)=\omega$.

  9. 9.

Similar requirements apply for arithmetic sequences. For example,

$k\mathbb{N}=\\{k,2k,3k,\dots\\}\implies
m(k\mathbb{N})=\textstyle{\frac{1}{k}}\omega$ (1)

  10. 10.

The size of the set of perfect squares should be ‘reasonable:’

$\mathbb{N}^{(2)}=\\{1,4,9,16,\dots\\}\implies
m(\mathbb{N}^{(2)})=\sqrt{\omega}$

The objective is to define the magnum in such a manner that the above
properties hold.

##  3 Magnum Spaces

###  3.1 Isobaric Equivalence

######  Definition 1.

A fenestration $\mathscr{W}=\\{W_{k}:k\in\mathbb{N}\\}$ is a collection of
disjoint finite sets or ‘windows’, all of equal length $L$, that partition
$\mathbb{N}$, such that

$\mathbb{N}=\biguplus_{k\in\mathbb{N}}W_{k}\qquad\mbox{where}\qquad
W_{k}=\\{(k-1)L+1,\dots,kL\\}\,.$ (2)

To indicate that order is important, we write each window as a sequence,

$\mathbf{W}_{k}=\bigl{\langle}(k-1)L+1,\dots,kL\bigr{\rangle}\,.$

The fenestration $\mathscr{W}$ induces a partition of any set
$A\in\mathscr{P}(\mathbb{N})$ into a countable union of disjoint, finite
subsets, or ‘blocks’, $A_{k}=A\cap W_{k}$, with $A=\uplus_{k}A_{k}$. We define
a sequence of weights

$\mathbf{w}_{A}=\bigl{\langle}\\#A_{1},\\#A_{2},\dots,\\#A_{k},\dots\bigr{\rangle}\,,$
(3)

whose $k$-th term give the number of elements of $A$ in window $W_{k}$.

By a ‘block permutation’ we mean a transformation
$\boldsymbol{\Pi}:\mathbb{N}\to\mathbb{N}$ in which the components of each
window $\mathbf{W}_{k}$ are permuted but where there are no exchanges between
windows:

$\boldsymbol{\Pi}\,\mathbb{N}=\biguplus_{k\in\mathbb{N}}\boldsymbol{\pi}_{k}\mathbf{W}_{k}\,,$

where $\boldsymbol{\pi}_{k}:\mathbf{W}_{k}\to\mathbf{W}_{k}$ is a permutation
on $\mathbf{W}_{k}$. It is clear that any set having weights $\mathbf{w}_{A}$
can be obtained from $A$ by a suitable block permutation.

We now introduce an equivalence relation — isobaric equivalence — between sets
having equal weight sequences.666The term _isobary_ is from $\iota\sigma o\
\beta\alpha\rho o\varsigma$, ‘equal weight’. We denote isobary by writing
$A_{1}\cong A_{2}$. We will show later that any two isobaric sets have equal
magnums.

######  Definition 2.

Two sets $A_{1}$, $A_{2}$ in $\mathscr{P}(\mathbb{N})$ are _isobaric_ if, for
some $L\in\mathbb{N}$, they have equal weight sequences, that is, if
$\mathbf{w}_{A_{1}}=\mathbf{w}_{A_{2}}$.

Isobary is obviously reflexive and symmetric: $A_{1}\cong A_{1}$ and
$A_{1}\cong A_{2}\iff A_{2}\cong A_{1}$. It is also transitive: If $A_{1}\cong
A_{2}$ with window length $L$ and $A_{2}\cong A_{3}$ with window length $M$,
then $A_{1}\cong A_{3}$ with window length $LM$. Thus, $\cong$ is an
equivalence relation.

###  3.2 Beta-algebras over $\mathbb{N}$

We define a beta-algebra over the natural numbers to be a family of subsets of
$\mathbb{N}$ that is closed under finite unions, complements and isobaric
equivalence.

######  Definition 3.

A family $\mathscr{B}$ of subsets of $\mathbb{N}$ is a $\beta$-algebra if

  1. 1.

The union of any pair of sets $A_{1}$ and $A_{2}$ in $\mathscr{B}$ is in
$\mathscr{B}$,

  2. 2.

The complement of any set $A$ in $\mathscr{B}$ is in $\mathscr{B}$,

  3. 3.

If $A_{1}\in\mathscr{B}$ and $A_{2}\cong A_{1}$, then $A_{2}\in\mathscr{B}$.

A $\beta$-algebra is a set algebra in the usual sense, but with the additional
property of containing all sets isobaric to each member. For any family
$\mathscr{F}\subset\mathscr{P}(\mathbb{N})$, we denote by $\beta(\mathscr{F})$
the smallest $\beta$-algebra containing $\mathscr{F}$. It is the intersection
of all the $\beta$-algebras over $\mathbb{N}$ that contain $\mathscr{F}$.

It is of interest to know if we can construct a chain of $\beta$-algebras
extending to the power set $\mathscr{P}(\mathbb{N})$. The simplest example of
a $\beta$-algebra is the ‘trivial’ $\beta$-algebra
$\mathscr{B}_{0}=\\{\varnothing,\mathbb{N}\\}$. Adjoining a singleton
$\\{n\\}$ to $\mathscr{B}_{0}$ we define

$\mathscr{B}_{1}:=\beta(\mathscr{B}_{0}\cup\\{\\{n\\}\\})\,.$

Clearly, all singletons are in $\mathscr{B}_{1}$, as are all doubletons and,
indeed, all finite sets. $\mathscr{B}_{1}$ also contains the complements of
these and so comprises all finite and all cofinite subsets of $\mathbb{N}$.

We now adjoin the set $2\mathbb{N}$ of even numbers, which is neither finite
nor cofinite:

$\mathscr{B}_{2}:=\beta(\mathscr{B}_{1}\cup\\{2\mathbb{N}\\})\,.$ (4)

We note that $\mathscr{B}_{0}\subset\mathscr{B}_{1}\subset\mathscr{B}_{2}$.
All sets isobaric to $2\mathbb{N}$ must be in $\mathscr{B}_{2}$. But much more
is true.

######  Lemma 1.

$\mathscr{B}_{2}=\mathscr{P}(\mathbb{N})$.

###### Proof.

Let $A$ be any subset of $\mathbb{N}$. We wish to show that
$A\in\mathscr{B}_{2}$. We consider a partition of $2\mathbb{N}$ with windows
of fixed length $L=2$: with obvious notation,

$2\mathbb{N}=\\{\ \|\ \cdot\ 2\ \|\ \cdot\ 4\ \|\ \cdot\ 6\ \|\ \cdot\ 8\ \|\
\cdot\ 10\ \|\ \cdot\ 12\ \|\ \cdot\ 14\ \|\ \cdot\ 16\ \|\ \cdot\ 18\ \|\
\cdot\ 20\ \|\ \cdots\\}\,.$

We can construct new sets isobaric to $2\mathbb{N}$ by ‘flipping’ elements
within a selection of windows; in this case, flipping is effected by
subtracting $1$ from an element. The set $\widetilde{2\mathbb{N}}$ is formed
by flipping all elements of $2\mathbb{N}$ that are not elements of $A$. Then
$A_{\mathrm{E}}=2\mathbb{N}\cap\widetilde{2\mathbb{N}}$ is the set of all even
elements of $A$. For the odd numbers,

$2\mathbb{N}-1=\\{\ \|\ 1\ \cdot\ \|\ 3\ \cdot\ \|\ 5\ \cdot\ \|\ 7\ \cdot\
\|\ 9\ \cdot\ \|\ 11\ \cdot\ \|\ 13\ \cdot\ \|\ 15\ \cdot\ \|\ 17\ \cdot\ \|\
19\ \cdot\ \|\ \cdots\\}\,,$

flipping is effected by adding $1$ to an element. The set
$\widetilde{2\mathbb{N}-1}$ is formed by flipping all elements of
$2\mathbb{N}-1$ that are not elements of $A$. Clearly,
$A_{\mathrm{O}}=(2\mathbb{N}-1)\cap(\widetilde{2\mathbb{N}-1})$ is the set of
all odd elements of $A$. Consequently, $A_{\mathrm{E}}\cup A_{\mathrm{O}}=A$,
so that $A\in\mathscr{B}_{2}$.

We conclude that all subsets of the power set are in the $\beta$-algebra, so
that $\mathscr{B}_{2}=\mathscr{P}(\mathbb{N})$. ∎

###  3.3 Magnum Spaces

The surnatural numbers $\boldsymbol{\mathsf{Nn}}$ are the non-negative omnific
integers, $\boldsymbol{\mathsf{Nn}}=\boldsymbol{\mathsf{Oz}}^{+}\cup\\{0\\}$.
The omnific integers and surnatural numbers are briefly described in Appendix
1.

######  Definition 4.

A _magnum space_ over $\mathbb{N}$ is a triplet $(\mathbb{N},\mathscr{B},m)$
consisting of the set $\mathbb{N}$, a $\beta$-algebra $\mathscr{B}$ of subsets
of $\mathbb{N}$ and a function $m:\mathscr{B}\to\boldsymbol{\mathsf{Nn}}$, the
_magnum_ , such that,

  1. 1.

$m(\\{x\\})=1$ for all singletons $\\{x\\}\in\mathscr{B}$.

  2. 2.

For all $A_{1},A_{2}\in\mathscr{B}$, if $A_{1}\cong A_{2}$ then
$m(A_{1})=m(A_{2})$.

  3. 3.

$m(A_{1}\uplus A_{2})=m(A_{1})+m(A_{2})$ for all disjoint sets in
$\mathscr{B}$.

For any $A\in\mathscr{B}$, since
$m(A)=m(A\uplus\varnothing)=m(A)+m(\varnothing)$, it follows that
$m(\varnothing)=0$.

The Euclidean Principle (also known as Aristotle’s Principle) holds for all
magnum spaces:

######  Theorem 1.

If $A_{1},A_{2}$ are in $\mathscr{B}$ with $A_{2}\subset A_{1}$, then
$m(A_{2})<m(A_{1})$.

###### Proof.

Suppose $A\neq\varnothing$ and let $x\in A$. Then
$m(A)=m((A\setminus\\{x\\})\uplus\\{x\\})=m(A\setminus\\{x\\})+m(\\{x\\})=m(A\setminus\\{x\\})+1\geq
1$. Thus, $m(A)>0$ for all nonvoid sets $A$. Since $A_{1}\setminus A_{2}$ is
nonvoid, $m(A_{1})=m((A_{1}\setminus A_{2})\uplus A_{2})=m(A_{1}\setminus
A_{2})+m(A_{2})>m(A_{2})$. ∎

Examples. (a) For any $X$ we have the trivial $\beta$-algebra
$\mathscr{B}=\\{\varnothing,X\\}$. Then $m(\varnothing)=0$, and $m(X)$ may be
given any value in $\boldsymbol{\mathsf{Nn}}$.
(b) Let $X$ be a finite set and $\mathscr{B}=\mathscr{P}(X)$. Since any
$A\in\mathscr{B}$ is a finite disjoint union of singletons, it follows that
$m(A)=\\#A$.
(c) Let $\mathscr{B}_{1}$ be the $\beta$-algebra of finite and cofinite
subsets of $\mathbb{N}$. Then defining $m(A)=\\#A$ for finite $A$ and
$m(A)=\omega-\\#(A^{c})$ for cofinite $A$, $(\mathbb{N},\mathscr{B}_{1},m)$ is
a magnum space.

We remark that Definition 4 bears some similarity to the definition of
_elementary numerosities._ Benci and DiNasso [BeNa03] introduced the concept
of numerosity as a generalisation of finite cardinality that also applies to
infinite sets. Like magnums, numerosity respects the Euclidean Principle, that
“the whole is greater than the part”: the numerosity of a proper subset is
strictly smaller than the numerosity of the whole set.

##  4 Genetic Definition of Magnum

We have defined a magnum space having the general qualities of finite
additivity, isobaric equivalence and unit magnum for singletons (see Def. 4 in
§3.3). We now wish to construct a magnum space for subsets of natural numbers
using the _genetic approach._ It is possible to define the magnums of sets in
a manner analogous to that in which the surreal numbers themselves are
defined.

Given a set $A$ we wish to define two sets $L_{A}$ and $R_{A}$ such that the
surreal number $\\{L_{A}\ |\ R_{A}\\}$ is the magnum of $A$. The challenge is
to construct a function $m$ from the power set of the natural numbers to the
_surnatural numbers_ , that is, the positive (non-negative) omnific integers,

$m:\mathscr{P}(\mathbb{N})\to\boldsymbol{\mathsf{Nn}}\,,$

in such a manner that $(\mathbb{N},\mathscr{P}(\mathbb{N}),m)$ is a magnum
space. We should then be able to conclude that the Euclidean Principle holds,
that $m(\varnothing)=0$ and that $m(A)=\\#A$ for $A$ finite.

###  4.1 Pre-magnums and the Magnum Form

We seek a general expression for the magnum of a set $A$ in the form

$m(A)=\\{m(B):B\in\mathscr{M},B\subset A\ |\ m(C):C\in\mathscr{M},A\subset
C\\}\,,$ (5)

where $\mathscr{M}$ is the family of all sets whose magnums are to be defined.
We note that the form (5) automatically guarantees that the Euclidean
Principle applies, because $B\subset A\subset C$ implies $m(B)<m(A)<m(C)$.
However, we cannot use (5) to define or evaluate $m(A)$, since it requires
knowledge of the magnums of all the other sets in $\mathscr{M}$. We need to
construct the magnums in a recursive or incremental fashion.

In the genetic approach, when defining the magnum of a set on day $\alpha$, we
use the magnums of ‘old’ sets to generate the magnums of ‘new’ sets. For each
ordinal number $\alpha$, Conway [onag, pg. 29] defines the sets $M_{\alpha}$,
$N_{\alpha}$ and $O_{\alpha}$:

  * •

$M_{\alpha}$ is the set of numbers born on or before day $\alpha$ (Made
numbers),

  * •

$N_{\alpha}$ is the set of numbers born first on day $\alpha$ (New numbers),
and

  * •

$O_{\alpha}$ is the set of numbers born before day $\alpha$ (Old numbers).

Note that $M_{\alpha}=N_{\alpha}\uplus\,O_{\alpha}=O_{\alpha+1}$. In a similar
manner, we define three collections of sets:

  * •

$\mathscr{M}_{\alpha}$ are the sets whose magnums are determined on or before
day $\alpha$,

  * •

$\mathscr{N}_{\alpha}$ are the sets whose magnums are determined on day
$\alpha$, and

  * •

$\mathscr{O}_{\alpha}$ are the sets whose magnums are determined before day
$\alpha$.

The last two families combine to give the first: $\mathscr{M}_{\alpha}$ =
$\mathscr{N}_{\alpha}\uplus\,\mathscr{O}_{\alpha}=\mathscr{O}_{\alpha+1}$.

Since, for a given $A$, we do not know on which day its magnum is determined,
we introduce a sequence of numbers, one for each ordinal $\gamma$, defined by

$m_{\gamma}(A)=\\{m(B):B\in\mathscr{O}_{\gamma},B\subset A\ |\
m(C):C\in\mathscr{O}_{\gamma},A\subset C\\}\,.$ (6)

In this recursive construction, the proper subsets $B$ and supersets $C$ of
$A$ range over all sets whose magnums have been defined prior to day $\gamma$.
As the day number increases, the value of the pre-magnum $m_{\gamma}(A)$ of
$A$ may change; $m_{\gamma}(A)$ is the nearest we can get to $m(A)$ on day
$\gamma$ [Sim17]. When a stage $\gamma=\alpha$ is reached where
$m_{\gamma}(A)$ cannot undergo further changes, we define the magnum of $A$ to
be

$m(A)=m_{\alpha}(A)$ (7)

and call $\alpha$ the birthday of $m(A)$.

###  4.2 Day-by-day Construction of $m$

We construct magnums _one day at a time_ , as for the surreal numbers
themselves. Our strategy is to start with the least element of
$\boldsymbol{\mathsf{Nn}}$, that is $0=\\{\ |\ \\}$ and proceed from each
number to its successor(s), considering magnums in the same order in which the
underlying surnatural numbers are defined. At each stage, we examine candidate
sets that might have these numbers as their magnums. In some cases, this is
obvious; in other cases, further investigation is needed to identify the
appropriate set or sets. Proceeding genetically, we obtain magnums for all the
finite subsets of $\mathbb{N}$ by day $\omega$ and for all the cofinite
subsets of $\mathbb{N}$ by day $2\omega$.

Day 0: At this point, no magnums have been defined so that, for _any_ set $A$,
we must have $L_{A}=R_{A}=\varnothing$. Thus, $m(A)=\\{\ \varnothing\ |\
\varnothing\ \\}=0$. Unless $A$ is empty, it has some proper subset $B$, and
the magnum of $B$ will be negative, which is outside the codomain. Thus, the
only possibility is $A=\varnothing$ and $m(A)=0$.

Day 1: The only magnum number available for day 1 is $0$, so the only possible
new value is $\\{\ 0\ |\ \varnothing\ \\}=1$. For any set $A$ other than a
singleton, there will exist a proper subset $B$ with $\varnothing\subset
B\subset A$. We conclude that, for all $a\in\mathbb{N},\ m(\\{a\\})=1$.

Days 2, 3, …: The only magnums available for day $n$ are
$\\{0,1,\dots,n-1\\}$, so the only possible new value is $\\{0,1,\dots,n-1\ |\
\varnothing\ \\}=n$. Since one of our desiderata is that, for finite sets,
$m(A)=\\#A$, it is natural to consider singletons on day 1, doubletons on day
2, tripletons on day 3, and so on. Thus we will find that

$m(\\{a\\})=1\,,\ m(\\{a_{1},a_{2}\\})=2\,,\quad\dots\quad
m(\\{a_{1},a_{2},\dots,a_{n}\\})=n\,,\quad\cdots\,.$

We easily see that all of these cases concur with the form (6). In this way,
we define the magnums for all finite subsets of $\mathbb{N}$. Moreover, for
all such sets, $m(A)=\\#A$.

Day $\boldsymbol{\omega}$: The obvious choice for day $\omega$ is
$A=\mathbb{N}$, when we obtain our first infinite magnum:

$m(\mathbb{N})=\\{1,2,3,\dots\ |\ \varnothing\ \\}=\omega\,.$ (8)

Since we are confining attention to the power-set of the natural numbers,
$\mathbb{N}$ is the only set that gets a magnum on day $\omega$.

In the construction of the surreal numbers, all dyadic rationals are born
before day $\omega$ and all remaining ‘real’ surreal numbers emerge on that
day. However, the only surnatural number to emerge on day $\omega$ is
$\omega$. From this point onwards, $\omega$ may appear on the right side of
the magnum form.

Days $\boldsymbol{\omega}+1$ to $2\boldsymbol{\omega}$: We now “work
downwards” from $\omega$. For $A=\mathbb{N}\setminus\\{a\\}$, the only proper
superset is $\mathbb{N}$, so, for any $a$, $m(A)=\\{\mathbb{N}\ |\
\omega\\}=\omega-1$. More generally, if we remove $n$ elements from
$\mathbb{N}$, a set of magnum $\omega-n$ results. We have now defined the
magnums of all finite and all cofinite sets in $\mathscr{P}(\mathbb{N})$.

Day $2\boldsymbol{\omega}$: On day $2\omega$, we remove an infinite number of
elements of $\mathbb{N}$, so, the elements of
$\\{\omega,\omega-1,\omega-2,\dots\\}$ will be in $R_{A}$. There are several
possibilities:

  * •

If we have removed _all_ elements of $\mathbb{N}$, the magnum must be $\\{\
\varnothing\ |\ \omega,\omega-1,\omega-2,\dots\\}=0$ the magnum of
$\varnothing$.

  * •

If we have left only a finite number, $k$, of elements, the magnum is
$\\{1,2,\dots,k-1|\ \omega,\omega-1,\omega-2,\dots\\}=k$.

  * •

If we have removed an infinite number and also left an infinite number, then
both $A$ and $\mathbb{N}\setminus A$ are infinite, and the new magnum777Since
we confine attention to $\mathscr{P}(\mathbb{N})$, the number $2\omega$ is
inadmissible as a magnum. is

$m(A)=\\{\ \mathbb{N}\ |\
\omega,\omega-1,\omega-2,\dots\\}=\tfrac{\omega}{2}\,.$

But what sets have magnum $\omega/2$? One obvious candidate is

$m(2\mathbb{N})=m(\\{2,4,6,\dots\\})=\\{\ \mathbb{N}\ |\
\omega,\omega-1,\omega-2,\dots\\}=\tfrac{\omega}{2}\,.$

The set of odd natural numbers, the complement of $2\mathbb{N}$, also has
magnum ${\omega}/{2}$, so $m(2\mathbb{N}-1)=\omega/2$, as have all sets
isobaric to $2\mathbb{N}$.

Day $2\boldsymbol{\omega}+1$ to $3\boldsymbol{\omega}$: We now have a new
limit magnum $\omega/2$, which may appear on either the left or right side of
the magnum form. The immediate successors of $\omega/2$ are $\\{\ \mathbb{N}\
|\ \frac{\omega}{2}\ \\}=\frac{\omega}{2}-1$ and $\\{\ \frac{\omega}{2}\ |\ \
\\}=\frac{\omega}{2}+1$. The first provides the magnum for sets formed by
removing a single even number from $2\mathbb{N}$, for example,
$2\mathbb{N}\setminus\\{2\\}$. The second provides the magnum for sets formed
by adding a single odd number to $2\mathbb{N}$, for example,
$2\mathbb{N}\uplus\\{1\\}$. Both $\frac{\omega}{2}-1$ and $\frac{\omega}{2}+1$
are surnatural numbers but only one successor of each is such. We can form
sets with magnums for these: at day $2\omega+n$ we have

$m(2\mathbb{N}\setminus\\{e_{1},e_{2},\dots,e_{n}\\})=\tfrac{\omega}{2}-n\,,\qquad
m(2\mathbb{N}\uplus\\{o_{1},o_{2},\dots,o_{n}\\})=\tfrac{\omega}{2}+n\,,$

where $\\{e_{1},e_{2},\dots,e_{n}\\}$ and $\\{o_{1},o_{2},\dots,o_{n}\\}$ are
sets of distinct even and odd integers.

Day $3\boldsymbol{\omega}$: Following the pattern of creation of surreal
numbers, where the dyadic fractions all appear before any other numbers, we
find a new number on day $3\omega$:

$\\{1,2,3,\dots\ |\ \tfrac{\omega}{2}\ \\}=\tfrac{\omega}{4}\,.$

There are many sets that may have magnum $\omega/4$, but the canonical choice
is $4\mathbb{N}=\\{4,8,12,\dots\ \\}$.

Day $3\boldsymbol{\omega}+1$ to $\boldsymbol{\omega}^{2}$: We continue the
process, with a new limit ordinal entering each $\omega$ days until, on day
$\omega^{2}$, we have all surreal numbers of the form
$((2m+1)/2^{k})\omega\pm\ell$, which may be used on either the left or right
side of the magnum form. We can form sets with magnums for these: at day
$4\omega+n$ we have

$m(4\mathbb{N}\setminus\\{2e_{1},2e_{2},\dots,2e_{n}\\})=\tfrac{\omega}{4}-n\,,\qquad
m(4\mathbb{N}\uplus\\{o_{1},o_{2},\dots,o_{n}\\})=\tfrac{\omega}{4}+n\,,$

where $\\{e_{1},e_{2},\dots,e_{n}\\}$ and $\\{o_{1},o_{2},\dots,o_{n}\\}$ are
sets of distinct even and odd integers.

Day $\boldsymbol{\omega}^{2}$ On day $\omega^{2}$ we have
$m(k\mathbb{N})=\frac{\omega}{k}$ for all $k\in\mathbb{N}$, so
$\mathscr{O}_{\omega^{2}}$ contains these sets and their isobaric equivalents.
We can also form

$\\{1,2,3,\dots\ |\
\tfrac{\omega}{2},\tfrac{\omega}{4},\tfrac{\omega}{8}\dots\
\\}=\sqrt{\omega}\,.$

We expect that the set of squares of the natural numbers,
$\mathbb{N}^{(2)}=\\{1,4,9,16,\dots\\}$, has magnum $\sqrt{\omega}$; this will
be confirmed in §8.

In Appendix 2, a calendar of some of the principle sets whose magnums are
defined on or before day $\omega^{2}$ is presented.

Days beyond $\boldsymbol{\omega}^{2}$: The process may be continued with
numbers like $\omega^{1/4}$, $\omega^{1/n}$ and indeed $\omega^{1/\omega}$
appearing. More generally, surnatural numbers of the form $\sum
r_{y}\cdot\omega^{y}$ with $y\in(0,1]$ arise; the construction process never
ceases. Ultimately, we have assigned magnums to a large family $\mathfrak{M}$
of sets. It remains to show that $\mathfrak{M}$ comprises the entire power set
of $\mathbb{N}$, and that, with the magnum defined genetically, the triple
$(\mathbb{N},\mathscr{P}(\mathbb{N}),m)$ is a magnum space.

In §6, we examine a definition of the magnum of a set $A$ in terms of the
counting function of $A$, and show that the conditions for a magnum space are
satisfied. This will require us to assume an additional axiom, the Transfer
Axiom.

##  5 Using Natural Density to Define Magnums

In this section we examine the suitability of the natural density of a set $A$
for defining a magnum for that set. In some cases, this yields the expected
value but in others it does not. Moreover, there are many sets for which the
density is undefined. In §6, we will consider a more general method, using the
counting sequence.

#### The Defining Sequence

Henceforth, we assume that the elements of $A$ are listed in order of
increasing size, with the _defining sequence_ $\alpha_{A}:k\mapsto a_{k}$. The
graph of $\alpha_{A}$ is the set of points
$\\{(1,\alpha_{A}(1)),\dots,(k,\alpha_{A}(k)),$ $\dots\\}$ and, since
$\alpha_{A}$ is strictly increasing, the function $\alpha_{A}$ has an inverse,
defined by $\alpha_{A}^{-1}:a_{k}\mapsto k$, with graph
$\\{(\alpha_{A}(1),1),\dots,(\alpha_{A}(k),k),\dots\\}$. The function
$\alpha_{A}^{-1}$ is defined on the set $A$. It may be extended to the domain
$\mathbb{N}$, either by linear interpolation or by using a monotone algebraic
expression if that is available.

#### The Counting Sequence

We now define the counting sequence $\kappa_{A}(n)$ to be the number of
elements of $A$ less than or equal to $n$. There is an intimate connection
between $\kappa_{A}$ and the defining sequence $\alpha_{A}$, such that
$\kappa_{A}(n)$ is the integer part of the inverse of $\alpha_{A}$ evaluated
at argument $n$,

$\displaystyle\kappa_{A}(n)=\lfloor\alpha_{A}^{-1}(n)\rfloor\,.$ (9)

The floor operator ensures that $\kappa_{A}(n)$ is a natural number for all
$n\in\mathbb{N}$. It also ensures that the value of $\kappa_{A}(n)$ is
independent of the manner in which $\alpha_{A}^{-1}$ is interpolated between
points of $A$.

If $A$ is finite, it is convenient to extend $\alpha_{A}^{-1}$ and
$\kappa_{A}$ to $\mathbb{N}$. The obvious way to do this is to define

$\alpha_{A}^{-1}(n)=N\quad\text{and}\quad\kappa_{A}(n)=N\quad\text{for\ all\
}n\geq\alpha_{A}(N)$ (10)

where $N$ is the cardinality of $A$. This leads to the magnum $m(A)=N$. If $A$
is infinite, then $\kappa_{A}(n)$ increases without bound.

#### The Density Sequence

The _natural density_ of a sequence $A:\mathbb{N}\to\mathbb{N}$ is defined, if
it exists, as the (real variable) limit of the density sequence:

$\rho_{A}=\lim_{n\to\infty}\rho_{A}(n)\,.$

The density sequence may be defined in two ways. The first uses the counting
function:

$\rho_{A}(n)=\frac{\kappa_{A}(n)}{n}\,.$ (11)

The alternative definition [Ten95] uses the defining function $\alpha_{A}(n)$:

$\boldsymbol{\varrho}_{A}(n)=\frac{n}{\alpha_{A}(n)}\,.$ (12)

We note that $\rho_{A}(n)$ has terms with denominators for every natural
number, whereas the denominators of $\boldsymbol{\varrho}_{A}(n)$ are confined
to elements of $A$. Thus, $\boldsymbol{\varrho}_{A}(n)$ is a subsequence of
$\rho_{A}(n)$ so that, if $\rho_{A}(n)$ converges to $\rho_{A}$ then so does
$\boldsymbol{\varrho}_{A}(n)$. The connection between the two sequences is
$\boldsymbol{\varrho}_{A}(n)=\rho_{A}(\alpha_{A}(n))$.

#### A Graphical Example

Figure 1: The initial values of the defining sequence $\alpha_{A}(n)$ (red
circles), its inverse $\alpha_{A}^{-1}(n)$ (blue triangles), and the counting
sequence $\kappa_{A}(n)$ (black diamonds) for the set $A=3\mathbb{N}\cup
4\mathbb{N}$.

To illustrate the sequences defined above, we consider a simple example, the
set $A=3\mathbb{N}\cup 4\mathbb{N}$. We will prove later that
$m(A)=m(3\mathbb{N})+m(4\mathbb{N})-m(12\mathbb{N})=\frac{\omega}{2}$. In Fig.
1 we show the initial values of the defining sequence $\alpha_{A}(n)$ (red
circles), the inverse of this function $\alpha_{A}^{-1}(n)$ (blue triangles),
and the counting sequence $\kappa_{A}(n)$ (black diamonds) for the set $A$.
The $n$-th term of the natural density sequence $\rho_{A}(n)$ is the slope of
the line from the origin to the point $(n,\kappa_{A}(n))$. The slope of the
line from the origin to the point $(n,\alpha_{A}(n))$ is the _sparsity_ ,
$\boldsymbol{\sigma}_{A}(n)=1/\boldsymbol{\varrho}_{A}(n)$.

#### Defining $m(A)$ by using the Density

For sets whose density is defined, we can use the density to obtain a surreal
expression for $m(A)$. In the case where $\rho_{A}\neq 0$, we may
_provisionally_ define the magnum of $A$ as

$m(A){}\overset{?}{:=}\rho_{A}\cdot\omega\,.$ (13)

Some examples will illustrate the evaluation of magnums using this approach.

$\bullet$ For $A=\mathbb{N}$, the defining function $\alpha_{\mathbb{N}}$ is
the identity function: $\alpha_{\mathbb{N}}(n)=\iota(n)=n$. The counting
function for $\mathbb{N}$ is $\kappa_{\mathbb{N}}(n)=n$ and the density
sequence is constant with $\rho_{\mathbb{N}}(n)=1$, so that its (real) limit
is $\rho_{\mathbb{N}}=1$. From (13) it follows that $m(\mathbb{N})=\omega$.

$\bullet$ For $A=2\mathbb{N}=\\{2,4,6,\dots\\}$ we have
$\rho_{2\mathbb{N}}(2k-1)=\frac{k-1}{2k-1}$ and
$\rho_{2\mathbb{N}}(2k)=\frac{k}{2k}$ with limiting value
$\rho_{2\mathbb{N}}=\frac{1}{2}$. Thus,

$m(2\mathbb{N})=\rho_{2\mathbb{N}}\cdot\omega=\frac{\omega}{2}\,.$

$\bullet$ For $A=2\mathbb{N}-1=\\{1,3,5,\dots\\}$ we have
$\rho_{2\mathbb{N}-1}(2k-1)=\frac{k}{2k-1}$ and
$\rho_{2\mathbb{N}-1}(2k)=\frac{k}{2k}$ with limiting value
$\rho_{2\mathbb{N}-1}=\frac{1}{2}$. Thus,

$m(2\mathbb{N}-1)=\rho_{2\mathbb{N}-1}\cdot\omega=\frac{\omega}{2}\,.$

$\bullet$ For $A=k\mathbb{N}=\\{k,2k,\dots,nk,\dots\\}$ it is easy to show
that $\rho_{k\mathbb{N}}=1/k$ and it follows that $m(k\mathbb{N})=\omega/k$.

#### Difficulties defining magnums using density

The above method avoids the direct extension of $\kappa_{A}(n)$, for which the
conditions are, as yet, imprecise. The density sequence $\rho(n)=\kappa(n)/n$
is _always defined_ , although the limit exists only in some cases. However,
the usual real-variable limit is insufficiently discriminating. To illustrate
the problems, let us consider two sets, $A=2\mathbb{N}\uplus\\{1\\}$ and
$A=\\{n^{2}:n\in\mathbb{N}\\}$.

$\bullet$ For $A=2\mathbb{N}\uplus\\{1\\}$ we find $\rho_{A}=\frac{1}{2}$ and
$m(A)=\omega/2$. The method fails to catch the addition of the term
$\alpha_{A}(1)=1$. We should have hoped for $m(A)=\frac{\omega}{2}+1$.

$\bullet$ For $A=\\{n^{2}:n\in\mathbb{N}\\}$ we have $\rho_{A}(n)\approx
1/\sqrt{n}$ so that $\rho_{A}=0$. The limiting process has obliterated all
information about the sequence, giving $m(A)=0$. The usual limit loses
information that could produce higher-order corrections to the magnum. We
really need a finer-grain description of the limit.

There are several paths to explore. For example, instead of considering
$\rho_{A}(n)={\kappa_{A}(n)}/{n}$, we might define
$\sigma(n)={\kappa_{A}(n)}/{f(n)}$, choosing $f(n)$ such that
$\sigma_{A}=\lim_{n\to\infty}\sigma_{A}(n)$ is finite and positive. However,
considering a set like
$A=\\{n^{2}:n\in\mathbb{N}\\}\cup\\{n^{3}:n\in\mathbb{N}\\}$, we need
$\sigma(n)$ to have components asymptotic to both $\sqrt{n}$ and
$\sqrt[3]{n}$.

It is clear that the definition (13) is insufficiently discriminating and
cannot be used as a general definition of a magnum. We are therefore led to
consider the direct evaluation of $\kappa_{A}(\omega)$.

##  6 Extending the Domain: the Transfer Axiom

For any infinite set $A=\\{a_{1},\dots,a_{k},\dots\\}\subseteq\mathbb{N}$, the
counting function $\kappa_{A}(n)$ is defined on the set of natural numbers. We
aim to show that the domain of $\kappa_{A}$ can be extended to the surnatural
numbers. We wish to define the magnum of the set $A$ to be

$\displaystyle m(A)=\kappa_{A}(\omega)=\lfloor\alpha_{A}^{-1}(\omega)\rfloor$
(14)

(for surreal arguments $s$ we define $\lfloor s\rfloor$ to be the (omnific)
integer part of $s$). To evaluate (14), we must know the value of the counting
function for argument $\omega$. For many simple sets, the defining function is
given by a formula defined on $\mathbb{N}$ but having an obvious extension to
larger sets. For example, if $\alpha_{A}(n)=n^{2}$ it is natural to assume an
extension to $\alpha_{A}(\omega)=\omega^{2}$.

There are difficulties evaluating limits in $\boldsymbol{\mathsf{No}}$. In
[onag, pg. 43], Conway remarks that, since there is a vast class of numbers
greater than all elements of $\mathbb{N}$ and less than $\omega$, one cannot
conclude that the limit of the sequence $(1,2,3,\dots)$ is $\omega$. Thus, we
avoid the concept of limits and consider extensions of the domain on which
functions are evaluated.

According to the _Encyclopedia of Mathematics_ [EoM16], a Transfer Principle
allows us to extend assertions from one algebraic system $\mathit{T1}$ to
another such system $\mathit{T2}$. Every elementary statement, such as a
formula, equation or inequality in the first-order language of $\mathit{T1}$
is true for the extension $\mathit{T2}$. For a discussion of the Transfer
Principle in the context of nonstandard analysis, see [Kei13, pp. 27–30,
906–909] and [Kei76].

Keisler [Kei76, pg. 9] sets out a set of five axioms for the hyperreal number
system. These include a transfer axiom: _“Given two systems of formulas $S$,
$T$ with the same variables, if every real solution of $S$ is a solution of
$T$, then every hyperreal solution of $S$ is a solution of $T$.”_ Keisler also
constructs such a system as an ultrapower of the real numbers, proving that a
hyperreal system exists and, in the process, proving the transfer axiom.

Ehrlich [Eh12] examines the relationship between surreals and nonstandard
analysis, and shows that the hyperreals are contained in the surreal numbers
$\boldsymbol{\mathsf{No}}$, in the sense that “any hyperreal number system is
…isomorphic to an initial subfield of $\boldsymbol{\mathsf{No}}$” [Eh12, pg.
35]. In his Theorem 20 [Eh12, pg. 37], Ehrlich announces that there is a
unique hyperreal system, isomorphic to $\boldsymbol{\mathsf{No}}$, for which
Keisler’s five axioms hold.

In the current context, we need to evaluate the counting function for argument
$\omega$. We observe that any function $f:\mathbb{N}\to\mathbb{N}$ specified
by an explicit algebraic formula can be extended to a function
$f:\boldsymbol{\mathsf{Nn}}\to\boldsymbol{\mathsf{Nn}}$ whose value for
$\nu\in\boldsymbol{\mathsf{Nn}}$ is obtained by applying the formula with
argument $\nu$, that is, $f(\nu)$. Functions of this sort may be said to have
a _natural extension_. This is certainly true for simple algebraic functions.
It is also true for a selection of transcendental functions including $\log n$
and $\exp(n)$.

We also require that properties holding for sets of natural numbers can be
‘transferred’ to larger sets of surnatural numbers. To this end, we formulate
a Transfer Axiom that will enable us to extend properties from the natural
numbers to the surnaturals. In essence, it states that the natural extension
of a function has the same properties as the original function.

There has been discussion of the transfer principle on MathOverflow, the
question-and-answer site for professional mathematicians. On this site,
Ehrlich has written: “It follows immediately from …[his Theorem 12] that the
ordered field of surreal numbers admits a relational extension to a model of
nonstandard analysis and, hence, that in such a relational extension the
transfer principle does indeed
hold.”888https://mathoverflow.net/questions/91646/.

The Transfer Axiom is usually formulated in a general way, such as:

Transfer Axiom (General Form). _Properties expressed by formulas or statements
that hold for all natural numbers can be _transferred_ to hold also for
surnatural numbers._

This axiom ensures us that formulas and statements that hold for sets of
natural numbers are also valid for larger sets of surnatural numbers.
Identities $f(n)=0$ that hold for all $n\in\mathbb{N}$ also hold for $\omega$.
It is clear that polynomial functions, rational functions and algebraic
functions can be extended in an obvious way: simply replace $n$ by $\omega$.

For our immediate purposes, a restricted form of the axiom will be sufficient:

Transfer Axiom (Special Form). _For all monotone functions
$f:\mathbb{N}\to\mathbb{N}$, there is an extension
$f:\boldsymbol{\mathsf{Nn}}\to\boldsymbol{\mathsf{Nn}}$, such that $[\forall
n\in\mathbb{N}:f_{1}(n)\leq
f_{2}(n)]\implies[\forall\nu\in\boldsymbol{\mathsf{Nn}}:f_{1}(\nu)\leq
f_{2}(\nu)]$._

It is clear that the implication also holds with ‘$\leq$’ replaced by ‘$<$’ or
by ‘$=$’. In what follows, we will assume the Special Transfer Axiom.

The defining function $\alpha(n)$, its inverse $\alpha^{-1}(n)$ and the
counting function $\kappa(n)$ are defined for all sets of natural numbers.
They are all monotone and non-decreasing on $\boldsymbol{\mathsf{Nn}}$ and
bounded above by $\omega$. We recall the Monotone Convergence Theorem for
functions of a real variable: _Let $(a_{n})_{{n\in\mathbb{N}}}$ be a monotone
sequence of real numbers. This sequence has a finite limit if and only if it
is bounded. If unbounded, the sequence has a limit in the extended real
numbers $\overline{\mathbb{R}}$._

In the absence of a corresponding theorem for the surreals, the Transfer Axiom
guarantees that, for all $A\subseteq\mathbb{N}$, the sequences $\alpha(n)$,
$\alpha^{-1}(n)$ and $\kappa(n)$ can all be extended to
$\boldsymbol{\mathsf{Nn}}$, so that $\alpha(\omega)$, $\alpha^{-1}(\omega)$
and $\kappa(\omega)$ are defined. The magnum of $A$ will now be defined as
indicated in (14) above:

######  Definition 5.

$\displaystyle m(A)=\kappa_{A}(\omega)=\lfloor\alpha_{A}^{-1}(\omega)\rfloor$
(15)

It follows that the family $\mathfrak{M}$ of sets having a magnum is equal to
$\mathscr{P}(\mathbb{N})$. We will continue to denote by $\mathfrak{M}$ the
family of sets for which magnums are defined as, in later work, this family
may be extended beyond $\mathscr{P}(\mathbb{N})$.999As a digression, we
mention the implications of the Transfer Axiom for the surreal values of
recurring decimal numbers. For the real numbers, $0.999\dots$ is equal to $1$.
For the surreals, this is not the case. We consider the sequence
$\\{\alpha(n)\\}$ where  $\alpha(n)=\underbrace{0.999\dots 9}_{n\
\mathrm{terms}}=1-10^{-n}\,.$ The Transfer Axiom implies that
$\alpha(\omega)=1-10^{-\omega}<1$. Many more examples might be given.

##  7 Theorems Following from the Counting Function

We aim to show that the triplet $(\mathbb{N},\mathfrak{M},m)$ is a magnum
space. Following Definition 4 in §3.3, we must show that
$m:\mathfrak{M}\to\boldsymbol{\mathsf{Nn}}$ is finitely additive, that
singletons have unit magnum and that isobaric sets have equal magnums. We
start with a lemma for the magnums of finite sets.

######  Lemma 2.

If $A$ is a finite set in $\mathfrak{M}$, then $m(A)=\\#A$. In particular all
singletons have unit magnum: $m(\\{a\\})=1$.

###### Proof.

Let $\kappa_{A}(n)$ be the counting function of $A$, extended as indicated in
(10). Then $\kappa(\omega)=\\#A$ and therefore $m(A)=\\#A$. Obviously,
$m(\varnothing)=0$ and $m(\\{a\\})=1$. ∎

#### Finite Additivity of the Magnum

We now establish finite additivity for the magnums of sets in
$\mathfrak{M}=\mathscr{P}(\mathbb{N})$.

######  Theorem 2.

Let $A_{1},A_{2},...,A_{n}$ be mutually disjoint sets in $\mathfrak{M}$. Then
the magnum of their union is the sum of their magnums:

$m\left(\biguplus_{k=1}^{n}A_{k}\right)=\sum_{k=1}^{n}m(A_{k})\,.$

###### Proof.

We first consider the case of two sets, $A_{1}$ and $A_{2}$, and show that
$m(A_{1}\uplus A_{2})=m(A_{1})+m(A_{2})$. Let $\chi_{1}(n)$ and $\chi_{2}(n)$
be the indicator functions of $A_{1}$ and $A_{2}$. Let the counting functions
of these sets be $\kappa_{1}(n)$ and $\kappa_{2}(n)$ and their magnums be
given by $m(A_{1})=\kappa_{1}(\omega)$ and $m(A_{2})=\kappa_{2}(\omega)$.
Since $A_{1}\cap A_{2}=\varnothing$, there is no $n$ for which
$\chi_{1}(n)=\chi_{2}(n)=1$. We write $\chi_{\uplus}$ and $\kappa_{\uplus}$
for the indicator function and counting function of $A_{1}\uplus A_{2}$. Then
$\chi_{\uplus}(n)=\chi_{1}(n)+\chi_{2}(n)$ and, consequently,
$\kappa_{\uplus}(n)=\kappa_{1}(n)+\kappa_{2}(n)$. Therefore, using the
transfer principle,
$\kappa_{\uplus}(\omega)=\kappa_{1}(\omega)+\kappa_{2}(\omega)$ or
$m(A_{1}\uplus A_{2})=m(A_{1})+m(A_{2})$. It is clear that, by finite
iteration, the more general result,
$m\left(\biguplus_{k=1}^{n}A_{k}\right)=\sum_{k=1}^{n}m(A_{k})$, follows. ∎

######  Corollary 1.

(a) The Euclidean Principle holds: $A_{1}\subset A_{2}\implies
m(A_{1})<m(A_{2})$.
(b) Let $A^{\mathrm{C}}=\mathbb{N}\setminus A$. Then
$m(A^{\mathrm{C}})=\omega-m(A)$.
(c) If $a\in A$ then $m(A\setminus\\{a\\})=m(A)-1$.
(d) If $b\notin A$ then $m(A\uplus\\{b\\})=m(A)+1$.
(e) If $a\in A$ and $b\notin A$ then
$m((A\setminus\\{a\\})\uplus\\{b\\})=m(A)$.
(f) For $n$ distinct elements $\\{a_{1},a_{2},\dots,a_{n}\\}$ of $A$ we have
$m(A\setminus\bigcup_{k=1}^{n}\\{a_{k}\\})=m(A)-n$.
(g) For $n$ distinct elements $\\{b_{1},b_{2},\dots,b_{n}\\}$ not in $A$ we
have $m(A\uplus\bigcup_{k=1}^{n}\\{b_{k}\\})=m(A)+n$.
(h) If $B\subset A$ then $m(A\setminus B)=m(A)-m(B)$.
(i) For any $B$, $m(A\setminus B)=m(A)-m(A\cap B)$.

###### Proof.

(a) It is clear that $m(A)>0$ for all $A\neq\varnothing$. Since
$A_{2}=A_{1}\uplus(A_{2}\setminus A_{1})$, it follows that
$m(A_{2})=m(A_{1})+m(A_{2}\setminus A_{1})>m(A_{1})$.

(b) Since $\mathbb{N}=A\uplus A^{\mathrm{C}}$, we have, by Theorem 2,
$m(\mathbb{N})=m(A)+m(A^{\mathrm{C}})$, and the result follows immediately.

(c) $A=(A\setminus\\{a\\})\uplus\\{a\\}$. Therefore,
$m(A)=m(A\setminus\\{a\\})+m(\\{a\\})=m(A\setminus\\{a\\})+1$.

(d) If $b\notin A$ then $A\cap\\{b\\}=\varnothing$ and
$m(A\uplus\\{b\\})=m(A)+m(\\{b\\})=m(A)+1$.

(e) This follows immediately from (c) and (d).

(f) This follows inductively from $m(\\{a_{1},a_{2},\dots,a_{n}\\})=n$ and
(c).

(g) This follows inductively from $m(\\{b_{1},b_{2},\dots,b_{n}\\})=n$ and
(d).

(h) $A=(A\setminus B)\uplus B$. So, $m(A)=m(A\setminus B)+m(B)$ or
$m(A\setminus B)=m(A)-m(B)$.

(i) $A=(A\setminus B)\uplus(A\cap B)$. So, $m(A)=m(A\setminus B)+m(A\cap B)$
or $m(A\setminus B)=m(A)-m(A\cap B)$.

∎

Parts (a)-(d), (h) and (i) are essential in the genetic definition of magnums.
Using part (e), we can replace any element of a set $A$ by an element not in
this set. From parts (f) and (g), any finite number of elements can be
replaced in this way.

There is an interesting contrast between parts (h) and (i), which concern
differences between magnums, and the corresponding properties of differences
between measures. If two sets are both of infinite measure, the measure of
their difference is undefined.

We can prove a general result, with non-disjoint sets, that implies finite
additivity:

######  Theorem 3.

If $A_{1}$ and $A_{2}$, and also their union and intersection, are elements of
$\mathfrak{M}$, then the following relationship holds:

$m(A_{1}\cup A_{2})+m(A_{1}\cap A_{2})=m(A_{1})+m(A_{2})\,.$ (16)

###### Proof.

Let $\kappa_{1}$ and $\kappa_{2}$ be the counting functions of $A_{1}$ and
$A_{2}$, and $\kappa_{\cup}$ and $\kappa_{\cap}$ the counting functions of
$A_{1}\cup A_{2}$ and $A_{1}\cap A_{2}$. It follows easily that
$\kappa_{\cup}(n)=\kappa_{1}(n)+\kappa_{2}(n)-\kappa_{\cap}(n)$. Evaluating
$\kappa_{\cup}+\kappa_{\cap}=\kappa_{1}+\kappa_{2}$ for argument $\omega$,
justified by the transfer principle, yields (16). ∎

It is worthwhile showing that, without calling on Theorem 2, we can prove
directly that the Euclidean Principle holds for sets in $\mathfrak{M}$.

######  Theorem 4.

Let $A_{1}$ and $A_{2}$ be elements of $\mathfrak{M}$ such that $A_{1}\subset
A_{2}$. Then $m(A_{1})<m(A_{2})$.

###### Proof.

Let $\chi_{1}(n)$ and $\chi_{2}(n)$ be the indicator functions of $A_{1}$ and
$A_{2}$, and $\kappa_{1}(n)=\sum_{k=1}^{n}\chi_{1}(k)$ and
$\kappa_{2}(n)=\sum_{k=1}^{n}\chi_{2}(k)$ their counting functions. Since
$a\in A_{1}\implies a\in A_{2}$, we must have $\chi_{1}(n)\leq\chi_{2}(n)$ for
all $n$. Moreover, there exists an $N\in\mathbb{N}$ such that
$\chi_{1}(N)<\chi_{2}(N)$. Therefore, we must have
$\kappa_{1}(n)\leq\kappa_{2}(n)$ for all $n$ and $\kappa_{1}(n)<\kappa_{2}(n)$
for all $n\geq N$. Appealing to the transfer principle,
$\kappa_{1}(\omega)<\kappa_{2}(\omega)$ or $m(A_{1})<m(A_{2})$. ∎

#### Windows and Isobaric Equivalence

We now prove the “Window Theorem” which states that isobaric sets have equal
magnums.

######  Theorem 5.

Let $A_{1}$ be a set with magnum $m(A_{1})$ and $A_{2}$ a set isobaric to
$A_{1}$. Then the two sets have equal magnums, $m(A_{1})=m(A_{2})$.

###### Proof.

Assume $A_{1}$ and $A_{2}$ are isobaric for a fenestration with window length
$L$. Let $(w_{k})$ be the sequence of weights for the set $A_{1}$. Then
$\kappa_{1}(nL)=\sum_{k=1}^{n}w_{k}$. But, since the two sets are isobaric, we
also have $\kappa_{2}(nL)=\sum_{k=1}^{n}w_{k}$, so that
$\kappa_{2}(nL)=\kappa_{1}(nL)$. Then, invoking the transfer axiom and
substituting the surnatural number $\omega/L$ for $n$, we have
$\kappa_{2}(\omega)=\kappa_{1}(\omega)$, or $m(A_{2})=m(A_{1})$. ††margin:  !
††margin:  ! ∎

We tabulate the initial values of $\kappa(n)$ for the set $A_{1}=2\mathbb{N}$
of even numbers, the set $A_{2}=2\mathbb{N}-1$ of odd numbers and the set
$A_{3}=2\mathbb{N}+1$ of odd numbers greater than $1$.

| $n$ |  | $\mathit{1}$ | $\mathit{2}$ |  | $\mathit{3}$ | $\mathit{4}$ |  | $\mathit{5}$ | $\mathit{6}$ |  | $\mathit{7}$ | $\mathit{8}$ |  | $\mathit{9}$ | $\mathit{10}$ | …
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$A_{1}=2\mathbb{N}$ | $\kappa_{1}(n)$ |  | 0 | 1 |  | 1 | 2 |  | 2 | 3 |  | 3 | 4 |  | 4 | 5 | …
$A_{2}=2\mathbb{N}-1$ | $\kappa_{2}(n)$ |  | 1 | 1 |  | 2 | 2 |  | 3 | 3 |  | 4 | 4 |  | 5 | 5 | …
$A_{3}=2\mathbb{N}+1$ | $\kappa_{3}(n)$ |  | 0 | 0 |  | 1 | 1 |  | 2 | 2 |  | 3 | 3 |  | 4 | 4 | …

First, we compare $A_{1}$ and $A_{2}$: since $\kappa_{1}(2n)=\kappa_{2}(2n)$,
substituting $\omega/2$ for $n$ gives $\kappa_{1}(\omega)=\kappa_{2}(\omega)$
or $m({A_{1}})=m({A_{2}})$. Next, we compare $A_{2}$ and $A_{3}$: since
$\kappa_{3}(2n)=\kappa_{2}(2n)-1$, substituting $\omega/2$ for $n$ gives
$m({A_{3}})=m({A_{2}})-1$. Finally, we compare $A_{1}$ and $A_{3}$: since
$\kappa_{3}(2n)=\kappa_{1}(2n)-1$, substituting $\omega/2$ for $n$ gives
$\kappa_{3}(\omega)=\kappa_{1}(\omega)-1$ or $m({A_{3}})=m({A_{1}})-1$.

Summarising, we have the following results, consistent with earlier findings:

$m(2\mathbb{N})=\frac{\omega}{2}\,,\qquad
m(2\mathbb{N}-1)=\frac{\omega}{2}\,,\qquad
m(2\mathbb{N}+1)=\frac{\omega}{2}-1\,.$

These results indicate that _only windows of fixed length are appropriate_.
Consider a fenestration $\mathscr{W}$ for which this is not so:
$W_{1}=\\{1,2,3\\}\,,\quad W_{2}=\\{4,5\\}\,,\quad
W_{3}=\\{6,7\\}\,,\dots\,,W_{n}=\\{2n,2n+1\\}\,,\dots\ $. The two sets
$A_{1}=2\mathbb{N}$ and $A_{3}=2\mathbb{N}+1$ are isobaric with respect to
$\mathscr{W}$, but, as we have seen, $m(A_{1})=\frac{\omega}{2}$ whereas
$m(A_{3})=\frac{\omega}{2}-1$. Thus, the requirement that all windows in
$\mathscr{W}$ are of equal length is essential.

If the windows are all of fixed length $L$ and two sets are related by
$\kappa_{1}(Ln)=\kappa_{2}(Ln)$, we can evaluate them, replacing $n$ by
$\omega/L\in\boldsymbol{\mathsf{Nn}}$ (permitted by the transfer principle) to
get $\kappa_{1}(\omega)=\kappa_{2}(\omega)$ or $m(A_{1})=m(A_{2})$. This is
the principle of isobaric equivalence. ††margin:  ! ††margin:  !

######  Theorem 6.

The Triplet $(\mathbb{N},\mathfrak{M},m)$ is a magnum space

###### Proof.

We have proved that $m:\mathfrak{M}\to\boldsymbol{\mathsf{Nn}}$ is finitely
additive, that all singletons have unit magnum and that any two sets that are
isobaric have equal magnums. Thus, all the conditions for the triplet
$(\mathbb{N},\mathfrak{M},m)$ to be a magnum space are satisfied. ∎

##  8 Application of Theorems to Sample Sets

Using the Transfer Axiom and the above theorems, we can evaluate the magnums
of a large class of sets. A few examples are given here:

$\displaystyle
A=k\mathbb{N}=\\{k,2k,3k,\dots,\\}=\\{kn:n\in\mathbb{N}\\}\qquad$
$\displaystyle\kappa_{A}(n)=\lfloor{n}/{k}\rfloor$ $\displaystyle\qquad
m(A)={\omega}/{k}$ $\displaystyle
A=\mathbb{N}^{(2)}=\\{1,4,9\dots\\}=\\{n^{2}:n\in\mathbb{N}\\}\qquad$
$\displaystyle\kappa_{A}(n)=\lfloor\sqrt{n}\rfloor$ $\displaystyle\qquad
m(A)=\sqrt{\omega}$ $\displaystyle
A=\mathbb{N}^{(k)}=\\{1,2^{k},3^{k}\dots\\}=\\{n^{k}:n\in\mathbb{N}\\}\qquad$
$\displaystyle\kappa_{A}(n)=\lfloor\sqrt[k]{n}\rfloor$ $\displaystyle\qquad
m(A)=\sqrt[k]{\omega}$ $\displaystyle
A=\\{k^{1},k^{2},k^{3},\dots\\}=\\{k^{n}:n\in\mathbb{N}\\}\qquad$
$\displaystyle\kappa_{A}(n)=\lfloor\log_{k}n\rfloor$ $\displaystyle\qquad
m(A)=\lfloor\log_{k}\omega\rfloor\,.$

Magnums for many other sets can be found. For example, consider the general
arithmetic sequence $A=\\{\alpha(n):n\in\mathbb{N}\\}$ where
$\alpha(n)=kn+\ell$ for arbitrary $k\in\mathbb{N}$ and $\ell>-k$. The inverse
function is $\alpha^{-1}(n)=(n-\ell)/k$ so that
$\kappa_{A}(n)=\lfloor\alpha^{-1}(n)\rfloor$ has the natural extension

$\kappa_{A}(\omega)=\biggl{\lfloor}\frac{\omega}{k}-\frac{\ell}{k}\biggr{\rfloor}=\frac{\omega}{k}+O(1)\,.$
(17)

The even numbers arise from $k=2$ and $\ell=0$, and (17) yields
$m(2\mathbb{N})=\omega/2$. The odd numbers arise from $k=2$ and $\ell=-1$, and
(17) yields $m(2\mathbb{N}-1)=\omega/2$. The odd numbers greater than $1$
arise from $k=2$ and $\ell=+1$, and (17) yields
$m(2\mathbb{N}+1)=\omega/2-1$.101010Caution: The floor function and negation
do not commute: for example, $-\lfloor\frac{1}{2}\rfloor=0$ whereas
$\lfloor-\frac{1}{2}\rfloor=-1$.

Now consider the general geometric sequence $A=\\{a\,r^{n-1}\\}$ for arbitrary
$a$ and $r$ in $\mathbb{N}$. The inverse of $\alpha(n)$ is
$\alpha^{-1}(n)=\log(rn/a)/\log r=\log_{r}(rn/a)$, so that
$\kappa_{A}(n)=\lfloor\alpha^{-1}(n)\rfloor$ has the natural extension

$\kappa_{A}(\omega)=\lfloor{\log_{r}({r\omega}/{a})}\rfloor=\lfloor{\log_{r}\omega}\rfloor+O(1)\,.$
(18)

Thus, if $a=1$ and $r=2$ we have $A=\\{1,2,4,8,\dots\\}$ and
$m(A)=\log_{2}2\omega$.

Magnums can be assigned to combinations of sets for which the definition in
terms of natural density fails. For example, using Theorem 16, the set
$A=3\mathbb{N}\cup 4\mathbb{N}$ shown in Fig. 1 has magnum

$m(3\mathbb{N}\cup
4\mathbb{N})=[m(3\mathbb{N})+m(4\mathbb{N})-m(12\mathbb{N})]=\left[\frac{\omega}{3}+\frac{\omega}{4}-\frac{\omega}{12}\right]=\frac{\omega}{2}\,.$

This also follows from the observation that $3\mathbb{N}\cup 4\mathbb{N}\cong
2\mathbb{N}$ with windows of length $12$.

The set $A=\mathbb{N}^{(2)}\cup\mathbb{N}^{(3)}$, containing all squares and
cubes, has magnum

$m(\mathbb{N}^{(2)}\cup\mathbb{N}^{(3)})=[m(\mathbb{N}^{(2)})+m(\mathbb{N}^{(3)})-m(\mathbb{N}^{(6)})]=[\sqrt[2]{\omega}+\sqrt[3]{\omega}-\sqrt[6]{\omega}]\,.$

Figure 2: Proportion of numbers less than $n$ that have an odd count of binary
digits. Note that the $x$-axis is logarithmic.

We have obtained the magnums of many subsets of $\mathbb{N}$, but it must be
noted that there are many sets for which we are unable to calculate the
magnums. As an example, consider the set $\mathbf{Od}_{2}$, whose elements are
all those numbers having an odd number of binary digits. The natural density
of this set oscillates between values that asymptote to $\frac{1}{3}$ and
$\frac{2}{3}$, never tending to a limit (see Fig. 2). Moreover, the variation
between one extreme and the next increases by a factor of $2$ for each stage.
It seems probable that some additional assumption or axiom is required to
determine the magnum of $\mathbf{Od}_{2}$ uniquely.

Although we have confined attention to subsets of $\mathbb{N}$, the theory
developed here is amenable to extension beyond this limit. To give a few
examples, it is intuitively reasonable to expect results such as
$m(\mathbb{N}\times\\{1,2\\})=2\omega$ and
$m(\mathbb{N}\times\mathbb{N})=\omega^{2}$. Many more examples could be given,
but we will not consider them here.

##  9 Equivalence between the Counting Function and the Magnum Form

We have defined magnums genetically in terms of magnum forms (§4) and also in
terms of the counting function (§6):

  * •

Genetic: $m(A)$ is constructed from the form $\\{L\ |\ R\\}$.

  * •

Functional: $m(A)$ is defined from the counting function:
$m(A)=\kappa(\omega)$.

We must now show that these two definitions are consistent.

For finite sets it is easily seen that the two approaches agree. Let
$A=I_{k}=\\{1,2,\dots,k\\}$. Then $m(A)=\\{0,1,2,\dots k-1\ |\ \\}=k$ and
$\kappa(k)=k$, which is extended with constant value $k$ so that
$\kappa(\omega)=k$. For cofinite sets the equivalence can also be shown. Let
$A=\mathbb{N}\setminus I_{k}$.

$\kappa(n)=\lfloor\alpha^{-1}(n)\rfloor=\alpha^{-1}(n)=\begin{cases}0,&n\leq
k\\\ n-k,&n>k\end{cases}$

which extends to $\kappa(\omega)=\omega-k$. The genetic magnum form takes the
same value:

$m(A)=\\{0,1,2,\dots\ |\ \omega,\omega-1,\dots,\omega-k+1\\}=\omega-k\,.$

So, for all finite and cofinite sets, the two methods yield equivalent
results. We also know that for $A=\mathbb{N}$, both yield $m(A)=\omega$ and
for the odd and even numbers both give $m(A)=\frac{\omega}{2}$.

More generally, suppose the magnum of a set $A$ is given by
$m(A)=\kappa_{A}(\omega)$. By definition, $m(A)\in\boldsymbol{\mathsf{Nn}}$.
Therefore,

$m(A)=\\{m(A)-1\ |\ m(A)+1\\}\,.$

For any nonempty set $A$, the largest proper subset $B$ of $A$ has magnum
$m(A)-1$ while the smallest proper superset $C$ has magnum $m(A)+1$. For
example, let $a\in A$ and $d\not\in A$ and let $B=A\setminus\\{a\\}$ and
$C=A\uplus\\{d\\}$. Then we can write

$m(A)=\\{m(B)\ |\ m(C)\\}\,.$

Clearly, including other subsets of $A$ on the left or supersets on the right
will not alter the value of the magnum of $A$.

We conclude that the two representations of the magnum of $A$, as
$m(A)=\kappa_{A}(\omega)$ and as

$m(A)=\\{m(B):B\subset A\text{ and }B\in\mathfrak{M}\ |\ m(C):C\supset A\text{
and }C\in\mathfrak{M}\\}$

are compatible.

##  10 Concluding Remarks

Cardinality fails to discriminate between sets that are “obviously” of
different sizes. We have defined a quantity — the set magnum — for sets of
natural numbers that accords with our intuition about relative sizes. The
codomain of the magnum is the class of surnatural numbers (the non-negative
omnific integers). The Euclidean Principle holds: for any set, the magnum of a
proper subset is strictly less than the magnum of the set itself.

Two approaches, the genetic assignment of magnums and the extension of
counting sequences using the Transfer Axiom, have been shown to be compatible.
The magnum $m(A)$ is found to possess the ten desiderata identified in advance
in §2. Several theorems have been proved, establishing that
$(\mathbb{N},\mathfrak{M},m)$ is a magnum space as defined in §3.

Although we have concluded that all subsets of $\mathbb{N}$ have magnums,
there are many sets whose magnums we are unable to calculate. We mentioned
above the set $\mathbf{Od}_{2}$ whose elements have an odd number of binary
digits. It seems probable that some additional assumption or axiom is required
to determine the magnum of $\mathbf{Od}_{2}$ uniquely.

We have confined attention to sets $A\in\mathscr{P}(\mathbb{N})$. In future
work, we hope to extend the theory developed here to a larger family of
countable sets. We note that Benci et al. [BeNa19] define numerosities on a
family
$\mathfrak{U}=\bigcup_{n\in\mathbb{N}}(\bigcup_{k=1}^{n}\mathbb{N}^{k})$,
which is a _universe_ , closed under subsets, disjoint unions and Cartesian
products. It should be possible to extend magnums in this way. The extension
of numerosities to sets of rational numbers is also accomplished in [BeNa19],
although this requires additional axioms. The extension of magnums to sets of
rational numbers requires further research.

## Acknowledgement

We are grateful to Dr Richard Elwes and Dr Vincenzo Mantova of the University
of Leeds for bringing to our attention the work of Prof Vieri Benci and
colleagues.

## References

  * [1]
  * [BeNa03] Benci, Vieri and Mauro Di Nasso, 2003: Numerosities of labelled sets: a new way of counting. _Advances in Maths._ , 173, 50–67.
  * [BeNa19] Benci, Vieri and Mauro Di Nasso, 2019: _How to Measure the Infinite: Mathematics with Infinite and Infinitesimal Numbers_. 317pp. World Scientific, Singapore. ISBN: 9-789-812-83637-3.
  * [onag] Conway, John Horton, 1976: _On Numbers and Games._ Academic Press, London. 2nd Edn., CRC Press, Taylor and Francis Group. 242pp. ISBN: 978-1-568-81127-7.
  * [EoM16] Encyclopedia of Mathematics, 2016. _Transfer principle._ http://encyclopediaofmath.org/
  * [He08] Heath, Sir Thomas Little, 1908: _The thirteen books of Euclid’s Elements._ University Press, Cambridge. Second edition: University Press, Cambridge, 1925. Reprint: Dover Publ., New York, 1956.
  * [Eh12] Ehrlich, Philip, 2012: The absolute arithmetic continuum and the unification of all numbers great and small. _Bull. Symb. Logic_ , 18, (1), 1–45. https://www.jstor.org/stable/41472439.
  * [Eh22] Ehrlich, Philip, 2022: The Surreal numbers and their aftermath. Pp. 1146–1150 in Ryba, Alex, 2022: Conway’s Mathematics After Conway. _Notices Amer. Math. Soc._ , 69, (7), 1145–1155.
  * [Kei76] Keisler, H. Jerome, 1976: _Foundations of Infinitesimal Calculus_ Prindle, Weber & Schmidt, Inc., 20 Newbury Street, Boston. Online version (updated 2022) freely available: https://people.math.wisc.edu/~hkeisler/foundations.pdf
  * [Kei13] Keisler, H. Jerome, 2013: _Elementary Calculus: An Infinitesimal Approach_ (3rd ed.), New York: Dover Publications, ISBN 978-0-486-48452-5. Online version freely available: https://people.math.wisc.edu/~hkeisler/calc.html
  * [Knu74] Knuth, D. E., 1974: _Surreal Numbers_. Addison-Wesley, 119pp. ISBN. 978-0-201-03812-5.
  * [Lyn19] Lynch, Peter, 2019: Magnums: Counting Sets with Surreals. Recreational Mathematics Colloquium VI: Proceedings, 123–138. August 2019, Jorge Nuno Silva (Ed.).
  * [NAMS] NAMS: Doctoral Degrees Conferred, 2016–2020. _Notices Amer. Math. Soc._ , 65(7), 969–999; 66(7), 1118–1149; 67(8), 1214–1249; 70(5), 830–865.
  * [PCM08] Gowers, Timothy et al. (Eds.), 2008: _The Princeton Companion to Mathematics._ Princeton Univ. Press, 1034 pp. ISBN: 978-0-691-11880-2.
  * [Risi20] de Risi, Vincenzo, 2020: Euclid’s common notions and the theory of equivalence. _Foundations of Science_ , 10.1007/s10699-020-09694-w.
https://hal.science/hal-03058635

  * [Ryba22] Ryba, Alex, 2022: Conway’s Mathematics After Conway. _Notices Amer. Math. Soc._ , 69, (7), 1145–1155.
  * [Sim17] Simons, Jim, 2017: _Meet the Surreal Numbers._ Notes for Lecture at Mathematical Association Annual Conference. https://www.m-a.org.uk/resources/downloads/4H-Jim-Simons-Meet-the-surreal-numbers.pdf
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## Appendix 1: Surreal Numbers and Surnatural Numbers

The class of surreal numbers, denoted by $\boldsymbol{\mathsf{No}}$,
discovered by Conway around 1970, is the largest possible ordered field
(Conway, 2001, referenced as [onag]). The basic arithmetical operations —
addition, subtraction, multiplication, division and roots — are defined in
$\boldsymbol{\mathsf{No}}$.

The surreal numbers are constructed inductively. We give a brief sketch,
without proofs or details, of some of their key properties. A full account is
given in [onag], an elementary discussion in [Knu74] and a tutorial in
[Sim17].

Every number $x$ is defined as a pair of sets, the left set and the right set:

$x=\\{X^{L}\ |\ X^{R}\\}$

where every element of $X^{L}$ is less than every element of $X^{R}$, and $x$
is the _simplest_ or earliest appearing number between these two sets. We
start with $0$, defined as $\\{\ \ |\ \ \\}$. Then

$\\{0\ |\ \ \\}=1\qquad\\{1\ |\ \ \\}=2\qquad\\{2\ |\ \ \\}=3\qquad\dots$

Negative numbers are defined inductively as $-x=\\{-X^{R}\ |\ -X^{L}\\}$. So

$\\{\ \ |\ 0\\}=-1\qquad\\{\ \ |\ 1\\}=-2\qquad\\{\ \ |\ 2\\}=-3\qquad\dots$

The dyadic fractions (rationals of the form $m/2^{n}$) appear as

$\\{0\ |\ 1\\}=\textstyle{\frac{1}{2}}\qquad\\{1\ |\
2\\}=\textstyle{\frac{3}{2}}\qquad\\{0\ |\
\textstyle{\frac{1}{2}}\\}=\textstyle{\frac{1}{4}}\qquad\\{\textstyle{\frac{1}{2}}\
|\ 1\\}=\textstyle{\frac{3}{4}}\qquad\dots$

After an infinite number of stages, when all dyadic fractions have emerged,
all the remaining rationals and real numbers appear. The first infinite and
infinitesimal numbers are defined at this stage:

$\omega=\\{0,1,2,3,\dots\ \ |\ \\}\,,\qquad 1/\omega=\\{\ 0\ |\
1,\textstyle{\frac{1}{2},\frac{1}{3},}\dots\ \\}\,.$

We can then introduce

$\omega+1=\\{0,1,2,3,\dots\ \omega\ |\ \\}\qquad\omega-1=\\{0,1,2,3,\dots\ \
|\ \ \omega\\}$ $2\omega=\\{0,1,2,3,\dots\ \omega,\omega+1,\omega+2,\dots\ |\
\\}\qquad\textstyle{\frac{1}{2}}\omega=\\{0,1,2,3,\dots\ \ |\ \
\omega,\omega-1,\omega-2,\dots\\}$

and many other more exotic numbers. They behave beautifully:
$\boldsymbol{\mathsf{No}}$ is a totally ordered field, indeed, the largest
such field. Since we are considering subsets of $\mathbb{N}$, we are concerned
_primarily_ with surreals less than or equal to $\omega$.

The sum of two surreal numbers

$x_{1}=\\{X^{L}_{1}\ |\ X^{R}_{1}\\}\qquad\mbox{and}\qquad x_{2}=\\{X^{L}_{2}\
|\ X^{R}_{2}\\}$

is defined to be

$x_{1}+x_{2}=\\{x_{1}+X^{L}_{2},X^{L}_{1}+x_{2}\ |\
x_{1}+X^{R}_{2},X^{R}_{1}+x_{2}\ \\}$

where, for example, $x_{1}+X^{L}_{2}$ is the set of numbers $x_{1}+x^{L}_{2}$
for all elements $x^{L}_{2}$ of $X^{L}_{2}$. The product of two surreal
numbers is defined to be

$\displaystyle x_{1}x_{2}$ $\displaystyle=$
$\displaystyle\\{x_{1}^{L}x_{2}+x_{1}x_{2}^{L}-x_{1}^{L}x_{2}^{L},x_{1}^{R}x_{2}+x_{1}x_{2}^{R}-x_{1}^{R}x_{2}^{R}\
|\ $ $\displaystyle\ \
x_{1}^{L}x_{2}+x_{1}x_{2}^{R}-x_{1}^{L}x_{2}^{R},x_{1}^{R}x_{2}+x_{1}x_{2}^{L}-x_{1}^{R}x_{2}^{L}\\}\,.$

#### The Omnific Integers

Conway ([onag], Ch. 5) defines the class $\boldsymbol{\mathsf{Oz}}$ of
_omnific integers_ : $x$ is an omnific integer if $x=\\{x-1|x+1\\}$. Omnifics
are the appropriate class of integers in the context of the surreals. They
include all the real integers and all ordinal numbers, and every surreal
number $x$ is the quotient of two omnific integers.

Any surreal number $x$, may be expressed uniquely in the normal form,

$x=\sum_{\beta<\alpha}r_{\beta}\cdot\omega^{y_{\beta}}\,,$ (19)

where $\alpha$ is an ordinal, the numbers $r_{\beta}$ are real and the numbers
$y_{\beta}$ form a descending sequence. The number expressed by (19) is an
omnific integer if and only if $r_{\beta}=0$ for $y_{\beta}<0$ and
$r_{0}\in\mathbb{Z}$ [onag, Ch. 5, Th. 1].

There are no infinitesimals in $\boldsymbol{\mathsf{Oz}}$. Every surreal
number is distant at most $1$ from some omnific integer, and
$\boldsymbol{\mathsf{No}}$ is the fraction field of
$\boldsymbol{\mathsf{Oz}}$. For surreal arguments $s$ we define $\lfloor
s\rfloor$ to be the (omnific) integer part of $s$. If the normal form of $s$
is $s=\sum_{\beta<\alpha}\omega^{y_{\beta}}\cdot r_{\beta}$ then $\lfloor
s\rfloor=\sum_{0<\beta<\alpha}\omega^{y_{\beta}}\cdot r_{\beta}+\lfloor
r_{0}\rfloor$ .

For an illuminating discussion of the omnific integers, see [Sim17, §7].

#### The Surnatural Numbers

If we confine attention to subsets of $\mathbb{N}$, we need nothing larger
than $\omega$ to define the magnums. Moreover, we assume that the magnum
function maps the power set of the natural numbers into the positive omnific
integers or _surnatural numbers_ ,
$\boldsymbol{\mathsf{Nn}}=\boldsymbol{\mathsf{Oz}}^{+}\cup\\{0\\}$:

$m:\mathscr{P}(\mathbb{N})\to\boldsymbol{\mathsf{Nn}}\,.$ (20)

This rules out negative and fractional values for $m(A)$, but admits numbers
$r\cdot\omega^{y}$, where $r$ is an arbitrary positive real number and
$y\in(0,1]$. For example, $\omega/3$ and $\omega^{1/2}$ are admissible.

To be explicit, a surnatural number may be expressed in the normal form (19)
where $\alpha$ is an ordinal, the numbers $r_{\beta}$ are non-negative real
numbers, the numbers $y_{\beta}$ form a descending sequence of non-negative
reals, and $r_{0}$ is a non-negative integer. Numbers with $r_{0}=0$ are
called purely infinite surnatural numbers.

It is noteworthy that, since all numbers $r\cdot\omega^{\beta}$ are
surnaturals, it follows that $\omega$ is divisible by all numbers in
$\mathbb{N}$, and indeed in $\mathbb{R}$. Thus, $\omega$ is an even number,
since $\omega/2\in\boldsymbol{\mathsf{Nn}}$, a multiple of $3$ since
$\omega/3\in\boldsymbol{\mathsf{Nn}}$, and so on. Indeed, for all
$k\in\mathbb{N}$, $k!$ divides $\omega$. Moreover,
$\sqrt[k]{\omega}\in\boldsymbol{\mathsf{Nn}}$, so $\omega$ is a perfect
square, a perfect cube, and so on.

#### Surreal Numbers & Current Research

Since their invention/discovery around 1970, surreal numbers have attracted
relatively little attention. In _The Princeton Companion to Mathematics_
[PCM08], a comprehensive review of mathematics extending to more than 1,000
pages, there is no index entry for ‘surreal numbers’. J.H.Conway’s name is
indexed, but all four references are to his work in group theory.

In the _Notices of the American Mathematical Society_ from 2018 to 2023
[NAMS], thesis titles were listed for a total of about 8,000 doctoral degrees
awarded in the mathematical sciences between July 1, 2016 and June 30, 2020,
as reported by about 250 departments in 200 universities in the United States.
There were no occurrences of the word ‘surreal’ in these lists.

Although Conway regarded surreals as his most significant creation, they have
attracted relatively little attention; given their appealing properties and
their remarkable elegance, this is surprising. A small number of researchers
have worked on surreal numbers, and they have also been of interest to
recreational mathematicians. For a review of progress over the past half-
century, see [Eh22] in [Ryba22].

## Appendix 2: Calendar of Magnums for Sets in $\mathscr{P}(\mathbb{N})$

On what day is the magnum of a subset of $\mathbb{N}$ first defined? To answer
this, we consider the ordinals as they arise.

Day 0: The magnum of $\varnothing$ is defined to be $0$.

Day 1: The magnum of the singleton $\\{n\\}$ is defined for all
$n\in\mathbb{N}$ to be $1$.

Day 2: The magnum of the doubleton $\\{m,n\\}$ is defined for all
$m,n\in\mathbb{N}$ to be $2$.

Day n: The magnums of all sets with $n$ elements are defined to be $n$.

Thus, all finite subsets of $\mathbb{N}$ are defined on finite days. Their
magnums are all the finite ordinal numbers.

Day $\omega$: The set $\mathbb{N}$ is given a magnum on this day:
$m(\mathbb{N})=\omega$, the first infinite ordinal. From now on, both ordinals
and other surnatural numbers will be used.

Day $\omega+1$: All “co-singletons”, sets of the form
$\mathbb{N}\setminus\\{k\\}$, are assigned the value $\omega-1$.

Day $\omega+n$: All sets with complements having magnum $n$ are assigned the
value $\omega-n$.

Before day $2\omega$, all finite and cofinite sets, together with
$\mathbb{N}$, have magnums assigned.

Day $2\omega$: The new magnum is $\omega/2$. The obvious candidate for this
magnum is $2\mathbb{N}$. In fact, non-denumerably many other sets have magnums
assigned on this day. For any natural number $L$, we consider all sets
isobaric to $2\mathbb{N}$ for a fenestration of $\mathbb{N}$ having window
length $L$. Even for $L=2$, we have an uncountable collection of sets isobaric
to $2\mathbb{N}$.

The question is: _have we now got all the sets_ with magnum $\omega/2$? To
define the magnum of a candidate set $A$ that is neither finite nor cofinite,
we consider the magnum form:

$m(A)=\\{m(B):B\in\mathscr{O}_{2\omega}\land B\subset A\ |\
m(C):C\in\mathscr{O}_{2\omega}\land C\supset A\\}\,,$

The only (old) magnums available on day $2\omega$ are $\\{m:m\in\mathbb{N}\\}$
and $\\{\omega-n:n\in\mathbb{N}\\}$. Thus all the admissible sets $B\subset A$
are finite and all the admissible sets $C\supset A$ are cofinite. The set $A$
and will certainly have infinite subsets and supersets, but the magnums of
these may not be yet assigned.

The set $2\mathbb{N}$ certainly has subsets and supersets whose magnums have
not yet appeared. For example, $4\mathbb{N}\subset 2\mathbb{N}$ and
$2\mathbb{N}\uplus(4\mathbb{N}+1)\supset 2\mathbb{N}$. We can rule out their
membership of $\mathscr{N}_{2\omega}$ since, by the Euclidean Principle,
proper subsets and proper supersets of a set cannot have magnums equal to it.

A calendar of some subsets of $\mathbb{N}$ acquiring magnums up to day
$\omega^{2}$ is shown in Table 1.

Table 1: Calendar of days on which some simple sets are given magnums. $j$, $k$, $\ell$, $m$ and $n$ are natural numbers. The symbol $\cong$ indicates all sets isobaric to those listed before it. Day # | Magnum | Sets
---|---|---
$0$ | $0$ | $\varnothing$
$1$ | $1$ |  $\\{k\\}$, $\cong$
$2$ | $2$ |  $\\{k,\ell\\}$, $\cong$
$3$ | $3$ |  $\\{k,\ell,m\\}$, $\cong$
$\cdots$ | $\cdots$ | $\cdots$
$n$ | $n$ |  $\\{m_{1},m_{2},\dots,m_{n}\\}$, $\cong$
$\cdots$ | $\cdots$ | $\cdots$
$\omega$ | $\omega$ | $\mathbb{N}$
$\omega+1$ | $\omega-1$ |  $\mathbb{N}\setminus\\{k\\}$, $\cong$
$\omega+2$ | $\omega-2$ |  $\mathbb{N}\setminus\\{k,\ell\\}$, $\cong$
$\cdots$ | $\cdots$ | $\cdots$
$\omega+n$ | $\omega-n$ |  $\mathbb{N}\setminus\\{m_{1},m_{2},\dots,m_{n}\\}$, $\cong$
$\cdots$ | $\cdots$ | $\cdots$
$2\omega$ | $\frac{\omega}{2}$ |  $2\mathbb{N}$, $2\mathbb{N}-1$, $\cong$
$2\omega+1$ | $\frac{\omega}{2}+1$ |  $2\mathbb{N}\uplus\\{k\\}$, $\cong$
| $\frac{\omega}{2}-1$ |  $2\mathbb{N}\setminus\\{k\\}$, $\cong$
$2\omega+2$ | $\frac{\omega}{2}+2$ |  $2\mathbb{N}\uplus\\{k,\ell\\}$, $\cong$
| $\frac{\omega}{2}-2$ |  $2\mathbb{N}\setminus\\{k,\ell\\}$, $\cong$
$\cdots$ | $\cdots$ | $\cdots$
$3\omega$ | $\frac{\omega}{4}$ |  $4\mathbb{N}$, $\cong$
| $\frac{3}{4}\omega$ |  $\mathbb{N}\setminus 4\mathbb{N}$, $\cong$
$3\omega+1$ | $\frac{\omega}{4}+1$ |  $4\mathbb{N}\uplus\\{k\\}$, $\cong$
| $\frac{\omega}{4}-1$ |  $4\mathbb{N}\setminus\\{k\\}$, $\cong$
| $\frac{3}{4}\omega+1$ |  $(\mathbb{N}\setminus 4\mathbb{N})\uplus\\{k\\}$, $\cong$
| $\frac{3}{4}\omega-1$ |  $(\mathbb{N}\setminus 4\mathbb{N})\setminus\\{k\\}$, $\cong$
$\cdots$ | $\cdots$ | $\cdots$
$4\omega$ | $\frac{1}{8}\omega$ |  $8\mathbb{N}$, $\cong$
| $\frac{3}{8}\omega$ |  $8\mathbb{N}\uplus(8\mathbb{N}-1)\uplus(8\mathbb{N}-1)$ $\cong$
| $\frac{5}{8}\omega$ |  $\mathbb{N}\setminus[8\mathbb{N}\uplus(8\mathbb{N}-1)\uplus(8\mathbb{N}-1)]$, $\cong$
| $\frac{7}{8}\omega$ |  $\mathbb{N}\setminus 8\mathbb{N}$, $\cong$
$\cdots$ | $\cdots$ | $\cdots$
$\omega^{2}$ | $\frac{j}{k}\omega$ |  $\biguplus_{m=0}^{j-1}(k\mathbb{N}-m)$, $\cong$
| $\cdots$ | $\cdots$
| $\sqrt{\omega}$ | $\mathbb{N}^{(2)}$