# Euclidean numbers and numerosities

Vieri Benci, Lorenzo Luperi Baglini Dipartimento di Matematica, Università
degli Studi di Pisa, Via F. Buonarroti 1/c, 56127 Pisa, Italy, e-mail:
<EMAIL_ADDRESS>di Matematica, Università degli Studi di
Milano, Via C. Saldini 50, 20133 Milano, Italy, e-mail:
<EMAIL_ADDRESS>supported by grant P 30821-N35 of the Austrian
Science Fund FWF.

###### Abstract

Several different versions of the theory of numerosities have been introduced
in the literature. Here, we unify these approaches in a consistent frame
through the notion of set of labels, relating numerosities with the Kiesler
field of Euclidean numbers. This approach allows to easily introduce, by means
of numerosities, ordinals and their natural operations, as well as the
Lebesgue measure as a counting measure on the reals.

###### Contents

  1. 1 Introduction
  2. 2 The Euclidean numbers
    1. 2.1 Non Archimedean fields
    2. 2.2 Construction of the Euclidean numbers
    3. 2.3 Labelled sets
  3. 3 The general theory of numerosities
    1. 3.1 Definition and first properties
  4. 4 Ordinal numbers and numerosities
    1. 4.1 The ordinal numerosities
    2. 4.2 Sums and products of ordinals
    3. 4.3 Numerosities and Cantor ordinals
  5. 5 Numerosities of some denumerable sets
    1. 5.1 The general strategy
    2. 5.2 Numerosity of the natural numbers
    3. 5.3 Numerosity of the integers
    4. 5.4 Numerosity of the rationals
  6. 6 Numerosities of non-denumerable sets
    1. 6.1 A suitable labelling
    2. 6.2 Cardinal numbers and numerosities
  7. 7 Numerosity and measures
    1. 7.1 The general theory
    2. 7.2 Numerosity of the subsets of $\mathbb{R}^{N}$
    3. 7.3 Numerosity and nonstandard measures

##  1 Introduction

The techniques of nonstandard analysis allow to construct several different
hyperreal fields which, for many pratical purposes, are equivalent. However,
there is a unique hyperreal field which is isomorphic to the closed real field
having the cardinality of the first strongly inaccessible111$\kappa$ is
strongly inaccessible if it is uncountable, it is not a sum of fewer than
$\kappa$ cardinals smaller than $\kappa$ and, for all $\alpha<\kappa$,
$2^{\alpha}<\kappa$. uncountable cardinal number. Such a field has been
introduced in [19] and we refer to it as to the Keisler field.

Given a ring of sets $\mathcal{R}$ (closed for cartesian product) and a non
Archimedean field $\mathbb{K}$, the numerosity is a function

$\mathfrak{num}:\mathcal{R}\rightarrow\mathbb{K}$ (1)

which satisfies the following properties:

  * •

Finite sets principle: if $A$ is a finite set, then
$\mathfrak{num}\left(A\right)=|A|$ ($|A|$ denotes the cardinality of $A$);

  * •

Euclid’s principle: if $A\subset B$, then
$\mathfrak{num}\left(A\right)<\mathfrak{num}\left(B\right)$;

  * •

Sum principle: if $A\cap B=\varnothing$, then $\mathfrak{num}\left(A\cup
B\right)=\mathfrak{num}\left(A\right)+\mathfrak{num}\left(B\right)$;

  * •

Product principle: $\mathfrak{num}\left(A\times
B\right)=\mathfrak{num}\left(A\right)\cdot\mathfrak{num}\left(B\right).$

The notion of numerosity has been introduced in [1],[5] and developed in
several directions ([4],[6],[7],[8],[9],[14],[16],[17],[21])222See also [10]
for a historical survay of the ideas related to the measure of the size of
infinite sets.. Since its beginning, numerosity theory has been strictly
related to some hyperreal field, namely the field $\mathbb{K}$ in (1) must be
hyperreal.

The aim of this paper is to relate the theory of numerosity to the Keisler
field in such a way that most of the properties investigated in the previous
papers are preserved and unified in a consistent frame.

In particular, we want at least the following three properties to be
satisfied:

  * •

(Consistency with the theory of cardinal numbers) if $A,B\subset\mathbb{A}$
then

$\left|A\right|<\left|B\right|\Rightarrow\mathfrak{num}\left(A\right)<\mathfrak{num}\left(B\right).$

where $\left|E\right|$ denotese the cardinality of $E$.

  * •

(Consistency with the theory of ordinal numbers) if $\mathbf{COrd}$ is the set
of the Cantor ordinal numbers smaller than the first inaccessible uncountable
cardinal number and $\mathbf{Num}$ is the set of numerosities, then there is a
map

$\Psi:\mathbf{COrd}\rightarrow\mathbf{Num}$

such that

    1. 1.

$\Psi(\sigma)=\mathfrak{num}\left(\\{\Psi(\tau)\mid\tau<\sigma\\}\right)$;

    2. 2.

$\Psi\left(\sigma\oplus\tau\right)=\Psi\left(\sigma\right)+\Psi\left(\tau\right);$

    3. 3.

$\Psi\left(\sigma\otimes\tau\right)=\Psi\left(\sigma\right)\cdot\Psi\left(\tau\right),$

where $\oplus$ and $\otimes$ denote the natural operations between ordinal
numbers (see Section 4.2).

  * •

(Consistency with the Lebesgue measure) if $E\subset\mathbb{R}$ is a Lebesgue
measurable set, then

$m_{L}\left(E\right)=st\left(\frac{\mathfrak{num}\left(E\right)}{\mathfrak{num}\left(\left[0,1\right)\right)}\right),$
(2)

where $m_{L}\left(E\right)$ denotes the Lebesgue measure of $E$ and $st(\xi)$
denotes the standard part of $\xi$.

To this aim we build a field which, following [9], we will call field of
Euclidean numbers. This field is isomorphic to the Keisler field and its
construction presents an extra structure that allows to build a numerosity
theory which satisfies, among others, the above requests.

##  2 The Euclidean numbers

In this section we introduce the field of Euclidean numbers. As we are going
to show, this is a hyperreal field constructed by means of a minor
modification of the usual superstructure construction, so to implement a
development of the theory of numerosity with certain useful peculiarities (see
Remark 7).

###  2.1 Non Archimedean fields

Here, we recall the basic definitions and some facts regarding non Archimedean
fields. In the following, ${\mathbb{K}}$ will denote an ordered field. We
recall that such a field contains (a copy of) the rational numbers. Its
elements will be called numbers.

######  Definition 1.

Let $\mathbb{K}$ be an ordered field. Let $\xi\in\mathbb{K}$. We say that:

  * •

$\xi$ is infinitesimal if, for all positive $n\in\mathbb{N}$,
$|\xi|<\frac{1}{n}$;

  * •

$\xi$ is finite if there exists $n\in\mathbb{N}$ such that $|\xi|<n$;

  * •

$\xi$ is infinite if, for all $n\in\mathbb{N}$, $|\xi|>n$ (equivalently, if
$\xi$ is not finite).

######  Definition 2.

An ordered field $\mathbb{K}$ is called non Archimedean if it contains an
infinitesimal $\xi\neq 0$.

Infinitesimal numbers can be used to formalize the notion of ”infinitely
close”:

######  Definition 3.

We say that two numbers $\xi,\zeta\in{\mathbb{K}}$ are infinitely close if
$\xi-\zeta$ is infinitesimal. In this case we write $\xi\sim\zeta$.

Clearly, the relation ”$\sim$” of infinite closeness is an equivalence
relation.

######  Theorem 4.

If $\mathbb{K\supset R}$ is an ordered field, then it is non Archimedean and
every finite number $\xi\in\mathbb{K}$ is infinitely close to a unique real
number $r\sim\xi$, called the the standard part of $\xi$.

The standard part can be regarded as a function:

$st:\left\\{x\in\mathbb{K}\mid x\ \text{is
finite}\right\\}\rightarrow\mathbb{R}.$ (3)

Moreover, with some abuse of notation, we can extend $st\ $to all $\mathbb{K}$
by setting

$st\left(\xi\right)=\left\\{\begin{array}[]{cc}+\infty&\text{if\ }\xi\
\text{is a positive infinite number;}\\\ -\infty&\text{if\ }\xi\ \text{is a
negative infinite number.}\end{array}\right.$

###  2.2 Construction of the Euclidean numbers

Given any set $E$ we let $\mathbb{V}(E)$ be the superstructure on $E$, namely
the family of sets which is inductively defined as follows:

$\displaystyle\mathbb{V}_{0}(E)$ $\displaystyle=$ $\displaystyle E;$
$\displaystyle\mathbb{V}_{n+1}(E)$ $\displaystyle=$
$\displaystyle\mathbb{V}_{n}(E)\cup\mathbf{\wp}\left(\mathbb{V}_{n}(E)\right);$
$\displaystyle\mathbb{V}(E)$ $\displaystyle=$
$\displaystyle\bigcup\limits_{n=0}^{\infty}\mathbb{V}_{n}(E).$

If an object $x\in\mathbb{V}_{n+1}(E)\setminus\mathbb{V}_{n}(E)$ we say that
its rank is $n+1$, and we write $rank(x)=n+1$. With the usual identifications
of pairs with Kuratowski pairs and functions and relations with their graphs,
we have that $\mathbb{V}(E)$ contains all the usual mathematical objects that
can be constructed from $E$. Moreover, notice that if $E$ is finite then also
each finite level $V_{n}(E)$ of the superstructure on $E$ is finite.

Now we let $\mathbb{A}$ be a set of atoms whose cardinality $\kappa$ is the
first strongly uncountable inaccessible cardinal number, and we assume that
$\mathbb{R}\subset\mathbb{A}$. The mathematical universe we will consider in
this paper is

$\Lambda=\left\\{E\in\mathbb{V}(\mathbb{A})\ |\ E\text{ is an atom or a set
such that}\ |E|<\kappa\right\\}$

where $|E|$ denotes the cardinality of $E.$

We let $\mathfrak{L}$ be the family of finite subsets of $\Lambda$:

$\mathfrak{L}=\mathbf{\wp}_{fin}(\Lambda).$

$\mathfrak{L}$, ordered by the inclusion relation $\subseteq$, is a directed
set; if $E$ is any set, we call net (with values in $E$) any function

$\varphi:\mathfrak{L}\rightarrow E.$

From now on, we will denote by $\sqsubseteq$ a partial order relation over
$\Lambda$ that extends the inclusion, namely such that
$\forall\lambda,\mu\in\mathfrak{L,}$

$\lambda\subseteq\mu\Rightarrow\lambda\sqsubseteq\mu.$

We assume that also $\left(\mathfrak{L},\sqsubseteq\right)$ is a directed set;
for the moment we will not make any other assumption on $E$. One of the main
task of this paper is to define $\sqsubseteq$ in such a way to get a
numerosity theory which satisfies the requests described in the introduction.

Let

$\mathfrak{F}\left(\mathfrak{L},\mathbb{R}\right)=\left\\{\varphi\in\mathbb{R}^{\mathfrak{L}}\
|\ \exists A\in\Lambda,\ \varphi\left(\lambda\cap
A\right)=\varphi\left(\lambda\right)\right\\}$

be endowed with the natural operations

$\displaystyle\left(\varphi+\psi\right)(\lambda)$ $\displaystyle=$
$\displaystyle\varphi(\lambda)+\psi(\lambda);$
$\displaystyle\left(\varphi\cdot\psi\right)(\lambda)$ $\displaystyle=$
$\displaystyle\varphi(\lambda)\cdot\psi(\lambda)$

and the partial ordering

$\varphi\geq\psi\Leftrightarrow\forall\lambda\in\mathfrak{L,\
}\varphi(\lambda)\geq\psi(\lambda).$

The field of Euclidean numbers is defined as follows444This construction can
be seen as an extension of $\alpha$-theory, see e.g. [3].:

######  Definition 5.

The field of Euclidean numbers $\mathbb{E\supset R}$ is a field so that there
exists a surjective map

$J:\mathfrak{F}\left(\mathfrak{L},\mathbb{R}\right)\rightarrow\mathbb{E}$

with the following properties:

  1. (i)

Ring homomorphism: $J$ is a ring homomorphism, namely for all
$\varphi,\psi\in\mathfrak{F}\left(\mathfrak{L},\mathbb{R}\right)$

     * •

$J\left(\varphi+\psi\right)=J\left(\varphi\right)+J\left(\psi\right);$

     * •

$J\left(\varphi\cdot\psi\right)=J\left(\varphi\right)\cdot
J\left(\psi\right).$

  2. (ii)

Monotonicity: for all
$\varphi\in\mathfrak{F}\left(\mathfrak{L},\mathbb{R}\right)$, for all
$r\in\mathbb{R}$, if eventually $\varphi(\lambda)\geq r$ (namely there exists
$\lambda_{0}\in\mathfrak{L}$ such that $\forall\lambda\sqsupseteq\lambda_{0},\
\varphi(\lambda)\geq r$), then

$J\left(\varphi\right)\geq r.$

Let us show that such a field exists555Readers with a basic knowledge of
nonstandard analysis will recognize immediately that our construction is a
minor modification of the usual limit ultrapower construction..

###### Proof.

Let $\mathcal{U}$ be a fine ultrafilter on $\mathfrak{L}$, namely a filter of
sets such that

  * •

Maximality: $Q\in\mathcal{U}\Leftrightarrow\mathfrak{L}\backslash
Q\notin\mathcal{U}$;

  * •

Finess: $\forall\lambda\in\mathfrak{L},\ Q\left[\lambda\right]\in\mathcal{U}$,
where

$Q\left[\lambda\right]:=\left\\{\mu\in\mathfrak{L}\ |\
\mu\sqsupseteq\lambda\right\\}.$ (4)

The existence of $\mathcal{U}$ is a well known and easy consequence of Zorn’s
Lemma. We use $\mathcal{U}$ to introduce an equivalence relation on nets, by
letting for all
$\psi,\varphi\in\mathfrak{F}\left(\mathfrak{B},\mathbb{R}\right)$

$\varphi\approx_{\mathcal{U}}\psi\Longleftrightarrow\exists Q\in\mathcal{U}\
\forall\lambda\in Q,\ \varphi\left(\lambda\right)=\psi\left(\lambda\right).$

We set

$\widetilde{\mathbb{E}}:=\mathfrak{F}\left(\mathfrak{L},\mathbb{R}\right)/\approx_{\mathcal{U}}$

and we denote by $\left[\varphi\right]_{\mathcal{U}}$ the equivalence classes.
Now we take a injective map

$\Phi:\widetilde{\mathbb{E}}\rightarrow\mathbb{A}$

such that $\forall r\in\mathbb{R}$,

$\Phi\left(\left[c_{r}\right]_{\mathcal{U}}\right)=r$

were $c_{r}$ is the net identically equal to $r.$ Finally we set

$\mathbb{E}=\Phi\left(\widetilde{\mathbb{E}}\right).$

The operations on $\mathbb{E}$ can be easily defined by letting

$\Phi\left(\left[\varphi\right]_{\mathcal{U}}\right)+\Phi\left(\left[\psi\right]_{\mathcal{U}}\right)=\Phi\left(\left[\varphi+\psi\right]_{\mathcal{U}}\right);\
\ \
\Phi\left(\left[\varphi\right]_{\mathcal{U}}\right)\cdot\Phi\left(\left[\psi\right]_{\mathcal{U}}\right)=\Phi\left(\left[\varphi\cdot\psi\right]_{\mathcal{U}}\right).$

It is very well known (see e.g. [19]) and simple to show that, thanks to
$\mathcal{U}$ being an ultrafilter, $\mathbb{E}$ endowed with the above
operations is a field; moreover, it can be made an ordered field by endowing
it with the following ordering:

$\forall\varphi,\psi\in\mathfrak{F}\left(\mathfrak{L},\mathbb{R}\right),\
\Phi\left(\left[\varphi\right]_{\mathcal{U}}\right)\geq\Phi\left(\left[\psi\right]_{\mathcal{U}}\right):\Longleftrightarrow\exists
Q\in\mathcal{U},\ \forall\lambda\in Q\ \
\varphi\left(\lambda\right)\geq\psi\left(\lambda\right).\hskip
5.0pt\hbox{\hskip 5.0pt\vrule width=4.0pt,height=6.0pt,depth=1.5pt\hskip
1.0pt}$

######  Remark 6.

$\mathbb{E}$ is an hyperreal field whose cardinality is $\kappa$; such a field
is unique up to isomorphisms (see [19]); namely, changing ”$\sqsubseteq$” we
get an isomorphic hyperreal field. However, we will choose ”$\sqsubseteq$” in
such a way to get interesting interactions with other mathematical structures.

The number $J\left(\varphi\right)$ is called the $\Lambda$-limit of the net
$\varphi$ and will be denoted by

$J\left(\varphi\right)=\lim_{\lambda\uparrow\Lambda}\varphi(\lambda).$

The reason of this name and notation is that the operation

$\varphi\mapsto\lim_{\lambda\uparrow\Lambda}\varphi(\lambda)$

satisfies many of the properties of the usual Cauchy limit, but with the
stronger property of existing for every net. More exactly, it satisfies the
following properties:

  * •

Existence: Every net $\varphi:\mathfrak{L}\rightarrow\mathbb{R}$ has a unique
limit $L\in\mathbb{E}$.

  * •

Monotonicity: For all $r\in\mathbb{R}$ if eventually $\varphi(\lambda)\geq r$,
then

$\lim_{\lambda\uparrow\Lambda}\varphi(\lambda)\geq r;$

  * •

Sum and product: For all $\varphi,\psi:\mathfrak{L}\rightarrow\mathbb{R}$

$\displaystyle\lim_{\lambda\uparrow\Lambda}\varphi(\lambda)+\lim_{\lambda\uparrow\Lambda}\psi(\lambda)$
$\displaystyle=$
$\displaystyle\lim_{\lambda\uparrow\Lambda}\left(\varphi(\lambda)+\psi(\lambda)\right),$
$\displaystyle\lim_{\lambda\uparrow\Lambda}\varphi(\lambda)\cdot\lim_{\lambda\uparrow\Lambda}\psi(\lambda)$
$\displaystyle=$
$\displaystyle\lim_{\lambda\uparrow\Lambda}\left(\varphi(\lambda)\cdot\psi(\lambda)\right).$

Notice that, if $\lim_{\lambda\rightarrow\Lambda}\varphi(\lambda)$ denotes the
usual Cauchy limit of $\varphi$, the relationship between the Cauchy limit and
the $\Lambda$-limit is

$\lim_{\lambda\rightarrow\Lambda}\varphi(\lambda)=st\left(\lim_{\lambda\uparrow\Lambda}\varphi(\lambda)\right).$

######  Remark 7.

The notion of Eucliean field defined by Definition 5 has been used in several
papers with ”$\subseteq$” instead of ”$\sqsubseteq$” (e.g. [2], [11], [12]).
Now, we will explain the main technical reason for using an Euclidean field
rather than a ”generic” hyperreal field. A set $F\subset\Lambda^{\ast}$ is
called hyperfinite if

$F=\lim_{\lambda\uparrow\Lambda}F_{\lambda}=\left\\{\lim_{\lambda\uparrow\Lambda}\
\varphi(\lambda)\ |\ \varphi(\lambda)\in F_{\lambda}\right\\}$

where the sets $F_{\lambda}\in\Lambda$ are finite. Hyperfinite sets play a
crucial role in many applications of nonstandard analysis. If we use a
Euclidean field, we can associate to every set $E\in\Lambda$ a unique
hyperfinite set $E^{\circledast}$ defined as follows

$E^{\circledast}=\lim_{\lambda\uparrow\Lambda}E\cap\lambda=\left\\{\lim_{\lambda\uparrow\Lambda}\
\varphi(\lambda)\ |\ \varphi(\lambda)\in E\cap\lambda\right\\}.$

The set $E^{\circledast}$ satisfies the property666Here, as usual, we have set
$E^{\sigma}=\left\\{x^{\ast}\ |\ x\in E\right\\}.$

$E^{\sigma}\subset E^{\circledast}\subset E^{\ast}$

which is very useful in the applications. Moreover, using an Euclidean field
we can easily define the numerosity function over any set $E\in\Lambda$ by
setting (see Section 3)

$\mathfrak{num}\left(E\right):=\lim_{\lambda\uparrow\Lambda}\
\left|E\cap\lambda\right|.$ (5)

In this paper, replacing with ”$\subseteq$” with a suitable ”$\sqsubseteq$”,
the numerosity theory given by Equation (5) is consistent with the main
features of the numerosity theories present in the litterature (e.g. [1], [4],
[5], [6], [7], [8], [9], [14], [16], [17], [21]).

###  2.3 Labelled sets

The notion of labelled set has been introduced in [1] and [5] to construct a
numerosity theory for countable sets. Here we extend this notion to adapt it
to the study of numerosity theories for larger sets.

######  Definition 8.

We call label set a family of sets $\mathfrak{B}\subset\mathfrak{L}$ such that

  * (i)

$\forall\mathfrak{s,t}\in\mathfrak{B,}\ \mathfrak{s}\cap\mathfrak{t},\
\mathfrak{s}\cup\mathfrak{t}\in\mathfrak{B}$;

  * (ii)

$\forall\mathfrak{s}\in\mathfrak{B},\
\mathfrak{s}\cap\mathfrak{L}=\varnothing;$

  * (iii)

$\bigcup_{\mathfrak{s}\in\mathfrak{B}}\mathbb{V}\left(\mathfrak{s}\right)=\Lambda$.

Requirement (i) gives to $\mathfrak{B}$ a lattice structure, whilst
requirement (ii) entails that the elements of a label are either atoms or
infinite sets.

Having fixed the notion of ”labels”, we can now introduce the notion of
”labelling”:

######  Definition 9.

Let $\mathfrak{B}$ be a set of labels. We call $\mathfrak{B}$-labelling the
map

$\ell:\Lambda\rightarrow\mathfrak{B}$

defined as follows:

$\ell(a)=\bigcap_{\mu\in I_{a}}\mu,$

where $I_{a}=\\{\mu\in\mathfrak{B}\mid a\in\mathbb{V}(\mu)\\}$. For every
$a\in\mathfrak{L}$ we call $\ell(a)$ the label of $a$.

Roughly speaking, the label of an object $a\in\Lambda$ is a finite set whose
elements allow to define $a$ by the fundamental finitistic set operations.

There is plenty of sets of labels: just set

$\mathfrak{B}_{\max}=\left\\{\mathfrak{t}\in\mathfrak{L}\mid\
\mathfrak{t}\cap\mathfrak{L}=\emptyset\right\\}.$ (6)

Obviously, $\mathfrak{B}_{\max}$ is a label set; more importantly, every label
set $\mathfrak{B}$ is a subset of $\mathfrak{B}_{\max}$.

######  Example 10.

Let $a=\mathbb{N}$. Then, using the $\mathfrak{B}_{\max}$-labelling,

$\ell\left(\mathbb{N}\right)=\\{\mathbb{N}\\}.$

Now we will describe some properties of a set of labels $\mathfrak{B}$ and the
corresponding $\mathfrak{B}$-labelling:

######  Proposition 11.

Let $\mathfrak{B}$ be a set of labels, and let $\mathfrak{s}\in\mathfrak{B}$.
Then

  1. $(i)$

$\mathbb{V}(\mathfrak{s})$ is countable;

  2. $(ii)$

$\mathbb{V}(\mathfrak{s})\setminus\mathfrak{s}$ consists only of finite sets;

  3. $(iii)$

for all $\mathfrak{s},\mathfrak{t}\in\mathfrak{B},$ for all $m\in\mathbb{N}$
we have that

$\mathbb{V}_{m}(\mathfrak{s})\subseteq\mathbb{V}_{m}(\mathfrak{t})\Leftrightarrow\mathfrak{s}\subseteq\mathfrak{t};$

  4. $(iv)$

for all $\mathfrak{s},\mathfrak{t}\in\mathfrak{B},$
$\mathbb{V}(\mathfrak{s}\cap\mathfrak{t})=\mathbb{V}(\mathfrak{s})\cap\mathbb{V}(\mathfrak{t})$;

  5. $(v)$

If $a\in\mathfrak{L}$ and $\mathfrak{t}\in\mathfrak{B}\mathfrak{,\ }$then
$\left\\{a\right\\}\in\mathbb{V}(\mathfrak{t})\Leftrightarrow
a\in\mathbb{V}(\mathfrak{t})$.

###### Proof.

(i) $\mathfrak{s}$ is finite, hence by induction it trivially holds that
$\mathbb{V}_{n}\left(\mathfrak{s}\right)$ is finite for every
$n\in\mathbb{N}$. Therefore $\mathbb{V}\left(\mathfrak{s}\right)$ is
countable.

(ii) As $\mathfrak{s}$ is finite, by induction it is immediate to prove that
$\mathbb{V}_{n+1}(\mathfrak{s})\setminus\mathfrak{s}$ consists only of finite
sets, hence the thesis follows straightforwardly.

(iii) The implication $\Leftarrow$ is trivial. Let us prove the other
implication. If
$\mathbb{V}_{m}(\mathfrak{s})\subseteq\mathbb{V}_{m}(\mathfrak{t})$ then, in
particular, $\mathfrak{s}\in\mathbb{V}_{m}(\mathfrak{t})$. Now, if
$\mathfrak{s}=\\{a_{1},\dots,a_{n}\\}$, all $a_{1},\dots,a_{n}$ are either
atoms or infinite sets, hence by (i), $a_{1},\dots,a_{n}\in\mathfrak{t}$.

(iv) The inclusion $\subseteq$ is trivial. For the reverse inclusion, let
$\eta\in\mathbb{V}(\mathfrak{s})\cap\mathbb{V}(\mathfrak{t})$. In particular,
$\eta\in\mathbb{V}_{n}(\mathfrak{s})\cap\mathbb{V}_{m}(\mathfrak{t})$ and so,
if $l=\max\\{n,m\\}$, we have that
$\eta\in\mathbb{V}_{l}(\mathfrak{s})\cap\mathbb{V}_{l}(\mathfrak{t})$. We
proceed by induction on $l$ to show that
$\mathbb{V}_{l}(\mathfrak{s})\cap\mathbb{V}_{l}(\mathfrak{t})\subseteq\mathbb{V}(\mathfrak{s}\cap\mathfrak{t})$.

If $l=0$, then
$\eta\in\mathbb{V}_{0}(\mathfrak{s})\cap\mathbb{V}_{0}(\mathfrak{t})$ if and
only if $\eta=\mathfrak{s}=\mathfrak{t}$, and the desired inclusion trivially
holds.

Now let us suppose the inclusion to hold for $l\in\mathbb{N}$, and let
$\eta\in\mathbb{V}_{l+1}(\mathfrak{s})\cap\mathbb{V}_{l+1}(\mathfrak{t})$. If
$\eta\in\mathbb{V}_{l}(\mathfrak{s})\cap\mathbb{V}_{l}(\mathfrak{t})$ we are
done by inductive hypothesis; if not, there are
$A\in\mathbb{V}_{l}(\mathfrak{s}),B\in\mathbb{V}_{l}(\mathfrak{t})$ such that
$\eta\in\wp(A)\cap\wp(B)$. In particular, $\eta\in\wp(A\cap B)$. But $A\cap
B\in\mathbb{V}_{l}(\mathfrak{s})\cap\mathbb{V}_{l}(\mathfrak{t})$, so by
induction $A\cap B\in\mathbb{V}(\mathfrak{s}\cap\mathfrak{t})$, hence
$\eta\in\mathbb{V}(\mathfrak{s}\cap\mathfrak{t})$ as desired.

(v) The implication $\Leftarrow$ is trivial. For the reverse implication, let

$l=\min\\{n\in\mathbb{N}\mid a\in\mathbb{V}_{n}(\mathfrak{t})\\}.$

In particular, we have that $l\geq 1$. In fact, if $l=0$ then
$\\{a\\}=\mathfrak{t}$, and this cannot happen as
$\mathfrak{t}\cap\mathfrak{L}=\emptyset$. Hence
$a\in\mathbb{V}_{l-1}(\mathfrak{t})$ and we are done.

As we will see in Section 5, the freedom of choosing a particular set of
labels allows to impose certain additional arithmetical properties on
numerosities.

######  Proposition 12.

Let $\mathfrak{B}$ be a set of labels and let $\ell$ be a
$\mathfrak{B}$-labelling. The following properties hold:

  1. $(i)$

$\forall a,b\in\Lambda$, $a\subseteq b\Rightarrow\ell(a)\subseteq\ell(b);$

  2. $(ii)$

$\forall a\in\Lambda,$ $\ell(a)\supseteq\ell(\\{a\\}),$ and equality holds if
$a\in\mathfrak{L}$;

  3. $(iii)$

$\forall a,b\in\Lambda,$ $a\in b\Rightarrow\ell(\\{a\\})\subseteq\ell(b);$

  4. $(iv)$

$\forall\lambda\in\mathfrak{B},$
$\lambda\in\mathbb{V}\left(\ell(\lambda)\right);$

  5. $(v)$

$\forall\mathfrak{s}\in\mathfrak{B},\
\ell\left(\mathfrak{s}\right)=\mathfrak{s};$

  6. $(vi)$

$\forall\mathfrak{s}\in\mathfrak{B},\forall m\in\mathbb{N},$
$\ell\left(\mathbb{V}_{m}(\mathfrak{s})\right)=\mathfrak{s};$

  7. $(vii)$

$\forall a\in\Lambda,\forall\mathfrak{s}\in\mathfrak{B},$
$\ell\left(a\right)\subseteq\mathfrak{s}\Leftrightarrow
a\in\mathbb{V}(\mathfrak{s})$;

  8. $(viii)$

$\forall a,b\in\Lambda,$
$\ell\left(\\{a,b\\}\right)=\ell\left(a\right)\cup\ell(b)$;

  9. $(ix)$

$\forall a,b\in\Lambda,$
$\ell\left(\left(a,b\right)\right)=\ell(a)\cup\ell(b)$.

  10. $(x)$

$\forall E\in\Lambda$, $\forall\lambda\in\mathfrak{L}$ if we set

$E_{\lambda}:=\left\\{x\in E\ |\
\ell\left(x\right)\subseteq\ell\left(\lambda\right)\right\\}$

then

$E_{\lambda}=E\cap\mathbb{V}(\ell\left(\lambda\right));$

  11. $(xi)$

$\forall E\in\Lambda,$ the set $E_{\lambda}$ is finite.

###### Proof.

(i) If $a\subseteq b,$ then trivially $a\in\mathbb{V}(\mathfrak{t})$ whenever
$b\in\mathbb{V}(\mathfrak{t})$ and hence

$\ell(a)=\bigcap\left\\{\mathbb{V}(\mathfrak{t})\ |\
\mathfrak{t}\in\mathfrak{B},\
a\in\mathbb{V}(\mathfrak{t})\right\\}\subseteq\bigcap\left\\{\mathbb{V}(\mathfrak{t})\
|\ \mathfrak{t}\in\mathfrak{B},\
b\in\mathbb{V}(\mathfrak{t})\right\\}=\ell(b).$

(ii) Trivially, for all $\mathfrak{t}\in\mathfrak{B}$, if
$a\in\mathbb{V}(\mathfrak{t})$ then also $\\{a\\}\in\mathbb{V}(\mathfrak{t})$,
hence $\ell(\\{a\\})\subseteq\ell(\\{a\\}$. The second claim follows from the
fact that, by Proposition 11.(v),
$a\in\mathbb{V}(\mathfrak{t})\Leftrightarrow\\{a\\}\in\mathbb{V}(\mathfrak{t})$.

(iii) If $a\in b,$ then $\left\\{a\right\\}\subseteq b$ and by (i) and (ii),
we have that

$\ell(a)=\ell(\left\\{a\right\\})\subseteq\ell(b).$

(iv) By definition, $\forall\mu\in I_{a},$ $a\in\mathbb{V}(\mu);$ hence

$a\in\bigcap_{\mu\in I_{a}}\mathbb{V}(\mu)=\mathbb{V}\left(\bigcap_{\mu\in
I_{a}}\mu\right)=\mathbb{V}\left(\ell(a)\right).$

(v) We have that
$\mathfrak{s}=\mathbb{V}_{0}(\mathfrak{s})\in\mathbb{V}(\mathfrak{s});$ hence
$\mathfrak{s}\in I_{\mathfrak{s}}$ and so
$\ell(\mathfrak{s})\subseteq\mathfrak{s}.$ Moreover, if $\mathfrak{t}\in
I_{\mathfrak{s}}$, $\mathfrak{s}\in\mathbb{V}(\mathfrak{t})$ and since
$\mathfrak{s}$ is a label, $\mathfrak{s}\subset\mathfrak{t}$ and so

$\mathfrak{s}\subseteq\bigcap_{\mathfrak{t}\in
I_{\mathfrak{s}}}\mathfrak{t}=\ell(\mathfrak{s}).$

(vi) If $\mathfrak{s}\in\mathfrak{B},$ then
$\forall\mathfrak{t}\in\mathfrak{B},$ by Proposition 11.(iv) we have that

$\mathbb{V}_{m}(\mathfrak{s})\in\mathbb{V}(\mathfrak{t})\Leftrightarrow\mathfrak{s}\subseteq\mathfrak{t}\Leftrightarrow\mathfrak{s}\in\mathbb{V}(\mathfrak{t}).$

Then

$\displaystyle I_{\mathbb{V}_{m}(\mathfrak{s})}$ $\displaystyle=$
$\displaystyle\\{\mathfrak{t}\in\mathfrak{B}\mid\mathbb{V}_{m}(\mathfrak{s})\in\mathbb{V}(\mathfrak{t})\\}=\\{\mathfrak{t}\in\mathfrak{B}\mid\mathfrak{s}\subseteq\mathfrak{t}\\}$
$\displaystyle=$
$\displaystyle\\{\mathfrak{t}\in\mathfrak{B}\mid\mathfrak{s}\in\mathbb{V}(\mathfrak{t})\\}=I_{\mathfrak{s}},$

hence the thesis follows by the definition of $\mathfrak{B}$-labeling.

(vii) ($\Rightarrow$) If $\ell\left(a\right)\subseteq\mathfrak{s}$, then, by
(i),
$\mathbb{V}\left(\ell\left(a\right)\right)\subseteq\mathbb{V}\left(\mathfrak{s}\right);$
therefore, by (iv),$\ a\in\mathbb{V}\left(\mathfrak{s}\right).$

($\Leftarrow$) If $a\in\mathbb{V}(\mathfrak{s}),$ then $\mathfrak{s}\in I_{a}$
and so $\ell(a)\subseteq\mathfrak{s}$.

(viii) For $\mathfrak{t}\in I_{\left\\{a,b\right\\}},$ since
$\left\\{a,b\right\\}\in\mathbb{V}(\mathfrak{t})$, but
$\left\\{a,b\right\\}\notin\mathfrak{t}$, we have that
$a\in\mathbb{V}(\mathfrak{t})$ and $b\in\mathbb{V}(\mathfrak{t});$ then

$\displaystyle I_{\left\\{a,b\right\\}}$ $\displaystyle=$
$\displaystyle\\{\mathfrak{t}\in\mathfrak{B}\mid a\in\mathbb{V}(\mathfrak{t})\
\text{and\ }b\in\mathbb{V}(\mathfrak{t})\\}$ $\displaystyle=$
$\displaystyle\\{\mathfrak{t}\in\mathfrak{B}\mid
a\in\mathbb{V}(\mathfrak{t})\\}\cap\\{\mathfrak{t}\in\mathfrak{B}\mid
b\in\mathbb{V}(\mathfrak{t})\\}=I_{a}\cap I_{b}.$

Hence

$\displaystyle\ell(\left\\{a,b\right\\})$ $\displaystyle=$
$\displaystyle\bigcap_{\mathfrak{t}\in
I_{\left\\{a,b\right\\}}}\mathfrak{t}=\bigcap_{\mathfrak{t}\in I_{a}\cap
I_{b}}\mathfrak{t}=\\{x\in\Lambda\mid x\in\mathfrak{t}\ \ \text{and}\ \
\mathfrak{t}\in I_{a}\cap I_{b}\\}$ $\displaystyle=$
$\displaystyle\\{x\in\Lambda\mid(x\in\mathfrak{t}\ \text{and}\ \mathfrak{t}\in
I_{a})\ \text{or\ }(x\in\mathfrak{t}\ \text{and}\ \mathfrak{t}\in I_{b})\\}$
$\displaystyle=$ $\displaystyle\\{x\in\Lambda\mid(x\in\mathfrak{t}\
\text{and}\ \mathfrak{t}\in I_{a})\\}\cup\\{x\in\Lambda\mid(x\in\mathfrak{t}\
\text{and}\ \mathfrak{t}\in I_{b})\\}$ $\displaystyle=$
$\displaystyle\left(\bigcap_{\mathfrak{t}\in
I_{a}}\mathfrak{t}\right)\cup\left(\bigcap_{\mathfrak{t}\in
I_{b}}\mathfrak{t}\right)=\ell(a)\cup\ell(b).$

(ix) We have that

$\ell\left(\left(a,b\right)\right)=\ell\left(\\{a,\\{a,b\\}\\}\right)=\ell\left(a\right)\cup\ell\left(\\{a,b\\}\right)=\ell\left(b\right)\ell\left(a\right)\cup\ell\left(a\right)\cup\ell\left(b\right)=\ell\left(a\right)\cup\ell\left(b\right).$

(x) First set $\mathfrak{s=\ell}\left(\lambda\right).$ Let us first prove the
inclusion $\subseteq$. Let $x\in E$ be such that
$\ell(x)\subseteq\mathfrak{s}$. By the definition of labelling then
$x\in\mathbb{V}(\mathfrak{s})$, and the inclusion is proven. For the reverse
inclusion, let $x\in E\cap\mathbb{V}(\mathfrak{s})$. In particular, it must be
$\ell(x)\subseteq\mathfrak{s}$, and we are done.

(xi) If $E\in\Lambda$ then $E$ has a finite rank $n$, which means that
$E\cap\mathbb{V}(\mathfrak{s})=E\cap\mathbb{V}_{n}(\mathfrak{s})$, and the
conclusion follows by (x) as, by construction, $\mathbb{V}_{n}(\mathfrak{s})$
is finite.

The notion of $\mathfrak{B}$-labelling allows to equip $\mathfrak{L}$ with a
partial order structure $\sqsubseteq$:

######  Definition 13.

We set

$\mathfrak{L}_{0}\left(\mathfrak{B}\right):=\\{\mathbb{V}_{m}(\mathfrak{t})\
|\ \ m\in\mathbb{N}_{0},\ \mathfrak{t}\in\mathfrak{B}\\}$

and for every $\lambda,\mu\in\mathfrak{L}$, we set

$\lambda\sqsubseteq\mu\Leftrightarrow\lambda\subseteq\mathop{\displaystyle\bigcap}\left\\{\tau\in\mathfrak{L}_{0}\mathfrak{\
}|\ \mu\subseteq\tau\right\\}.$

Notice that, by definition

$\lambda\sqsubseteq\mu\Leftrightarrow I_{\lambda}\subseteq I_{\mu},$

where $I_{\lambda}$ has been introduced in Definition 9, and that

$\mathfrak{L}_{0}\left(\mathfrak{L_{0}}\left(\mathfrak{B}\right)\right)=\mathfrak{L}_{0}\left(\mathfrak{B}\right).$

Clearly $\sqsubseteq$ induces a lattice structure on
$\mathfrak{L}_{0}\left(\mathfrak{B}\right)$, since

$\lambda\vee\mu:=\mathop{\displaystyle\bigcap}\left\\{\tau\in\mathfrak{L}_{0}\mathfrak{\
}|\ \lambda\cup\mu\subseteq\tau\right\\};$
$\lambda\wedge\mu:=\mathop{\displaystyle\bigcup}\left\\{\tau\in\mathfrak{L}_{0}\mathfrak{\
}|\ \tau\subseteq\lambda\cap\mu\right\\}.$

Since $\lambda\subseteq\mu\Rightarrow\lambda\sqsubseteq\mu,$ we can use the
directed set
$\left(\mathfrak{L}_{0}\left(\mathfrak{B}\right),\sqsubseteq\right)$ to define
a field of Euclidean numbers as in Definition 5. From now on, $\mathbb{E}$
will denote such a field.

######  Remark 14.

Now the idea is to construct a suitable set of labels in such a way that the
relation $\sqsubseteq$ carry all the informations needed for a ”good”
numerosity theory. All these informations depend on $\sqsubseteq$ and not on
the choice of the ultrafilter used in the construction of $\mathbb{E}$.

##  3 The general theory of numerosities

Different versions of the notion of numerosity have already been studied in
several previous papers [1, 5, 6, 7, 8, 14, 16, 17]; we refer also to the book
[4] for a complete overview of the countable case. In this paper, we want to
show how the new definition of labels and of the Euclidean field allows to
easily provide the most interesting features of the theory of numerosities. In
particular, we show how numerosities can be used to simultaneously unify and
generalize objects and results coming from different areas, like (a version
of) Lagrange’s Theorem for groups, the Peano-Jordan measure and the Lebesgue
measure.

###  3.1 Definition and first properties

######  Definition 15.

Let $E$ be a set in $\Lambda$. We call numerosity of $E$ the Euclidean number

$\mathfrak{num}\left(E\right):=\lim_{\lambda\uparrow\Lambda}\
\left|E\cap\lambda\right|.$

The set of numerosities will be denoted by $\mathbf{Num}$.

The notion of numerosity allows to ”give a name” to some hyperreal number. We
set

$\alpha=\mathfrak{num}\left(\mathbb{N}\right);\
\beta=\mathfrak{num}\left(\left[0,1\right)\right).$ (8)

The numerosity of a set depends on the choice of the set of labels
$\mathfrak{B}$, as well as on the ultrafilter $\mathcal{U}$ on $\mathfrak{B}$
chosen to construct $\mathbb{E}$. However the properties which will be listed
below are independent of any choice.

######  Theorem 16.

Let $E,F$ be sets in $\Lambda$. Numerosities satisfy the following properties:

  1. $(i)$

Finite sets principle: if $E$ is a finite set, then
$\mathfrak{num}\left(E\right)=|E|$;

  2. $(ii)$

Euclid’s principle: if $E\subset F$ then
$\mathfrak{num}\left(E\right)<\mathfrak{num}\left(F\right)$;

  3. $(iii)$

Labels principle: if

$E_{\lambda}=\left\\{x\in E\ |\
\ell\left(x\right)\subseteq\ell\left(\lambda\right)\right\\}.$

then, if $\lambda\in\mathfrak{L}_{0}\left(\mathfrak{B}\right)$,
$E_{\lambda}=E\cap\lambda$ and hence

$\mathfrak{num}\left(E\right)=\lim_{\lambda\uparrow\Lambda}\
\left|E_{\lambda}\right|;$

  4. $(iv)$

Comparison principle: if $\Phi:E\rightarrow F$ is a bijection that preserves
labels, namely such that for all $x\in E$

$\ell(\Phi\left(x\right))=\ell\left(x\right),$

then $\mathfrak{num}\left(E\right)=\mathfrak{num}\left(F\right)$;

  5. $(v)$

Sum principle: if $E\cap F=\varnothing$ then $\mathfrak{num}\left(E\cup
F\right)=\mathfrak{num}\left(E\right)+\mathfrak{num}\left(F\right);$

  6. $(vi)$

Product principle: $\mathfrak{num}\left(E\times
F\right)=\mathfrak{num}\left(E\right)\cdot\mathfrak{num}\left(F\right)$;

  7. $(vii)$

Finite parts principle:
$\mathfrak{num}\left(\mathbf{\wp}_{fin}\left(E\right)\right)=2^{\mathfrak{num}\left(E\right)}$;

  8. $(viii)$

Finite functions principle: let $E$ be nonempty, and

$\mathfrak{F}_{fin}\left(X,E\right):=\left\\{f:D\rightarrow E\mid
D\in\mathbf{\wp}_{fin}(X)\right\\}.$

Then, if $a\in E$, we have

$\mathfrak{num}\left(\mathfrak{F}_{fin}\left(X,E\backslash\left\\{a\right\\}\right)\right)=\mathfrak{num}\left(E\right)^{\mathfrak{num}(X)}.$

###### Proof.

(i) If $|E|=n<\infty$, then for every $\lambda\in\mathfrak{L}$, we have
$\left|E\cap\lambda\right|=n$, and the thesis then follows by taking the
$\lambda$-limit.

(ii) If $E\subset F$, eventually
$\left|E\cap\lambda\right|<\left|F\cap\lambda\right|$, so
$\lim_{\lambda\uparrow\Lambda}\left|E\cap\lambda\right|<\lim_{\lambda\uparrow\Lambda}\left|E\cap\lambda\right|$.

(iii) Take $\lambda=\mathbb{V}_{m}(\mathfrak{s})$ with $m\geq rank(E)$ and
$\mathfrak{s\in B}$. Then, by Proposition 12.$(x)$

$E\cap\lambda=E\cap\mathbb{V}_{m}(\mathfrak{s})=E\cap\mathbb{V}(\mathfrak{s})=E_{\lambda}.$

(iv) By hypothesis we have that for all $\lambda\in\mathfrak{L}$
$|E_{\lambda}|=|F_{\lambda}|$, and so by the labels principle

$\mathfrak{num}\left(E\right)=\lim_{\lambda\uparrow\Lambda}\
\left|E\cap\lambda\right|=\lim_{\lambda\uparrow\Lambda}\
\left|F\cap\lambda\right|=\mathfrak{num}\left(F\right).$ (9)

(v) Just notice that $|E\cup F|_{\lambda}=|E_{\lambda}|+|F_{\lambda}|$ for
every $\lambda\in\mathfrak{L}$, hence the thesis follows by Definition 5.(2)
and by the labels principle.

(vi) Let $\lambda\in\mathfrak{L}$. By property (ix) in Proposition 12, we have
that $\left(E\times F\right)_{\lambda}=E_{\lambda}\times F_{\lambda}$, hence
$|\left(E\times F\right)_{\lambda}|=|E_{\lambda}\times
F_{\lambda}|=|E_{\lambda}|\cdot|F_{\lambda}|$, and the thesis then follows
immediately, again by the labels principle.

(vii) Let $\lambda=\mathbb{V}_{m}(\mathfrak{s})\in\mathfrak{L}_{0}$
($m>rank(E))$, and let
$a\in\mathbf{\wp}_{fin}\left(E\right)\cap\mathbb{V}(\mathfrak{s})$. Then by
Proposition 12.(x) we have that it must be
$a\in\mathbf{\wp}_{fin}\left(E_{\lambda}\right)$. Conversely, if
$a\in\mathbf{\wp}_{fin}\left(E_{\lambda}\right)$ it is immediate to see that
$a\in\mathbf{\wp}_{fin}\left(E\right)\cap\mathbb{V}(\mathfrak{s})$. Hence, by
Proposition 12 we have

$\left|\left[\mathbf{\wp}_{fin}\left(E\right)\right]_{\lambda}\right|=\left|\mathbf{\wp}_{fin}\left(E\right)\cap\mathbb{V}(\mathfrak{s})\right|=\left|\mathbf{\wp}_{fin}\left(E_{\lambda}\right)\right|=2^{\left|E_{\lambda}\right|},$

and so by the labels principle

$\mathfrak{num}\left(\mathbf{\wp}_{fin}\left(E\right)\right)=\lim_{\lambda\uparrow\Lambda}\
2^{\left|E_{\lambda}\right|}=2^{\mathfrak{num}\left(E\right)}.$

(viii) We set $\lambda=\mathbb{V}_{m}(\mathfrak{s})\in\mathfrak{L}_{0}$,
$m>rank(f).$ Let
$f\in\mathfrak{F}_{fin}\left(X,E\backslash\\{a\\}\right)\cap\mathbb{V}(\mathfrak{s})$,
and let $D$ be the domain of $f$. By identifying functions with Kuratowski
pairs, and by our definition of labellings on pairs, it is immediate to see
that
$f\in\mathfrak{F}_{fin}\left(X,E\backslash\\{a\\}\right)\cap\mathbb{V}(\mathfrak{s})$
if and only if $D\left(f\right)\subset
X\cap\mathbb{V}(\mathfrak{s})=X_{\lambda}$ and
$Im\left(f\right)\subset\left(E\backslash\\{a\\}\right)\cap\mathbb{V}(\mathfrak{s})=E_{\lambda}\backslash\\{a\\}.$
Therefore

$\mathfrak{F}_{fin}\left(X,E\backslash\\{a\\}\right)\cap\mathbb{V}(\mathfrak{s})=\mathfrak{F}_{fin}\left(X_{\lambda},E_{\lambda}\backslash\\{a\\}\right).$

Notice that

$\left|\mathfrak{F}_{fin}\left(X_{\lambda},E_{\lambda}\backslash\\{a\\}\right)\right|=\left|\mathfrak{F}\left(X_{\lambda},E_{\lambda}\right)\right|.$

In fact, the association
$g\in\mathfrak{F}_{fin}\left(X_{\lambda},E_{\lambda}\backslash\\{a\\}\right)\rightarrow\widetilde{g}\in\mathfrak{F}\left(X_{\lambda},E_{\lambda}\right)$,
with

$\widetilde{g}(x)=\begin{cases}g(x),&\text{if}\ x\in X_{\lambda};\\\
a,&\text{otherwise}\end{cases}$

is a bijection. Hence, again by the labels principle,

$\displaystyle\mathfrak{num}\left(\mathfrak{F}_{fin}\left(X,E\backslash\\{a\\}\right)\right)$
$\displaystyle=$ $\displaystyle\lim_{\lambda\uparrow\Lambda}\
\left|\mathfrak{F}_{fin}\left(X,E\backslash\\{a\\}\right)\cap\mathbb{V}(\lambda)\right|=\lim_{\lambda\uparrow\Lambda}\
\left|\mathfrak{F}\left(X_{\lambda},E_{\lambda}\right)\right|$
$\displaystyle=$ $\displaystyle\lim_{\lambda\uparrow\Lambda}\
\left|E_{\lambda}\right|^{\left|X_{\lambda}\right|}=\mathfrak{num}\left(E\right)^{\mathfrak{num}\left(X\right)}.$

##  4 Ordinal numbers and numerosities

In this section we will select a subset of the numerosities which we will call
ordinal numerosities (or simply ordinals). This set, equipped with its natural
order relation $<$, is isomorphic to the set of ordinal numbers. In Section
4.2 we will show that this correspondence is deeper than expected since it
preserves also the natural operations between ordinals.

###  4.1 The ordinal numerosities

Let $\mathbf{Num}$ be the set of numerosities.

######  Definition 17.

The set $\mathbf{Ord}\subset\mathbf{Num}$ of ordinal numerosities is defined
as follows: $\tau\in\mathbf{Ord}$ if and only if

$\tau=\mathfrak{num}\left(\Omega_{\tau}\right),$

where

$\Omega_{\tau}=\left\\{x\in\mathbf{Ord\ |\ }x<\tau\right\\}.$

It is easy to see by transfinite induction that this is a good definition. In
fact, it is immediate to check that

  * •

$0\in\mathbf{Ord}$;

  * •

if $\tau\in\mathbf{Ord,}$ then
$\tau+1=\mathfrak{num}\left(\Omega_{\tau}\cup\left\\{\tau\right\\}\right)\in\mathbf{Ord}$
(and hence $\mathbb{N}\subset\mathbf{Ord}$).

Moreover, if $\tau_{k}=\mathfrak{num}\left(\Omega_{k}\right),$ $k\in K,$
($\left|K\right|<\kappa$) are ordinal numerosities, then

$\tau:=\mathfrak{num}\left(\mathop{\displaystyle\bigcup}\limits_{k\in
K}\Omega_{k}\right)\in\mathbf{Ord.}$

In fact, this holds as $\mathop{\displaystyle\bigcup}\limits_{k\in
K}\Omega_{k}=\\{x\in\mathbf{Ord}\mid x<\tau\\}$: the inclusion
$\mathop{\displaystyle\bigcup}\limits_{k\in
K}\Omega_{k}\subseteq\\{x\in\mathbf{Ord}\mid x<\tau\\}$ holds trivially, as if
$x\in\mathop{\displaystyle\bigcup}\limits_{k\in K}\Omega_{k}$ then
$x\in\mathbf{Ord}$ and $x\in\Omega_{k}$ for some $k$, and so
$x<\tau_{k}<\tau$; conversely, if $x\in\mathbf{Ord}$ is such that $x<\tau$, if
$x\notin\mathop{\displaystyle\bigcup}\limits_{k\in K}\Omega_{k}$ we would have
that $\Omega_{x}\supseteq\mathop{\displaystyle\bigcup}\limits_{k\in
K}\Omega_{k}$, and so by taking numerosities we would get $x\geq\tau$, which
is absurd.

######  Definition 18.

If $\tau_{k},$ $k\in K,$ ($\left|K\right|<\kappa$) are ordinals, we set

$\underset{k\in
K}{\sup}\tau_{k}=\mathfrak{num}\left(\mathop{\displaystyle\bigcup}\limits_{k\in
K}\Omega_{\tau_{k}}\right),$

where $\tau_{k}=\mathfrak{num}\left(\Omega_{\tau_{k}}\right).$

Then $\tau=\sup_{k\in K}\tau_{k}$ is the least element in $\mathbf{Ord}$ equal
or greater than every $\tau_{k},$ namely $\tau\in\mathbf{Ord}$ and

$\forall k\in K,\ \tau\geq\tau_{k};$ (10) $\forall k\in K,\
\forall\xi\geq\tau_{k}\Rightarrow\xi\geq\tau.$ (11)

However $\tau$ is not the least element in $\mathbf{Num}$ greater or equal to
every $\tau_{k}\mathbf{.}$ In fact, as we have seen, if $\sup_{k\in
K}\tau_{k}$ is not a maximum, there are numerosities $\xi\in\mathbb{E}$ ,
greater that every $\tau_{k}$ and smaller than $\tau$, e.g. $\left(\sup_{k\in
K}\tau_{k}\right)-1$.

Our construction of the ordinal numbers is similar to the construction of Von
Neumann. However, whilst a Von Neumann ordinal $\tau$ is the set of all the
Von Neumann ordinals contained in $\tau$, in our construction an ordinal
$\tau$ is the numerosity of the set of ordinals smaller than $\tau$. Hence,
here, an ordinal number, as any other numerosity, is an atom.

Obviously, not all numerosities are ordinals: for example,
$\mathfrak{num}\left(\mathbb{N}\right)$ is not an ordinal. In fact, if
$\alpha=\mathfrak{num}\left(\mathbb{N}\right)$ were an ordinal then:

$\displaystyle\alpha$ $\displaystyle=$
$\displaystyle\mathfrak{num(}\left\\{x\in\mathbf{Ord}\ |\
x<\mathfrak{num}\left(\mathbb{N}\right)\right\\})=\mathfrak{num}(\mathbb{N}_{0})$
$\displaystyle=$
$\displaystyle\mathfrak{num}(\mathbb{N}\cup\left\\{0\right\\})=\alpha+1.$

In a similar way, one can prove that no infinite numerosity smaller than
$\mathfrak{num}\left(\mathbb{N}\right)$ is an ordinal. However, $\alpha+1$ is
an ordinal:

$\alpha+1=\mathfrak{num}\left(\mathbb{N}_{0}\right)=\mathfrak{num}\left(\\{x\in\mathbf{Ord}\
|\ x<\alpha\\}\right).$

Actually $\alpha+1$ is the smallest infinite ordinal. From now on, we will
call it $\omega$.

As we expect, $\mathbf{Ord}$ is a well ordered set; in fact is
$E\subset\mathbf{Ord,}$ the minimum is given by

$\min\ E=\sup\left\\{x\in\mathbf{Ord}\ |\ \forall a\in E,\ x\leq a\right\\}.$

###  4.2 Sums and products of ordinals

In this section we will show that the set of ordinal numerosities is closed
under sums and products, and we will show that there is relationship between
sums and products of ordinal numerosities and the natural operations between
Cantor ordinals.

First, we start by showing that the operations between numerosities are
consistent with the order structure over the ordinals.

######  Theorem 19.

For all ordinal numbers $\sigma,\tau\in\mathbf{Ord}$ we have that

$\displaystyle\mathfrak{num}(\Omega_{\sigma})+\mathfrak{num}(\Omega_{\tau})$
$\displaystyle=$ $\displaystyle\mathfrak{num}(\Omega_{\sigma+\tau});$
$\displaystyle\mathfrak{num}(\Omega_{\sigma})\cdot\mathfrak{num}(\Omega_{\tau})$
$\displaystyle=$ $\displaystyle\mathfrak{num}(\Omega_{\sigma\tau}).$

In particular, $\sigma+\tau\in\mathbf{Ord}$ and $\sigma\tau\in\mathbf{Ord}$.

###### Proof.

First let us prove that

$\mathfrak{num}(\Omega_{\sigma+\tau})=\mathfrak{num}(\Omega_{\sigma})+\mathfrak{num}(\Omega_{\tau})$

acting by induction on $\tau.$ If $\tau=0,$ then this relation is obvious. If
$\tau=\gamma+1,$ then

$\displaystyle\mathfrak{num}(\Omega_{\sigma+\tau})$ $\displaystyle=$
$\displaystyle\mathfrak{num}(\Omega_{\sigma+\gamma+1})=\mathfrak{num}(\Omega_{\sigma+\gamma}\cup\left\\{\sigma+\gamma+1\right\\})$
$\displaystyle=$
$\displaystyle\mathfrak{num}(\Omega_{\sigma+\gamma})+\mathfrak{num}(\left\\{\sigma+\gamma+1\right\\})=\mathfrak{num}(\Omega_{\sigma})+\mathfrak{num}(\Omega_{\gamma})+1$
$\displaystyle=$
$\displaystyle\mathfrak{num}(\Omega_{\sigma})+\mathfrak{num}(\Omega_{\gamma}\cup\left\\{\gamma+1\right\\})=\mathfrak{num}(\Omega_{\sigma})+\mathfrak{num}(\Omega_{\tau}).$

If $\tau=\ \sup_{k\in K}\tau_{k},\ $(where
$\tau_{k}=\mathfrak{num}\left(\Omega_{k}\right)$), is a limit ordinal, then

$\mathfrak{num}(\Omega_{\sigma+\tau})=\sup_{k\in K}\
\mathfrak{num}(\Omega_{\sigma+\tau_{k}})=\sup_{k\in K}\
\left[\mathfrak{num}(\Omega_{\sigma})+\mathfrak{num}(\Omega_{\tau_{k}})\right].$

Since
$\sigma+\tau_{k}=\mathfrak{num}(\Omega_{\sigma})+\mathfrak{num}(\Omega_{\tau_{k}})$
is an ordinal number, $\tau$ satisfies (10) and (11) and hence,

$\forall k\in K,\ \sigma+\tau\geq\sigma+\tau_{k};$ $\forall k\in K,\
\forall\xi\in\mathbf{Ord}\
\sigma+\xi\geq\sigma+\tau_{k}\Rightarrow\sigma+\xi\geq\sigma+\tau.$

Then,

$\sup_{k\in K}\ \left(\sigma+\tau_{k}\right)=\sigma+\sup_{k\in K}\ \tau_{k}$

and so

$\mathfrak{num}(\Omega_{\sigma+\tau})=\sigma+\sup_{k\in K}\
\tau_{k}=\mathfrak{num}(\Omega_{\sigma})+\sup_{k\in K}\
\left[\mathfrak{num}(\Omega_{\tau_{k}}\right]=\mathfrak{num}(\Omega_{\sigma})+\mathfrak{num}(\Omega_{\tau}).$

Similarly we act with the product. If $\tau=0,$ then this relation is obvious.
If $\tau=\gamma+1,$ then

$\displaystyle\mathfrak{num}(\Omega_{\sigma\tau})$ $\displaystyle=$
$\displaystyle\mathfrak{num}(\Omega_{\sigma\left(\gamma+1\right)})=\mathfrak{num}(\Omega_{\sigma\gamma+\sigma})=\mathfrak{num}(\Omega_{\sigma\gamma})+\mathfrak{num}(\Omega_{\sigma})$
$\displaystyle=$
$\displaystyle\mathfrak{num}(\Omega_{\sigma})\cdot\mathfrak{num}(\Omega_{\gamma})+\mathfrak{num}(\Omega_{\sigma})=\mathfrak{num}(\Omega_{\sigma})\left[\mathfrak{num}(\Omega_{\gamma})+1\right]$
$\displaystyle=$
$\displaystyle\mathfrak{num}(\Omega_{\sigma})\cdot\mathfrak{num}(\Omega_{\tau}).$

If $\tau=\ \sup_{k\in K}\tau_{k}\ $(where
$\tau_{k}=\mathfrak{num}\left(\Omega_{k}\right)$), is a limit ordinal, then

$\mathfrak{num}(\Omega_{\sigma\tau})=\sup_{k\in K}\
\mathfrak{num}(\Omega_{\sigma\tau_{k}})=\sup_{k\in K}\
\left[\mathfrak{num}(\Omega_{\sigma})\cdot\mathfrak{num}(\Omega_{\tau_{k}})\right].$

Since $\tau$ satisfies (10) and (11),

$\forall k\in K,\ \sigma\tau\geq\sigma\tau_{k};$ $\forall k\in K,\
\forall\xi\in\mathbf{Ord}\
\sigma\xi\geq\sigma\tau_{k}\Rightarrow\sigma\xi\geq\sigma\tau.$

Then

$\sup_{k\in K}\ \left(\sigma\tau_{k}\right)=\sigma\cdot\sup_{k\in K}\
\tau_{k},$

hence

$\mathfrak{num}(\Omega_{\sigma\tau})=\sigma\cdot\sup_{k\in K}\
\tau_{k}=\mathfrak{num}(\Omega_{\sigma})\cdot\mathfrak{num}(\Omega_{\tau}).\hskip
5.0pt\hbox{\hskip 5.0pt\vrule width=4.0pt,height=6.0pt,depth=1.5pt\hskip
1.0pt}$

###  4.3 Numerosities and Cantor ordinals

The relation with the Cantor definition of ordinal is the following: if
$\tau\in\mathbf{Ord,}$ $\Omega_{\tau}$ is a well ordered set and hence
$ot(\Omega_{\tau})$ (the order type of $\Omega_{\tau})$ is a Cantor ordinal.
From now on, to avoid confusion, we will denote the Cantor ordinals by
$\bar{\tau}$ and their set by $\mathbf{COrd}$. Whilst a Cantor ordinal is an
equivalence class of well-ordered sets, in our definition an ordinal is the
numerosity of a suitable well ordered set; in particular, if we let $\omega$
be the smallest infinite ordinal, then
$\omega=\mathfrak{num}\left(\mathbb{N}_{0}\right)$ and
$\bar{\omega}=ot(\mathbb{N}_{0})$.

Now, let us consider the map

$\Phi:\mathbf{Ord}\rightarrow\mathbf{COrd:\ \
}\Phi\left(\tau\right)=ot(\Omega_{\tau}):=\bar{\tau}$ (12)

which identifies the ”numerosity ordinals” with the ”Cantor ordinals”. So, by
construction $\Phi$ is an isomorphism between the ordered sets
$\left(\mathbf{Ord,<}\right)$ and $\left(\mathbf{COrd,<}\right).$

In general, the map does not preserve the operations $+,\cdot$, as $+$ and
$\cdot$ are commutative on $\mathbf{Num}\subset\mathbb{N}^{\ast}$ but not on
$\mathbf{COrd}$. However, the situation is more interesting if we consider the
natural operations $\oplus,\otimes$ between ordinals. We recall that each
ordinal $\bar{\sigma}$ has a unique _normal form_

$\bar{\sigma}=\sum_{n=0}^{m}\bar{\omega}^{j_{n}}a_{n}$

where $a_{n}\in\mathbb{N}$ and $n_{1}<n_{2}\Rightarrow j_{n_{1}}>j_{n_{2}}$.

By using the normal form, the _natural ordinal operations_ can be defined as
follows: given

$\bar{\sigma}=\sum_{n=0}^{m}\bar{\omega}^{j_{n}}a_{n}\,\,\ \mathrm{and}\,\,\
{\bar{\tau}}=\sum_{n=0}^{m}\bar{\omega}^{j_{n}}b_{n}$ (13)

we let

$\bar{\sigma}\oplus{\bar{\tau}}=\sum_{n=0}^{m}\bar{\omega}^{j_{n}}\left(a_{n}+b_{n}\right)\
\
\mathrm{and}\,\,\,\bar{\sigma}{\otimes\bar{\tau}}=\bigoplus_{n,h=0}^{m}a_{n}b_{h}\bar{\omega}^{j_{n}\oplus
j_{h}},$ (14)

where $a_{n}+b_{n}$ and $a_{l}b_{m}$ are the usual operations on natural
numbers.

In order to compare the operations between numerosities and the natural
ordinal operations, we extend a notion used for the Cantor ordinals to the
numerosities .

######  Definition 20.

An ordinal $\theta>0$ is called irreducible if

$\sigma,\tau,\gamma<\theta\Rightarrow\sigma\tau+\gamma<\theta$

If $\theta$ is irreducible then

$\sigma,\tau\in\Omega_{\theta}\Rightarrow\sigma+\gamma<\theta\ \text{and}\
\sigma\tau<\theta;$

we need to prove that $\sigma+\gamma\ $and$\ \sigma\tau\in\Omega_{\theta}$.

We denote by $\theta_{j}$, $j\in\mathbf{Ord}$ the sequence of irreducible
ordinals, namely

  * •

$\theta_{0}=\omega,\ $

  * •

$\theta_{j}=\min$ $\left\\{x\in\mathbf{Ord}\ \mathbf{|\ }\forall
m\in\mathbb{N}_{0},\ \forall k<j,\ x>\theta_{k}^{m}\right\\}.$

######  Proposition 21.

If $\tau\in\mathbf{Ord}$, we have that

$\tau<\theta_{j+1}\Leftrightarrow\tau=\sum_{k=0}^{m}b_{k}\theta_{j}^{k}$

with $b_{k}\in\Omega_{\theta_{j}}.$

###### Proof.

This proof is based only on the order structure of $\mathbf{Ord}$ and hence it
could be considered well known. However we will report it for completeness and
for the sake of the reader.

$\left(\Leftarrow\right)$ trivial.

$\left(\Rightarrow\right)$ If $\tau<\theta_{j+1},$ we take

$n=\max\left\\{m\in\mathbb{N}_{0}\ |\ \theta_{j}^{m}\leq\tau\right\\}$

Such an $m$ exists by the definition of $\theta_{j+1}.$ Then we set

$b_{m}=\sup\left\\{x\in\Omega_{\theta_{j}}\ |\
x\theta_{j}^{m}\leq\tau\right\\}$

and

$y_{j,m}=\tau-b_{m}\theta_{j}^{m}$

Then,

$\forall z\in\Omega_{\theta_{j}},\ y_{j,m}\leq z.$ (15)

Now, by inducion over $k=m-1,...,0$, we set

$b_{k}=\sup\left\\{x\in\Omega_{\theta_{j}}\ |\
\sum_{l=k+1}^{m}b_{l}\theta_{j}^{l}+x\theta_{j}^{k}\leq\tau\right\\}$

and

$y_{j,k}=\tau-\sum_{l=k}^{m}b_{l}\theta_{j}^{l}$

so we have that,

$\forall z\in\Omega_{\theta_{j}},\ y_{j,m}\leq z$ (16)

Now we claim that

$\tau-\sum_{k=0}^{m}b_{k}\theta_{j}^{k}=0$ (17)

In order to prove this we argue by iduction over
$j\in\mathbf{Ord}\cup\left\\{-1\right\\}$ by proving that

$y_{jk}=0.$ (18)

If $j=-1,\ \tau\in\Omega_{\theta_{0}}=\mathbb{N}_{0},$ then $\forall
n\in\mathbb{N}_{0},\ y_{00}\leq 0$ and hence $y_{00}=0$. If (18) holds
$\forall\tau\in\Omega_{\theta_{j}},$ then by (15) and (16), equality (18)
holds also for $\tau\in\Omega_{\theta_{j+1}}.$

######  Corollary 22.

If $\sigma,\tau\in\mathbf{Ord,}$ then $\sigma+\tau\in\mathbf{Ord}$ and
$\sigma\tau\in\mathbf{Ord}$

###### Proof.

By Prop. 21,

$\sigma=\sum_{k=0}^{n}a_{k}\theta_{j}^{k},\
\tau=\sum_{k=0}^{n}b_{k}\theta_{j}^{k}$

for some $j\in\mathbf{Ord}$ and hence

$\sigma+\tau=\sum_{k=0}^{n}\left(a_{k}+b_{k}\right)\theta_{j}^{k};\
\sigma\tau=\sum_{h,k=0}^{n}\left(a_{h}b_{k}\right)\theta_{j}^{h+k}.\hskip
5.0pt\hbox{\hskip 5.0pt\vrule width=4.0pt,height=6.0pt,depth=1.5pt\hskip
1.0pt}$

Now we describe the sequence of the irreducible ordinal numerosities: we set

  * •

$\theta_{0}=\bar{\omega}$

  * •

$\theta_{\bar{j}}=\sup\left\\{\theta_{\bar{k}}^{n}\ |\ n\in\mathbb{N},\
\bar{k}<\bar{j}\right\\}$

So we have that

$\displaystyle\theta_{0}$ $\displaystyle=$ $\displaystyle\omega,$
$\displaystyle\theta_{1}$ $\displaystyle=$ $\displaystyle\omega^{\omega},$
$\displaystyle\theta_{2}$ $\displaystyle=$
$\displaystyle\omega^{\omega^{\omega}}$ $\displaystyle\dots$
$\displaystyle\theta_{j+1}$ $\displaystyle=$
$\displaystyle\theta_{j}^{\omega}$ $\displaystyle\dots$
$\displaystyle\theta_{\omega}$ $\displaystyle=$ $\displaystyle\varepsilon_{0}$
$\displaystyle\dots$

and so on. Since the definiton of $\theta_{j}$ depends only on the order
structure of $\left(\mathbf{Ord,<}\right)$, then

$\Phi\left(\theta_{j}\right)=\bar{\theta}_{j}.$

It is well known and easy to check that any ordinal number
${\bar{\tau}}\in\mathbf{COrd,}\bar{\tau}<\bar{\theta}_{j+1},$ can be written
as follows:

${\bar{\tau}}=\mathop{\displaystyle\bigoplus}\limits_{n=0}^{m}\bar{a}_{n}\otimes\bar{\theta}_{j}^{n};\
\ \bar{a}_{n}<\bar{\theta}_{j}$

and the natural operations $\oplus,\otimes$ take the following form:

$\left(\mathop{\displaystyle\bigoplus}\limits_{n=0}^{m}\bar{a}_{n}\otimes\bar{\theta}_{j}^{n}\right)\oplus\left(\mathop{\displaystyle\bigoplus}\limits_{n=0}^{m}\bar{b}_{n}\otimes\bar{\theta}_{j}^{n}\right)=\mathop{\displaystyle\bigoplus}\limits_{n=0}^{m}\left(\bar{a}_{n}\oplus\bar{b}_{n}\right)\otimes\bar{\theta}_{j}^{n}$
$\,\bar{\sigma}{\otimes\bar{\tau}}=\mathop{\displaystyle\bigoplus}\limits_{n,h=0}^{m}\left(\bar{a}_{n}\otimes\bar{b}_{h}\right)\otimes\bar{\theta}_{j}^{n+h}$

######  Theorem 23.

The map (12) is an isomorphism between the semirings $(\mathbf{Ord,}+,\cdot)$
and $(\mathbf{COrd,}\oplus,\otimes),$ namely

$\displaystyle\Phi\left(\sigma+\tau\right)$ $\displaystyle=$
$\displaystyle\bar{\sigma}\oplus\bar{\tau}$
$\displaystyle\Phi\left(\sigma\tau\right)$ $\displaystyle=$
$\displaystyle\bar{\sigma}\otimes\bar{\tau}$

###### Proof.

Let $\tau=\sum_{k=0}^{m}b_{k}\theta_{j}^{k}$ be an ordinal numerosity. Then

$\displaystyle\sum\limits_{n=0}^{m}\bar{b}_{n}\otimes\bar{\theta}_{j}^{n}$
$\displaystyle=$ $\displaystyle
ot\left(\left\\{\mathop{\displaystyle\bigoplus}\limits_{n=0}^{m}\bar{a}_{n}\otimes\bar{\theta}_{j}^{n}\in\mathbf{COrd}\
|\
\mathop{\displaystyle\bigoplus}\limits_{n=0}^{m}\bar{a}_{n}\otimes\bar{\theta}_{j}^{n}<\mathop{\displaystyle\bigoplus}\limits_{n=0}^{m}\bar{b}_{n}\otimes\bar{\theta}_{j}^{n}\right\\}\right)$
$\displaystyle=$ $\displaystyle
ot\left(\left\\{\sum_{n=0}^{m}a_{n}\theta_{j}^{n}\in\mathbf{Ord}\ |\
\sum_{n=0}^{m}a_{n}\theta_{j}^{n}<\sum_{n=0}^{m}b_{n}\theta_{j}^{n}\right\\}\right)=ot\left(\Omega_{\tau}\right)=\bar{\tau},$

namely

$\Phi\left(\tau\right)=\Phi\left(\sum_{n=0}^{m}b_{n}\theta_{j}^{n}\right)=\mathop{\displaystyle\bigoplus}\limits_{n=0}^{m}\bar{b}_{n}\otimes\bar{\theta}_{j}^{n}=\bar{\tau}.$

Hence $\Phi$ is an isomorphism.

######  Remark 24.

Theorems 23 and 19 provide a new interpretation for the natural operations
$\oplus\ $and $\otimes$ namely

$\bar{\sigma}\oplus{\tau}=ot(\Omega_{\sigma+\tau})\ \
\mathrm{and}\,\,\,\bar{\sigma}{\otimes\bar{\tau}}=ot(\Omega_{\sigma\tau})$

This fact is somewhat surprising since the operation $+$ and $\cdot$ between
numerosities have been introduced in a natural way for the numerosity theory
and, _a priori_ , they should not have any realation with the natural
operations between ordinal numbers.

Notice, however, that not all operations are the same between numerosity
ordinals and Cantor ordinals: for example, let
$\bar{\varepsilon}_{0}=\bar{\theta}_{\bar{\omega}}$ be the Cantor ordinal that
corresponds to the numerosity ordinal $\theta_{\omega}$. If we use the ordinal
exponentiation, we have that

$\bar{\omega}^{\bar{\varepsilon}_{0}}=\bar{\varepsilon}_{0},$

whilst on the contrary, if we use the Euclidean exponentiation, we get that

$\omega^{\varepsilon_{0}}>\theta_{\omega}.$

In particular the equation

$\bar{\omega}^{x}=\bar{\varepsilon}_{0}$

in the world of Cantor ordinals has the solution $x=\bar{\varepsilon}_{0}$
while the equation

$\omega^{x}=\varepsilon_{0}$

in the world of Euclidean numbers, has the solution
$\xi=\log_{\omega}\varepsilon_{0}.$ $\xi$ is a well defined Euclidean number,
but it is not an ordinal number since

$\xi<\varepsilon_{0}=\mathfrak{num}\left(\mathop{\displaystyle\bigcup}\limits_{k<\omega}\Omega_{\theta_{k}}\right).$

However it is easy to prove that the ordinal exponentiation agrees with the
Euclidean exponentiation for numbers in $\Omega_{\theta_{\omega}}$.

##  5 Numerosities of some denumerable sets

There are many different ways of definining a label set according to
Definition 8. Different label sets might give different algebraical properties
to the numerosity; moreover, in some cases particular choices of the label
sets may lead to other concepts (e.g. Lebesgue measure for the reals). In this
and the next Sections, we want to show several examples of these facts.

###  5.1 The general strategy

Theorem 16 describes the fundamental properties of numerosities, which are
satisfied for all choices of the label set $\mathfrak{B}$ (and of the
ultrafilter $\mathcal{U}$). However, certain additional properties are
satisfied only for some choices of $\mathfrak{B}$: in fact, they depend on the
ultrafilter $\mathcal{U}$ over $\mathbf{\wp}_{fin}\left(\mathfrak{B}\right)$,
whose existence depends on Zorn’s lemma which cannot be explicit and hence it
is impossible to prove or disprove some of them. However, if we choose a
suitable label set $\mathfrak{B}$ (and, consequently we restrict the choice of
$\mathcal{U}$), it is possible to show that some properties, as the ones
mentioned in the Introduction, are satisfied independently of $\mathcal{U}$.
The goal of Section 5 is to show how a suitable choice of $\mathfrak{B}$
allows the numerosity function to satisfy interesting properties in many
specific cases.

The smaller the set $\mathfrak{B}$ is, the more properties are satisfied by
the numerosity function. So the idea is to begin with a set
$\mathfrak{B}_{\max}=\left\\{\mathfrak{t}\in\mathfrak{L\ }|\
\mathfrak{t}\cap\mathfrak{L}=\varnothing\right\\}$ and to construct smaller
label sets
$\mathfrak{B}_{\max}\supset\mathfrak{B}_{1}\supset\mathfrak{B}_{2}\supset\dots$
which provide a richer and richer structure to the theory. In this paper we
are interested in the numerosity of some specific subsets of $\mathbb{N}_{0}$,
$\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$ and so we will construct set of
labels
$\mathfrak{B}_{\max}\supset\mathfrak{B}\left(\mathbb{N}_{0}\right)\supset\dots\supset\mathfrak{B}\left(\mathbb{R}\right)$.
Each set of labels allows to enrich the theory with new theorems; all these
theorems are independent of the ultrafilter employed in the sense that every
ultrafilter which satisfies the finess property777The finess property has been
introduced in the proof of the existence of the field of Euclidean numbers.
does the job888Of course a smaller set of labels reduces the choice of the
ultrafilter. More precisely if $\mathfrak{B}_{1}\supset\mathfrak{B}_{2}$, an
ultrafilter constructed over $\mathfrak{B}_{2}$ makes $\mathfrak{B}_{1}$ to be
a qualified set..

The construction which we will present in the next sections is based on the
following definition:

######  Definition 25.

If $\mathfrak{D}\subset\mathfrak{B}_{\max}$is a directed set (wih respect to
$\subseteq$), we define

$\overline{\mathfrak{D}}=\mathfrak{G}\left(\\{\mathfrak{s\in
B}_{\max}\mid\exists\mathfrak{t}\in\mathfrak{D},\
\mathfrak{s}\sqsupseteq\mathfrak{t}\\}\right),$

where $\mathfrak{G}F$ denotes the smallest lattice containing $F.$

Notice that, by Definition, if $\mathfrak{D}\subset\mathfrak{B}_{\max}$ is a
directed set then

$\overline{\overline{\mathfrak{D}}}=\overline{\mathfrak{D}}.$

######  Lemma 26.

For every $\mathfrak{D}\subset\mathfrak{B}_{\max},\ \overline{\mathfrak{D}}$
is a label set.

###### Proof.

Let us check that $\overline{\mathfrak{D}}$ satisfies the properties of
Definition 8.

Property 8.(i) holds as $\overline{\mathfrak{D}}$ is a lattice by definition.

Property 8.(ii) holds as
$\overline{\mathfrak{D}}\subseteq\mathfrak{B}_{\max}$.

Property 8.(iii) holds as $\forall
a\in\Lambda,\exists\mathfrak{s}\in\mathfrak{B}_{\max},\
a\in\mathbb{V}\left(\mathfrak{s}\right)$ and hence, if you take any
$\mathfrak{t}\in\mathfrak{D,}$
$a\in\mathbb{V}\left(\mathfrak{s}\cup\mathfrak{t}\right);$ on the other hand
$\mathfrak{s}\cup\mathfrak{t\in}\overline{\mathfrak{D}}\mathfrak{\ }$and so
$\bigcup_{\mathfrak{s}\in\overline{\mathfrak{D}}}\mathbb{V}\left(\mathfrak{s}\right)=\Lambda$.

The numerosity of a set depends on the set of labels $\mathfrak{B}$ and an
ultrafilter $\mathcal{U}$ consistent with $\mathfrak{B.}$ As in this section
we will discuss also coherence properties between different label sets, we
will use the notation $\mathfrak{num}_{\mathfrak{B}}^{\mathcal{U}}$ to denote
the numerosity function obtained using labels in $\mathfrak{B}$ and the
ultrafilter $\mathcal{U}$ and similarly we denote by $\ell_{\mathfrak{B}}(x)$
the label relative to $\mathfrak{B}$ (see Definition 9). This notation will be
used only when there is danger of confusion, as multiple sets of labels are
used at once. We will keep to use to the simpler notation $\mathfrak{num}$
whenever there is no danger of such confusion.

By definition, the $\overline{\mathfrak{D}}$-labelling of $a\in\Lambda$ is
given by

$\ell_{\overline{\mathfrak{D}}}(a)=\bigcap\\{\mathfrak{s}\in\overline{\mathfrak{D}}\mid
a\in\mathbb{V}\left(\mathfrak{s}\right)\\}=\bigcap\\{\mathfrak{s}\in\mathfrak{B}_{\max}\mid
a\in\mathbb{V}\left(\mathfrak{s}\right)\ \text{and}\
\exists\mathfrak{t}\in\mathfrak{D},\ \mathfrak{s}\supseteq\mathfrak{t}\\};$

in particular, we have that

$\mathfrak{s}\in\mathfrak{D}\Rightarrow\ell_{\overline{\mathfrak{D}}}(\mathfrak{s})=\mathfrak{s}.$
(19)

######  Proposition 27.

If $\mathfrak{D}_{1}\subset\mathfrak{D}_{2}$ then
$\overline{\mathfrak{D}}_{1}\subset\overline{\mathfrak{D}}_{2}$, hence for all
ultrafilter $\mathcal{U}$ consistent with $\overline{\mathfrak{D}}_{1}$, for
every set $A$ in $\Lambda$

$\mathfrak{num}_{\overline{\mathfrak{D}}_{1}}^{\mathcal{U}}(A)=\mathfrak{num}_{\overline{\mathfrak{D}}_{2}}^{\mathcal{U}}(A).$

###### Proof.

The inclusion $\overline{\mathfrak{D}}_{1}\subset\overline{\mathfrak{D}}_{2}$
holds trivially from Definition 25. The consistency is immediate as if
$\mathcal{U}$ contains $\overline{\mathfrak{D}}_{1}$ and
$\overline{\mathfrak{D}}_{1}\subset\overline{\mathfrak{D}}_{2}$ then
necessarily $\mathcal{U}$ contains $\overline{\mathfrak{D}}_{2}$.

######  Lemma 28.

If $\lambda\in\overline{\mathfrak{D}},$ then $\lambda$ can be split as follows

$\lambda=\mathfrak{s}\cup\mathfrak{t}$

where $\mathfrak{s}\in\mathfrak{D}$ and $\mathfrak{t}$ is such that

$\forall\sigma\in\mathfrak{D},\ \mathfrak{t}\cap\sigma=\varnothing.$

###### Proof.

Given $\lambda\in\overline{\mathfrak{D}},$ we set

$\mathfrak{s}:=\mathop{\displaystyle\bigcup}\\{\mathfrak{u}\in\mathfrak{D}\mid\mathfrak{u}\subset\lambda\\}$

and

$\mathfrak{t}:=\lambda\backslash\mathfrak{s}.$

Then $\mathfrak{s}\in\overline{\mathfrak{D}}$, as $\overline{\mathfrak{D}}$ is
a lattice and the union defining $\mathfrak{s}$ is finite, and
$\forall\sigma\in\mathfrak{D},\ \mathfrak{t}\cap\sigma=\varnothing.$ In fact,
if we set $\mathfrak{u}=\mathfrak{t}\cap\sigma$ then, as
$\mathfrak{s}\supseteq\sigma$, we have

$\varnothing=\mathfrak{s}\cap\mathfrak{t}\supseteq\sigma\cap\mathfrak{t}=\mathfrak{u},$

hence $\mathfrak{u}=\varnothing$.

######  Remark 29.

We can look at the splitting given by Lemma 28 thinking of $\mathfrak{B\ }$as
a vector space over $\mathbb{Z}_{2};$ in this case we can write

$\mathfrak{B}=\mathfrak{D}\oplus\mathfrak{D}^{\perp}$

and the splitting $\lambda=\mathfrak{s}\cup\mathfrak{t\ }$implies that

$\mathfrak{s\in D\ }\text{and\ }\mathfrak{t}\in\mathfrak{D}^{\perp}.$

###  5.2 Numerosity of the natural numbers

In what follows, we set $\mathbb{N}:=\\{1,2,3,\dots\\}$ and
$\mathbb{N}_{0}:=\mathbb{N}\cup\\{0\\}$, and we let $\alpha$ denote the
numerosity of $\mathbb{N}$. We will consider numbers in $\mathbb{N}_{0}$, as
well as more generally in $\mathbb{R}$, as atoms. Our goal is to find a label
set $\mathfrak{B}(\mathbb{N}_{0})\subset\mathfrak{B}_{\max}$ so that we can
prove some properties of $\alpha$ and to describe the numerosity of some
subsets of $\mathbb{N}_{0}$ by functions of $\alpha$.

We define $\mathfrak{D}\left(\mathbb{N}_{0}\right)$ as follows:

$\lambda\in\mathfrak{D}(\mathbb{N}_{0})\Leftrightarrow\exists m\in\mathbb{N}\
\text{such that}\ \lambda=\left\\{0,\dots,m!^{m!}\right\\},$

and we set

$\mathfrak{B}(\mathbb{N}_{0}):=\overline{\mathfrak{D}(\mathbb{N}_{0})}.$

which, by Lemma 26, is a label set. By Definition, we have that for every
$n\in\mathbb{N}_{0}$,

$\ell(n)=\left\\{0,\dots,f(n)\right\\},$

where

$f(n):=\min\left\\{m!^{m!}\ |\ m\in\mathbb{N},\ m!^{m!}\geq n\right\\}.$

The main reason for such a peculiar labelling is to ensure the following
algebraical properties for $\alpha$:

######  Proposition 30.

Let $n\in\mathbb{N}$. Then

  1. $(i)$

$\mathfrak{num}(\left\\{nm\ |\ m\in\mathbb{N}\right\\})=\frac{\alpha}{n}$;

  2. $(ii)$

$\mathfrak{num}(\left\\{m^{n}\ |\
m\in\mathbb{N}\right\\})=\alpha^{\frac{1}{n}}$.

###### Proof.

(i) For $i=0,\dots,n-1$ let

$A_{i}=\\{m\in\mathbb{N}_{0}\mid m\equiv i\mod n\\}.$

Then for every $\lambda\sqsupseteq\\{0,1,\dots,n!^{n!}\\}$, with
$\lambda\in\mathfrak{B}\left(\mathbb{N}_{0}\right)$, for every $0\leq i,j<n$
we have

$|A_{i}\cap\lambda|=|A_{j}\cap\lambda|,$

as $\lambda\cap\mathbb{N}=\\{0,1,\dots,f(m)\\}$ for some $m\geq n$, and $n$
divides $f(m)$ for every such $m$. In particular, this shows that
$\mathfrak{num}\left(A_{i}\right)=\mathfrak{num}\left(A_{j}\right)$ for every
$0\leq i,j<n$, hence

$\alpha=\mathfrak{num}\left(\mathbb{N}_{0}\right)=\sum_{i=0}^{n-1}\mathfrak{num}\left(A_{i}\right)=n\cdot\mathfrak{num}\left(A_{0}\right).$

(ii) Let $\lambda\sqsupseteq\\{1,\dots,n!^{n!}\\}$, with
$\lambda\in\mathfrak{B}\left(\mathbb{N}_{0}\right)$. As noticed in (i) above,
it must be $\lambda\cap\mathbb{N}=\\{1,\dots,m!^{m!}\\}$ for some $m\geq n$.
If $a=m!^{\frac{m!}{n}}$, we can rewrite $\\{1,\dots,m!^{m!}\\}$ as
$\\{1,\dots,a^{n}\\}$. Hence $|\left\\{m^{n}\ |\
m\in\mathbb{N}\right\\}\cap\lambda|\
=a=|\mathbb{N}\cap\lambda|^{\frac{1}{n}}$. The thesis is reached by taking the
$\Lambda$-limit on the above equality.

######  Remark 31.

Of course, the choice of $\mathfrak{D}\left(\mathbb{N}_{0}\right)$ is not
intrinsic, and has been done so to make it possible to have the properties
listed in Proposition 30. Some additional motivations for this choice of
$\mathfrak{D}\left(\mathbb{N}_{0}\right)$ can be found in [4]; different
motivations have lead the authors of [9] to make the following different
choice:

$\lambda\in\mathfrak{D}_{1}(\mathbb{N}_{0})\Leftrightarrow\exists
m\in\mathbb{N}\ \text{such that}\ \lambda=\left\\{0,\dots,2^{m}-1\right\\}.$

This can be seen as a feature of this approach: different algebraical
properties of the numerosity can be rather easily obtained by changing the
label set.

###  5.3 Numerosity of the integers

We proceed as in the case of the natural numbers. We define
$\mathfrak{D}(\mathbb{Z})$ as follows:

$\lambda\in\mathfrak{D}\left(\mathbb{Z}\right)\Leftrightarrow\exists
m\in\mathbb{N}\ \text{such that}\
\lambda=\left\\{-m!^{m!},\dots,m!^{m!}\right\\}.$

Clearly
$\mathfrak{D}\left(\mathbb{Z}\right)\subset\mathfrak{B}(\mathbb{N}_{0})$ and
hence, by Lemma 26,

$\mathfrak{B}(\mathbb{Z}):=\overline{\mathfrak{B}\left(\mathbb{N}_{0}\right)\cap\mathfrak{D}(\mathbb{Z})}$

is a label set. Using this label basis for every $z\in\mathbb{Z}$,

$\ell(z)\cap\mathbb{Z}=\\{-n(z),\dots,n(z)\\},$

where

$n(z):=\min\left\\{m!^{m!}\ |\ m\in\mathbb{N},\ m!^{m!}\geq|z|\right\\}.$

Moreover, as $\mathfrak{B}(\mathbb{Z})\subseteq\mathfrak{B}(\mathbb{N})$, by
Proposition 27 the numerosities constructed with $\mathfrak{B}(\mathbb{Z})$
are coherent with those constructed with $\mathfrak{B}(\mathbb{N})$.

With this choice of $\mathfrak{D}\left(\mathbb{Z}\right),$
$\mathfrak{num}\left(\mathbb{Z}\right)=2\alpha+1$ and we have that

$\mathfrak{num}\left(\mathbb{Z}_{<0}\right)=\mathfrak{num}\left(\mathbb{Z}_{>0}\right)=\alpha;$
(20)

this equality agrees with the intuition that the positive numbers are as many
as the negative numbers.

Just as an example of a possible application, let us prove the following
result for subgroups of $\mathbb{Z}$, which reminds Lagrange’s Theorem for
finite groups:

######  Theorem 32.

Let $S:=m\mathbb{Z}$ be a subgroup of $(\mathbb{Z},+)$. Then

$\frac{\mathfrak{num}\left(\mathbb{Z}\right)}{\mathfrak{num}\left(S\right)}\sim
m=\mathfrak{num}\left(\mathbb{Z}_{m}\right).$ (21)

###### Proof.

By definition, $S=\\{mn\mid n\in\mathbb{Z}\\}$. We write $S=S_{+}\cup
S_{-}\cup\\{0\\}$, where

$S_{+}=\\{a\in S\mid a>0\\},S_{-}=\\{a\in S\mid a<0\\}.$

By Proposition (30) we know that
$\mathfrak{num}\left(S_{+}\right)=\frac{\alpha}{m}$, and it is trivial to show
that $\mathfrak{num}\left(S_{-}\right)=\mathfrak{num}\left(S_{+}\right)$.
Hence
$\mathfrak{num}\left(S\right)=\mathfrak{num}\left(S_{+}\right)+\mathfrak{num}\left(S_{-}\right)+1=2\frac{\alpha}{m}+1$.
As $\mathfrak{num}\left(\mathbb{Z}\right)=2\alpha+1,$ we have

$\frac{\mathfrak{num}\left(\mathbb{Z}\right)}{\mathfrak{num}\left(S\right)}=\frac{2\alpha+1}{\frac{2\alpha}{m}+1}=\frac{2\alpha+1}{\frac{1}{m}\left(2\alpha+m\right)}\sim
m,$

as $\alpha$ is infinite.

######  Remark 33.

Let us notice that, with our labelling, in the above Proposition we do not
have the equality

$\frac{\mathfrak{num}\left(\mathbb{Z}\right)}{\mathfrak{num}(S)}=m$ (22)

because not all lateral classes $\left[k\right]$ in the quotient have the same
numerosity:

$\mathfrak{num}\left(\left[k\right]\right)=\frac{2\alpha}{m}\ \ \text{if}\ \
k\neq 0;\ \ \mathfrak{num}\left(\left[0\right]\right)=\frac{2\alpha}{m}+1.$

If we want the equality in (21), then we can replace
$\mathfrak{D}\left(\mathbb{Z}\right)$ with
$\mathfrak{D}_{1}\left(\mathbb{Z}\right)$ defined as follows:

$\lambda\in\mathfrak{D}_{1}\left(\mathbb{Z}\right)\Leftrightarrow\exists
m\in\mathbb{N}\ \text{such that}\
\lambda=\left\\{-m!^{m!}+1,\dots,m!^{m!}\right\\}.$

In this case, we get (22), but $\mathfrak{num}\left(\mathbb{Z}\right)=2\alpha$
and the equality (20) is violated.

###  5.4 Numerosity of the rationals

The labelling of $\mathbb{Z}$ given in Section 5.3 can be extended in several
ways to the rationals. A natural one is obtained by setting

$\mathfrak{D}\left(\mathbb{Q}\right):=\left\\{\ \mathbb{H}_{n}\mid\exists
m\in\mathbb{N},\ n=m!^{m!}\right\\},$

where

$\mathbb{H}_{n}:=\left\\{\frac{a}{n}\mid a\in\mathbb{Z},\
-n^{2}<a<n^{2}\right\\}.$

By Lemma 26,

$\mathfrak{B}(\mathbb{Q}):=\overline{\mathfrak{D}(\mathbb{Q})}$

is a label set.

As $\mathfrak{D}\left(\mathbb{Q}\right)\subset\mathfrak{B}(\mathbb{Z})$, by
Proposition 27 the numerosities constructed with $\mathfrak{B}(\mathbb{Q})$
are coherent with those constructed with $\mathfrak{B}(\mathbb{Z})$. Using the
label basis $\mathfrak{B}(\mathbb{Q})$ we have that, for every
$q\in\mathbb{Q}$,

$\ell(q)\cap\mathbb{Q}=\mathbb{H}_{n(q)},$

where

$n(q):=\min\left\\{m!^{m!}\ |\ m\in\mathbb{N},\ m!^{m!}\geq|q|\right\\}.$

This labelling has been chosen in order to have the following results:

######  Proposition 34.

Using the labelling $\mathfrak{B}(\mathbb{Q})$, the following properties hold

  * $(i)$

for all $n\in\mathbb{N}_{0}$,
$\mathfrak{num}\left(\mathbb{Q}\cap\left[n,n+1\right)\right)=\alpha$;

  * $(ii)$

for all $p,q\in\mathbb{R}$ with $p<q$,
$\frac{\mathfrak{num}\left(\mathbb{Q}\cap\left[p,q\right)\right)}{\mathfrak{num}\left(\mathbb{Q}\cap\left[0,1\right)\right)}\sim(p-q)$;

  * $(iii)$

$\mathfrak{num}\left(\mathbb{Q}\right)=2\alpha^{2}+1$.

###### Proof.

(i) Take $H_{m}\in\mathfrak{B}_{\mathbb{Q}}$ with $m$ larger than $n+1$. Then
$|\left(\mathbb{Q}\cap\left[n,n+1\right)\right)\cap\mathbb{H}_{m}|\ =m$, hence
eventually $|\left(\mathbb{Q}\cap\left[n,n+1\right)\right)\cap\mathbb{H}_{m}|\
=|\mathbb{N}\cap\mathbb{H}_{m}|$, and the thesis follows by taking the
$\Lambda$-limit.

(ii) Take $\mathbb{H}_{m}\in\mathfrak{B}_{\mathbb{Q}}$ with $m$ larger than
$|p|,|q|$. Then $\left(\mathbb{Q}\cap\left[p,q\right)\right)=(p-q)m$ if
$p\in\mathbb{H}_{m}$, $\left(\mathbb{Q}\cap\left[p,q\right)\right)=(p-q)m-1$
if $p\notin\mathbb{H}_{m}$. By taking the $\Lambda$-limit we have that either
$\mathfrak{num}\left(\mathbb{Q}\cap\left[p,q\right)\right)=(p-q)\alpha-1$ or
$\mathfrak{num}\left(\mathbb{Q}\cap\left[p,q\right)\right)=(p-q)\alpha$, and
the thesis follows as, by (i),
$\mathfrak{num}\left(\mathbb{Q}\cap\left[0,1\right)\right)=\alpha$.

(iii) Let us first compute $\mathfrak{num}\left(\mathbb{Q}_{>0}\right)$. Let
$\lambda=\mathbb{H}_{n}\in\mathfrak{B}_{\mathbb{Q}}$. Then
$|\mathbb{H}_{n}\cap\mathbb{Q}_{>0}|=n^{2}$, hence if $\varphi$ is the
enumeration of $\mathbb{Q}_{>0}$ and $\psi$ is the enumeration of
$\mathbb{N}$, we have that $\varphi(\lambda)=\psi(\lambda)^{2}$, so

$\mathfrak{num}\left(\mathbb{Q}_{>0}\right)=\lim_{\lambda\uparrow\Lambda}\varphi(\lambda)=\lim_{\lambda\uparrow\Lambda}\psi^{2}(\lambda)=\left(\lim_{\lambda\uparrow\Lambda}\psi(\lambda)\right)^{2}=\alpha^{2}.$

Therefore, as each $\mathbb{H}_{n}$ is symmetrical with respect to $0$, we
also have that $\mathfrak{num}\left(\mathbb{Q}_{<0}\right)=\alpha^{2}$, and so

$\mathfrak{num}\left(\mathbb{Q}\right)=\mathfrak{num}\left(\mathbb{Q}_{<0}\right)+\mathfrak{num}\left(\mathbb{Q}_{>0}\right)+1=2\alpha^{2}+1.\hskip
5.0pt\hbox{\hskip 5.0pt\vrule width=4.0pt,height=6.0pt,depth=1.5pt\hskip
1.0pt}$

An example of a possible application, let us prove the following result:

######  Theorem 35.

Let $m_{PJ}$ denote the Peano-Jordan measure of a $m_{PJ}$-measurable set $E.$
Then

$m_{PJ}\left(E\right)=st\left(\frac{\mathfrak{num}\left(E\cap\mathbb{Q}\right)}{\mathfrak{num}\left(\left[0,1\right)\cap\mathbb{Q}\right)}\right)=st\left(\frac{1}{\mathfrak{\alpha}}\cdot\mathfrak{num}\left(E\cap\mathbb{Q}\right)\right).$
(23)

###### Proof.

If $E$ is an interval then the result follows from Proposition 34. We can
extend this result to a plurinterval $E=\mathop{\displaystyle\bigcup}E_{i}$ by
the Sum Principle (see Theorem 16.(v)). In general, if $E$ is
$m_{PJ}$-measurable, $\forall\varepsilon\in\mathbb{R}_{>0}$ there are two
plurintervals $A$ and $B$ such that

$A\subseteq E\subseteq B$ $\left|m_{PJ}(B)-m_{PJ}(E)\right|<\varepsilon\
\text{and}\ \left|m_{PJ}(E)-m_{PJ}(A)\right|<\varepsilon$ (24)

By the Euclid Principle (see Theorem 16.(ii)), we have that

$\mathfrak{num}\left(A\cap\mathbb{Q}\right)\subseteq\mathfrak{num}\left(E\cap\mathbb{Q}\right)\subseteq\mathfrak{num}\left(E\cap\mathbb{Q}\right),$

then

$m_{PJ}(A)\leq
st\left(\frac{1}{\mathfrak{\alpha}}\cdot\mathfrak{num}\left(E\cap\mathbb{Q}\right)\right)\leq
m_{PJ}(B).$

The conclusion follows by the inequality above, Equation (24) and the
arbitrariness of $\varepsilon$.

##  6 Numerosities of non-denumerable sets

###  6.1 A suitable labelling

Let $\mathbb{\hat{R}}^{N},$ $N\in\mathbb{N}$,
$\mathbb{\hat{R}}^{N}\subset\mathbb{A}$, be a family of sets such that

$\displaystyle\mathbb{\hat{R}}^{0}$ $\displaystyle=$
$\displaystyle\mathbb{R},$ $\displaystyle\mathbb{\hat{R}}^{N}$
$\displaystyle\subset$ $\displaystyle\mathbb{\hat{R}}^{N+1}$

and each $\mathbb{\hat{R}}^{N+1}$ is isomorphic to $\mathbb{R}^{N}$. This
akward distinction between $\mathbb{\hat{R}}^{N}$ and $\mathbb{R}^{N}$ is
useful since, in this contest, it is easier to deal with atoms and the points
of $\mathbb{R}^{N}$ are $N$-ples. Moreover, we need to assume that the
isomorphism

$\Psi:\mathbb{R}^{N}\rightarrow\mathbb{\hat{R}}^{N}$

preserves also the labels, namely, if
$\left(x_{1},\dots,x_{N}\right)\in\mathbb{R}^{N},$ then

$\ell\left[\Psi\left(x_{1},\dots,x_{N}\right)\right]=\max\left\\{\ell\left(x_{1}\right),\dots,\ell\left(x_{N}\right)\right\\}.$
(25)

If $A\in\wp\left(\mathbb{\hat{R}}^{N}\right)$ for some $N$, we denote by
$\mathcal{H}_{d}\left(A\right)$ the normalized $d$-dimensional Hausdorff
measure999The normalized $d$-dimensional Hausdorff measure is given by
$\mathcal{H}_{d}\left(A\right)=N_{d}H_{d}\left(A\right),$ where $H_{d}$ is the
usual Hausdorff measure and the normalization factor is such that
$\mathcal{H}_{d}\left(A\right)$ coincides with the usual Lebesgue $m_{d}$
measure for $d\in\mathbb{N}$. in $\mathbb{\hat{R}}^{N}$. We introduce on
$\wp\left(\mathbb{A}\right)$ the following order relation: given
$A,B\in\wp\left(\mathbb{A}\right)$, if $\left|A\right|\neq\mathfrak{c,}$ we
let $A\sqsubseteq B\Leftrightarrow\left|A\right|\leq\left|B\right|$; if
$\left|A\right|=\left|B\right|=\mathfrak{c,}$ we let

$A\sqsubseteq B\Leftrightarrow\
\mathcal{H}_{d}\left(A\cap\mathbb{\hat{R}}^{d}\right)\leq\mathcal{H}_{d}\left(B\cap\mathbb{\hat{R}}^{d}\right).$
(26)

If $A\sqsubseteq B$ and $B\sqsubseteq A,$ we will write $A\equiv B.$

We define $\mathfrak{D}\left(\mathbb{A}\right)$ as follows:
$\lambda\in\mathfrak{D}\left(\mathbb{A}\right)$ if and only if

$\lambda=\Xi\cup\mathfrak{A},$

where

  * •

$\Xi\in\mathfrak{B}\left(\mathbb{Q}\right)$;

  * •

$\mathfrak{A}\in\wp_{fin}\left(\wp\left(\mathbb{A}\right)\right);$

  * •

for all $A,B\in\mathfrak{A}$, the following property holds:

$A\sqsubseteq B\Rightarrow\left|A\cap\Xi\right|\leq\left|B\cap\Xi\right|;$
(27) $\left|A\right|>\aleph_{0}\Rightarrow\left|\Xi\cap
A\right|>\left|\mathbb{Q}\cap\Xi\right|^{2}.$ (28)

######  Lemma 36.

If $A_{1},\dots,A_{l}\subset\mathbb{A}$ and $F\subset\mathbb{A}$ is a finite
set, there exists $\Xi\in\wp_{fin}\left(\mathbb{A}\right)$ such that
$F\subseteq\Xi\mathbb{\ }$and

$\Xi\cup\left\\{A_{1},\dots,A_{l}\right\\}\in\mathfrak{D}\left(\mathbb{A}\right).$

###### Proof.

Let $A_{1},\dots,A_{l}\subset\mathbb{A}$ and $F\subset\mathbb{A}$ be given;
first we prove the Lemma in the case in which

$A_{j}\cap A_{k}=\varnothing\ \ \text{for}\ \ j\neq k.$ (29)

We order the $A_{j}$’s so that

$j<k\Rightarrow A_{j}\sqsubseteq A_{k},$

and we construct a sequence of labels

$\lambda_{k}=\Xi_{k}\cup\left\\{A_{1},\dots,A_{k}\right\\},\ k\leq l$

such that $\lambda_{k}\in\mathfrak{D}\left(\mathbb{A}\right),\
\lambda_{k}\subset\lambda_{k+1}$ and $\Xi_{k}\supseteq F$. We do it by
induction: for $k=1$, we set

$\lambda_{1}=\Xi_{1}\cup\left\\{A_{1}\right\\};\ \ \Xi_{1}=F.$

Trivially $\lambda_{1}\in\mathfrak{D}\left(\mathbb{A}\right)$, since there is
nothing to verify.

Now, if $k<l,$ in order to define $\Xi_{k+1},$ we consider four cases:

  * (i)

$\left|A_{k+1}\right|\leq\aleph_{0};$

  * (ii)

$\left|A_{k+1}\right|>\aleph_{0},$ and
$\mathcal{H}_{d}\left(A_{k+1}\cap\mathbb{\hat{R}}^{N}\right)=0;$

  * (iii)

$\left|A_{k+1}\right|>\aleph_{0},$ and
$\mathcal{H}_{d}\left(A_{k+1}\cap\mathbb{\hat{R}}^{N}\right)>0;$

  * (iv)

$\left|A_{k+1}\right|>\mathfrak{c}$.

(i) We take a finite set $F_{k+1}\subset A_{k+1}$ such that
$\left|F_{k+1}\right|>\left|A_{k}\cap\Xi_{k}\right|$ and we set

$\lambda_{k+1}=\Xi_{k+1}\cup\left\\{A_{1},\dots,A_{k},A_{k+1}\right\\},\ \
\Xi_{k+1}=\Xi_{k}\cup F_{k+1}.$

Then $\lambda_{k}\in\mathfrak{D}\left(\mathbb{A}\right)$ since (27) holds.

(ii) We take a finite set $F_{k+1}\subset A_{k+1}\backslash\mathbb{Q}$ such
that

$\left|F_{k+1}\right|>\max\left\\{\left|A_{k}\cap\Xi_{k}\right|,\
\left|\mathbb{Q}\cap\Xi_{k}\right|^{2}\right\\}$

and we set

$\lambda_{k+1}=\Xi_{k+1}\cup\left\\{A_{1},\dots,A_{k+1}\right\\},\ \
\Xi_{k+1}=\Xi_{k}\cup F_{k+1}.$

Then (27) trivially holds; moreover

$\left|A_{k+1}\cap\Xi_{k+1}\right|=\left|F_{k+1}\right|>\
\left|\mathbb{Q}\cap\Xi_{k}\right|^{2}.$

Then also (28) is satisfied.

(iii) if $\mathcal{H}_{d}\left(A_{k+1}\cap\mathbb{\hat{R}}^{N}\right)>0,$ then
$\left|A_{k+1}\right|\geq\mathfrak{c}$; if $\left|A_{k+1}\right|>\mathfrak{c}$
we are in case (iv); if $\left|A_{k+1}\right|=\mathfrak{c}$, we take
$F_{k+1}\subset\left(A_{k+1}\cap\mathbb{\hat{R}}^{N}\right)\backslash\left(\mathbb{Q}\cup\Xi_{k}\right)$
such that $\left|F_{k+1}\right|>\max\left\\{\left|A_{k}\cap\Xi_{k}\right|,\
\left|\mathbb{Q}\cap\Xi_{k}\right|^{2}\right\\}$ and we argue as in point
(ii).

(iv) $\left|A_{k+1}\right|>\mathfrak{c}$, we argue as in point (ii).

Now let us consider the case in which $A_{1},\dots,A_{l}$ does not satisfy
(29). In this case we take a finite family of sets
$\left\\{B_{1},\dots,B_{m}\right\\}$ which satisfies (29) and such that every
$A_{k}$ is the union of some $B_{j}$’s. Then, given
$\left\\{B_{1},\dots,B_{m}\right\\}$ and $\Xi,$ we have proved that there
exists

$\bar{\lambda}=\bar{\Xi}\cup\left\\{B_{1},\dots,B_{m}\right\\}\in\mathfrak{D}\left(\mathbb{A}\right),\
\text{with}\ F\subseteq\bar{\Xi}.$

Now, it is easy to chek that

$\lambda=\bar{\Xi}\cup\left\\{A_{1},\dots,A_{l},B_{1},\dots,B_{m}\right\\}\in\mathfrak{D}\left(\mathbb{A}\right)\hskip
5.0pt\hbox{\hskip 5.0pt\vrule width=4.0pt,height=6.0pt,depth=1.5pt\hskip
1.0pt}$

######  Lemma 37.

We have that
$\mathfrak{D}\left(\mathbb{Q}\right)\subset\mathfrak{D}\left(\mathbb{A}\right)$
and $\left(\mathfrak{D}\left(\mathbb{A}\right),\subseteq\right)$ is a directed
set.

###### Proof.

Given $\mathbb{H}_{n}\in\mathfrak{D}\left(\mathbb{Q}\right)$, eventually,
there is a set $A\in\mathfrak{A}$ such that
$\mathbb{H}_{n}\in\mathfrak{D}\left(\mathbb{A}\right)$ and so
$\mathfrak{D}\left(\mathbb{Q}\right)\subset\mathfrak{D}\left(\mathbb{A}\right).$

For $i=1,2$ let

$\lambda_{i}=\Xi^{i}\cup\left\\{A_{1}^{i},\dots,A_{l_{i}}^{i}\right\\}\in\mathfrak{D}_{0}\left(\mathbb{A}\right).$

We set

$F=\Xi^{1}\cup\Xi^{2};$

then, by Lemma (36), we can add points to $F$ and get a set
$\Xi\supset\Xi^{1}\cup\Xi^{2}$ so that (27) and (28) are satisfied.

Using Lemma 26, we define the label set

$\mathfrak{B}(\mathbb{A}):=\overline{\mathfrak{D}\left(\mathbb{A}\right)}.$
(30)

###  6.2 Cardinal numbers and numerosities

A property that is natural to expect, when one has a numerosity theory for all
sets in $\Lambda$, is that it must be coherent with cardinalities, namely it
must satisfy the following property:

Cantor property: if $A,B\subset\Lambda\backslash\mathbb{A}$ then

$\left|A\right|<\left|B\right|\Rightarrow\mathfrak{num}\left(A\right)<\mathfrak{num}\left(B\right).$
(31)

Using the labelling $\mathfrak{B}(\mathbb{A})$ defined by (30), the following
result holds:

######  Theorem 38.

If $A,B\subset\mathbb{A}$ then

$\left|A\right|<\left|B\right|\Rightarrow\mathfrak{num}\left(A\right)<\mathfrak{num}\left(B\right).$

###### Proof.

Given two sets $A,B\subset\mathbb{A}$ with $\left|A\right|<\left|B\right|$, we
take a label
$\lambda\supseteq\lambda_{0}:=\Xi\cup\left\\{A,B\right\\}\in\mathfrak{B}(\mathbb{A}).$
Then, by (27)

$\left|A\cap\lambda\right|=\left|A\cap\Xi\right|<\left|B\cap\Xi\right|=\left|B\cap\lambda\right|.$

The conclusion follows taking the $\Lambda$-limit

By Theorem 16, it follows that the numerosity function is well defined for
every set belonging to the family

$K=\left\\{E\in\mathbb{V}\left(F\right)\ |\
F\in\wp_{fin}\left(\mathbb{A}\right)\right\\},$

since $\mathfrak{D}(\mathbb{A})$ provides a label to the elements of $K.$ In
particular, by the Comparison Principle (Theorem 16.(iv)) and (25), we have
that for every set $E\subset\mathbb{R}^{N},$

$\mathfrak{num}\left(E\right)=\mathfrak{num}\left[\Psi\left(E\right)\right].$

Now, we want to extend the notion of numerosity to any set $A$ in $\Lambda$ in
such a way that the Cantor property (31) be satisfied. The simplest way to
realize this task is to consider the family of infinite sets

$\mathbb{S}:=\Lambda\backslash\left(\mathfrak{L}\cup\mathbb{A}\right)$

and to assign a label to each of them. We can take an injective map

$\Phi:\mathbb{S\rightarrow A}$

and set

$\ell(A)=\ell(\Phi\left(A\right)).$

Then, every set in $\Lambda\backslash\mathbb{A}$ has a label in
$\mathfrak{B}(\mathbb{A})$. By the Comparison Principle (Theorem 16.(iv)), we
get our desired final result:

######  Theorem 39.

If $A,B\in\Lambda\backslash\mathbb{A}$, then

$\left|A\right|<\left|B\right|\Rightarrow\mathfrak{num}\left(A\right)<\mathfrak{num}\left(B\right).$

##  7 Numerosity and measures

###  7.1 The general theory

Given a numerosity theory and a set $E\in\Lambda$, we put

$\mu_{\gamma}(E)=st\left(\frac{\mathfrak{num}\left(E\right)}{\mathfrak{\gamma}}\right),$

where $\gamma\in\mathbb{N}^{\ast}$. $\mu_{\gamma}$ is called numerosity
measure. As we will see, an interesting case occurs if you take
$\mathfrak{\gamma=num}\left(\left[0,1\right)\right)^{d}\ $with
$d\in\mathbb{R}_{\geq 0}$. In this case we will say that $\mu_{\gamma}$ is the
canonical $d$-dimensional numerosity measure.

######  Theorem 40.

The numerosity measure $\mu_{\gamma}$ satisfies the following properties:

  1. (i)

it is finitely additive: for all sets $A,B$

$\mu_{\gamma}\left(A\cup
B\right)=\mu_{\gamma}\left(A\right)+\mu_{\gamma}\left(B\right)-\mu_{\gamma}\left(A\cap
B\right);$

  2. (ii)

it is superadditive, namely given a denumerable partition
$\left\\{A_{n}\right\\}_{n\in\mathbb{N}}$ of a set $A\subset\mathbb{R}$, then

$\mu_{\gamma}\left(A\right)\geq\sum_{n=0}^{\infty}\mu_{\gamma}\left(A_{n}\right).$

###### Proof.

(i) This is a trivial consequence of the additivity of the numerosity.

(ii) By Theorem 16, we have that for all $N\in\mathbb{N}$,

$\mathfrak{num}\left(A\right)\geq\mathfrak{num}\left(\mathop{\displaystyle\bigcup}\limits_{n=0}^{N}A_{n}\right)=\sum_{n=0}^{N}\mathfrak{num}\left(A_{n}\right),$

hence

$st\left(\frac{\mathfrak{num}\left(A\right)}{\gamma}\right)\geq
st\left(\sum_{n=0}^{N}\frac{\mathfrak{num}\left(A_{n}\right)}{\gamma}\right)=\sum_{n=0}^{N}st\left(\frac{\mathfrak{num}\left(A_{n}\right)}{\gamma}\right);$

therefore,

$\mu_{\gamma}\left(A\right)\geq\sum_{n=0}^{N}\mu_{\gamma}\left(A_{n}\right).$

The conclusion follows taking the Cauchy limit in the above inequality for
$N\rightarrow\infty$.

###  7.2 Numerosity of the subsets of $\mathbb{R}^{N}$

In this section, we will show that $\mu_{\beta}$ agrees with the Lebesgue
measure, namely, if $E$ is a Lebesgue measurable set, then

$m_{L}\left(E\right)=\mu_{\beta}\left(E\right)=st\left(\frac{\mathfrak{num}\left(E\right)}{\beta}\right),$
(32)

where

$\beta:=\mathfrak{num}\left(\left[0,1\right)\right).$ (33)

First, let we show that this holds for intervals:

######  Theorem 41.

The numerosity measure $\mu_{\gamma}$ is translation invariant for any
$\gamma\in\mathbb{N}^{\ast}$. In particular, if $\gamma=\beta$ then for any
$\varepsilon=\frac{a}{b}\in[0,1)$ we have that
$\mu_{\beta}\left([0,\frac{a}{b})\right)=\frac{a}{b}$.

###### Proof.

Let $r\in\mathbb{R},E\subseteq\mathbb{R}$. By Property 27, as $E\equiv r+E$
(in the sense of the ordering $\sqsubseteq$), we have that for every
$\lambda\in\mathfrak{B}\left(\mathbb{A}\right),\lambda=\Xi\cup\mathfrak{A}$
necessarily $|E\cap\Xi|=|(E+r)\cap\Xi|$. By taking the $\Lambda$-limit, we get
our first claim.

As for the second, we just have to observe that
$[0,1)=[0,\frac{1}{b})\cup[\frac{1}{b},\frac{2}{b})\dots\cup[\frac{b-1}{b},1)$,
so by finite additivity and translation invariance we get
$\mu_{\beta}\left([0,\frac{1}{b})\right)=\frac{1}{b}$, and the thesis follows
as, similarly,
$[0,\frac{a}{b})=[0,\frac{1}{b})\cup[\frac{1}{b},\frac{2}{b})\dots\cup[\frac{a-1}{b},1)$.

Moreover, we have the following property:

######  Proposition 42.

The numerosity measure $\mu_{\beta}$ is subadditive on the $\sigma$-algebra of
Lebesgue measurable sets.

###### Proof.

Let $E\in\wp(\mathbb{R})$; wlog, we assume $E\in\wp\left(\mathbb{R}_{\geq
0}\right)$, as the result for a generic $E$ will then follow easily by
splitting $E=E^{+}\cup E^{-}$. Let

$E=\bigcup_{j\in\mathbb{N}}E_{j}$

be a partition of $E$, with all $E_{j}$’s Lebesgue measurable. Let
$\varepsilon=\frac{a}{b}\in[0,1)$; for $N$ large enough we have

$m_{L}\left(E\right)\leq
m_{L}\left(\bigcup_{j=1}^{N}E_{j}\right)+\varepsilon.$

Now $E\cap\left[-1,-1+\varepsilon\right]=\varnothing$, so

$m_{L}\left(E\right)\leq
m_{L}\left(\bigcup_{j=1}^{N}E_{j}\cup\left[-1,-1+\varepsilon\right]\right).$

By Property 36 of our labelling, where $d=1$, we have

$\mathfrak{num}\left(E\right)\leq\mathfrak{num}\left(\bigcup_{j=1}^{N}E_{j}\cup\left[-1,-1+\varepsilon\right]\right).$

By Theorem 41

$\mathfrak{num}\left(\bigcup_{j=1}^{N}E_{j}\cup\left[-1,-1+\varepsilon\right]\right)=\sum_{j=1}^{N}\mathfrak{num}\left(E_{j}\right)+\mathfrak{num}\left([-1,-1+\varepsilon]\right)\sim\sum_{j=1}^{N}\mathfrak{num}\left(E_{j}\right)+\varepsilon\beta,$

hence

$\mu_{\beta}\left(E\right)\leq\sum_{j=1}^{N}\mu_{\beta}\left(E_{j}\right)+\varepsilon\leq\sum_{j=1}^{\infty}\mu_{\beta}\left(E_{j}\right)+\varepsilon.$

The arbitrariness of $\varepsilon$ gives the desired inequality

$\mu_{\beta}\left(E\right)\leq\sum_{j=1}^{\infty}\mu_{\beta}\left(E_{j}\right).\hskip
5.0pt\hbox{\hskip 5.0pt\vrule width=4.0pt,height=6.0pt,depth=1.5pt\hskip
1.0pt}$

We can now prove our desired final result:

######  Theorem 43.

$\mu_{\beta}(E)=\mu_{L}(E)$ for all Lebesgue measurable sets
$E\subseteq\mathbb{R}$.

###### Proof.

By Theorems 40,41 and by Proposition 42 we have that $\mu_{\beta}$, restricted
to Lebesgue measurable sets, has the empty set property, it is countably
additive (as it is both subadditive and superadditive), it is invariant under
translation and it is normalized. Hence it must coincide with the Lebesgue
measure.

The arguments above could be generalized to prove that for any measurable set
$A\subset\mathbb{R}^{N}$ we have that

$m_{N}(A)=st\left(\frac{\mathfrak{num}\left(A\right)}{\beta^{N}}\right).$

Similarly, we can define the ”fractal measure” any fractal set
$A\subset\mathbb{R}^{N}$ as follows:

$\mathfrak{m}_{d}(A)=st\left(\frac{\mathfrak{num}\left(A\right)}{\beta^{d}}\right),\
\ \ d\in\left[0,N\right].$

We are not gonna study this fractal measure in detail here; however, it is not
difficult to chek that $\mathfrak{m}_{d}(A)$ concides with the normalized
Hausdorff measure $H_{d}$.

###  7.3 Numerosity and nonstandard measures

It is well known that the Lebesgue measure can be realized using a counting
procedure based on hyperfinite sets: this is, e.g., at the core of the
construction of Loeb measures, which is the most known and used of such
constructions. Loeb measures were introduced in mid-70’s, see [20]; see also
[22] for an overview of Loeb methods and applications, and [23] for an
overview of other applications of nonstandard analysis in measure theory. To
confront Loeb construction with our approach, here we shortly recall Loeb
construction following Goldblatt’s presentantion, see [18], Section 16.8.

Let $N$ be an infinite hypernatural number, and let
$S=\\{\frac{k}{N}\mid-N^{2}\leq k\leq N^{2},k$ hyperinteger$\\}$. Let
$\wp_{I}(S)$ the set of internal subsets of $S$, and for every
$A\in\wp_{I}(S)$ let

$m(A):=st\left(\frac{|A|}{N}\right),$

where $|A|$ denotes the internal cardinality of $A$. Then
$m:\wp_{I}\left(S\right)\rightarrow[0,+\infty]$ is a finitely additive measure
on $\wp_{I}\left(S\right)$. The Loeb measure is obtained by means of the usual
Carathéodory extension procedure applied to $m$ (we will denote also the Loeb
measure by $m$). What Loeb proved is that the Lebesgue measure can be seen as
a restriction of $m$, in the sense that for every Lebesgue measurable set $X$
the Lebesgue measure $m_{L}(X)$ is equal to the Loeb measure of the so-called
pre-shadow $st^{-1}(X)$ of $X$, namely

$m_{L}(X)=st\left(m\left(st^{-1}\left(X\right)\right)\right),$

where $st^{-1}(X)=\\{\xi\in S\mid st(\xi)\in X\\}$.

The similarity between our approach is that we have that, actually,
$\mu_{\beta}$ is obtained as the standard part of a quotient similar to Loeb’s
one. In fact,
$\mu_{\beta}\left(A\right)=st\left(\frac{|A^{\ast}\cap\Gamma|}{|[0,1)^{\ast}\cap\Gamma|}\right)$,
where:

  1. 1.

$|\cdot|$ denotes the internal cardinality of a set;

  2. 2.

$\Gamma$ is the hyperfinite set obtained by taking
$\lim_{\lambda\uparrow\Lambda}\lambda\cap\mathbb{R}$.

However, in our approach the use of Carathéodory extension procedure, as well
as of pre-shadows, is substituted with the choice of a particular labelling
set, which can be equivalently seen as a particular choice of the hyperfinite
set used in the quotient. A similar result in a general nonstandard setting
was first obtained by Bernstein and Wattenberg (see [13]; see also [15],
Section 2, for a comparison of Bernstein-Wattenberg’s result and Loeb
measures), who in fact proved that there exists hyperfinite subsets
$S\subseteq[0,1]^{\ast}$ such that for all Lebesgue measurable
$A\subseteq[0,1]$

$m_{L}(A)=st\left(\frac{|A^{\ast}\cap S|}{|S|}\right).$

As we said before, Theorem 43 provides a new proof of the above result by
taking

$S=\lim_{\lambda\uparrow\Lambda}\ \left(\lambda\cap\mathbb{R}\right).$

Finally, the problem of the relationship between numerosities and Lebesgue
measure in general has been addressed in [7, 8]. In these papers, the authors
introduced the notion of ”elementary numerosity” (see [7], Definition 1.1),
that we recall:

######  Definition.

An elementary numerosity on a set $\Omega$ is a function
$n:\wp\left(\Omega\right)\rightarrow[0,+\infty)$ defined on all subsets of
$\Omega$, taking values in the non-negative part of an ordered field
$\mathbb{F}\supseteq\mathbb{R}$, and such that the following two conditions
are satisfied:

  1. 1.

$n({x})=1$ for every point $x\in\Omega$;

  2. 2.

$n(A\cup B)=n(A)+n(B)$ whenever A and B are disjoint.

The main connection between the ”elementary numerosity” and Lebesgue measure
is given by the following result, which is one of the instances of Theorem 3.1
in [8]:

######  Theorem.

There exists an elementary numerosity
$\mathfrak{n}:\wp\left(\mathbb{R}\right)\rightarrow[0,+\infty)_{\mathbb{F}}$
such that
$m_{L}(X)=st\left(\frac{\mathfrak{n}\left(X\right)}{\mathfrak{n}\left([0,1)\right)}\right)$
for every Lebesgue measurable set $X$.

Once again, Theorem 43 provides another proof of the above result, as
$\mathfrak{num}$, when restricted to $\wp\left(\mathbb{R}\right)$ is, in fact,
an elementary numerosity on $\mathbb{R}$.

The interest of Theorem 43 lies on the fact that it is based on a numerosity
theory which satisfies many other additional properties.

## References

  * [1] Benci V., I numeri e gli insiemi etichettati, Conferenze del seminario di matematica dell’ Università di Bari, vol. 261, Laterza, Bari 1995.
  * [2] Benci V., _Ultrafunctions and generalized solutions,_ Adv. Nonlinear Stud. 13, (2013), 461–486, arXiv:1206.2257.
  * [3] Benci V., Di Nasso M., Alpha-theory: an elementary axiomatic for nonstandard analysis, Expo. Math. 21, (2003), pp. 355-386.
  * [4] Benci V., Di Nasso M., How to measure the infinite: Mathematics with infinite and infinitesimal numbers, World Scientific, Singapore, 2018.
  * [5] Benci, V., Di Nasso M., Numerosities of labelled sets: a new way of counting, Adv. Math. 21 (2003), pp. 505–67.
  * [6] Benci V., Di Nasso M., Forti M., An Aristotelian notion of size, Ann. Pure Appl. Logic 143 (2006), pp. 43–53.
  * [7] Benci V., Bottazzi E., Di Nasso M., Elementary numerosity and measures, Journal of Logic and Analysis, vol. 6 (2014).
  * [8] Benci V., Bottazzi E., Di Nasso M., Some applications of numerosities in measure theory, Rend. Lincei Mat. Appl. 26 (2015), pp. 1–11.
  * [9] Benci V., Bresolin L., Forti M., The Euclidean numbers, in press, arXiv:1702.04163.
  * [10] Benci V., Freguglia P, La matematica e l’infinito, Carocci Editore, Roma (2019).
  * [11] Benci V., Luperi Baglini L., _Ultrafunctions and applications,_ DCDS-S, Vol. 7, No. 4, (2014), 593-616. arXiv:1405.4152.
  * [12] Benci V., Luperi Baglini L., Squassina M., _Generalized solutions of variational problems and applications_ , ANONA, Vol. 9, (2018).
  * [13] Bernstein A.R., Wattenberg F., Nonstandard measure theory, in: Applications of model theory to algebra, analysis and probability (Ed. W. A. J. Luxemburg, Holt, Rinehart and Winston), New York (1969) pp. 171–185.
  * [14] Blass A., Di Nasso M., Forti M., Quasi-selective ultrafilters and asymptotic numerosities, Adv. Math. 231 (2012), pp. 1462–1486.
  * [15] Cutland H.J., Nonstandard Measure Theory and its Applications, Bull. London Math. Soc. 15 (1983), pp. 529–589, https://doi.org/10.1112/blms/15.6.529.
  * [16] Di Nasso M., Forti M., Numerosities of point sets over the real line, Trans. Amer. Math. Soc. 362 (2010), pp. 5355–5371.
  * [17] Forti M., Morana Roccasalvo G., Natural numerosities of sets of tuples, Trans. Amer. Math. Soc. 367 (2015), pp. 275–292
  * [18] Goldblatt R., Lectures on the Hyperreals: an Introduction to Nonstandard Analysis, Graduate Texts in Mathematics, Vol. 188, Springer, Berlin (1998).
  * [19] Keisler H. J., Foundations of Infinitesimal Calculus, Prindle, Weber & Schmidt, Boston (1976).
  * [20] Loeb P.A., Conversion from nonstandard to standard measure spaces and applications in probability theory, Trans. Amer. Math. Soc. 211 (1975), pp. 113–122.
  * [21] Mancosu P., Measuring the size of infinite collections of natural numbers: was Cantor’s theory of infinite number inevitable?, Rev. Symb. Logic 4 (2009), pp. 612–646.
  * [22] Ross D.A., Loeb measure and probability, NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. vol. 493, Kluwer Acad. Publ., Dordrecht, 1997, pp. 91–120.
  * [23] Ross D.A., Nonstandard measure constructions - Solutions and problems, in: Nonstandard Methods and Applications in Mathematics (N.J. Cutland, M. Di Nasso, D.A. Ross, eds.), Lect. Notes Log., Assoc. Symb. Logic 25 (2006), pp. 127-146.