# The Euclidean Universe

Vieri Benci  Dipartimento di Matematica, Università degli Studi di Pisa, Via
F. Buonarroti 1/c, Pisa, ITALY
I dedicate this work to my friend Eugenio who  more than 40 years ago, when we
were two  young guys in New York, he gave me a book  by Keisler and introduced
me to the beauty  of Nonstandard Analysis.

###### Contents

  1. 1 Introduction
  2. 2 The three Axioms
    1. 2.1 The Numerosity axiom
    2. 2.2 The Euclidean Field Axiom
    3. 2.3 The Center Axiom
  3. 3 Structure of the Euclidean numbers
    1. 3.1 The ordinal numerosities
    2. 3.2 The $\Lambda$-limit theorem
    3. 3.3 The real numbers
  4. 4 The Euclidean universe
    1. 4.1 $\Lambda$-limit of sets
    2. 4.2 $\Lambda$-limit of functions
    3. 4.3 Hyperfinite sets
  5. 5 Nonstandard Analysis
    1. 5.1 Nonstandard theories
    2. 5.2 The Normal Universe
    3. 5.3 The $\alpha$-theory
    4. 5.4 About the idea of continuum
  6. 6 Euclidean Calculus
    1. 6.1 Normal functions and sets
    2. 6.2 The notion of derivative
    3. 6.3 The integral
  7. 7 Consistency of the axioms
    1. 7.1 The construction of the field $\mathbb{E}$
    2. 7.2 Proof of the consistency of Axioms 1-3

##  1 Introduction

In this paper, we introduce a mathematical structure called Euclidean
Universe. This structure provides a basic framework for Non-Archimedean
Mathematics, namely the Mathematics based on infinite and infinitesimal
numbers.

The Euclidean Universe is defined by three axioms which have been chosen in
such a way to appear absolutely natural. The first axiom introduce the
infinite numbers as the numerosities of infinite sets in such a way that the V
Euclidean common notion

the whole is larger than the part

be preserved. In the second axiom we introduce the Euclidean line with the
following peculiarity: if any magnitude can be ”represented” by a point on the
Euclidean line, then also the infinite (and consequently the infinitesimal)
magnitudes have this right. Then the Euclidean line must be larger than the
real line. The last axiom is more technical and it is necessary to make the
Euclidean line to include (a copy of) the real numbers.

Actually, this paper can be considered a new introduction to the Non-
Archimedean Mathematics in the spirit of Veronese [20, 21] and Levi-Civita
[18]. Moreover, these axioms are sufficiently strong to include the basic
principles of Nonstandard Analysis such as the Leibniz Principle and the
Saturation Principle. In particular the Euclidean line incudes, as subfields,
infinitely many copies of the hyperreal numbers of any saturation less or
equal to the first inaccessible number (see Def. 1).

Then, the Euclidean Universe includes many results of Non-Archimedean
Mathematics obtained in the last 30 years.

For a recent historical and foundational analysis of the underlying ideas, we
refer to [11] and the references therein.

Acknowledgement: I want to express my gratitude to Prof. Marco Forti for the
many interesting discussions on these topics. Also a big thank is due to a
referee whose remarks made me to improve considerably this paper.

##  2 The three Axioms

###  2.1 The Numerosity axiom

The first axiom defines the notion of numerosity. The notion of numerosity was
introduced in [2, 6, 9] as a generalization of finite cardinality that also
applies to infinite sets. The main feature of numerosities is that they
preserve the spirit of the ancient Euclidean principle that the whole is
larger than the part; indeed, the numerosity of a proper subset is strictly
smaller than the numerosity of the whole set.

In principle it would be desirable to define the numerosity for the class of
all sets; however, in order to develop the theory, it is convenient to work in
a ”universe” which is a set closed with respect to the main set operations
provided that it is very large. In order to do this we recall a well known
notion in set theory:

######  Definition 1.

A cardinal number $\chi$ is inaccessible if it is not a sum of fewer than
$\chi$ cardinals that are less than $\chi$ and $\zeta<\chi$ implies
$2^{\zeta}<\chi$. $\chi$ is strongly inaccessible if it is inaccessible and
uncountable.

The first inaccessible cardinal number is $\aleph_{0}.$ The first strongly
inaccessible cardinal number will be denoted by $\kappa.$ The existence of
sets of strongly inaccessible cardinality is established by the Axiom of
Inaccessibility which is independent from ZFC. We will assume this axiom and
in particular we will assume that there exists a set of atoms111In set theory,
an atom $a$ is any entity that is not a set, namely $a$ is an atom if and only
if  $\forall x,\ x\notin a$ $\mathbb{A}$ having cardinality $\kappa.$ Then we
can define a ”universe” $\Lambda$ defined as follows:

$\Lambda=\left\\{X\in V(\mathbb{A})\ |\ \left|X\right|<\kappa\right\\}$ (1)

where for any set $A,$ $V(A)$ denotes the superstructure over $A$ namely

$V(A)=\bigcup_{n\in\mathbb{N}}V_{n}(A)$

with

$V_{0}(A)=A$

and, for every $n\in\mathbb{N}$,

$V_{n+1}(A)=V_{n}(A)\cup\mathfrak{\wp}\left(V_{n}(A)\right).$ (2)

We will refer to $\Lambda$ as to the accessible universe since its sets have
strongly accessible cardinality (and finite rank222The rank of $\varnothing\
$is $0$. The rank of a set $E\neq 0$ is the least ordinal number greater than
the rank of any element of the set.). $\Lambda$ can be split as follows

$\Lambda=\Lambda_{S}\cup\mathbb{A}$

where $\Lambda_{S}$ is a family of sets. Notice that $\mathbb{A}\notin\Lambda$
since $\left|\mathbb{A}\right|=\kappa.$

Now we are ready to state our first axiom:

######  Axiom 1.

(Numerosity axiom) The numerosity is a surjective map

$\mathfrak{num}:\Lambda_{S}\rightarrow\mathfrak{N},\ \ \
\mathfrak{N}\subset\mathbb{A}$

which satisfies the following properties: if $a,b\in\Lambda$ and
$A$,$B$,$A^{\prime}$,$B^{\prime}\in\Lambda_{S},$

  1. 1.

$\mathfrak{num}\left(\left\\{a\right\\}\right)=\mathfrak{num}\left(\left\\{b\right\\}\right)$

  2. 2.

if $A\subset B\ $strictly, then

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

  3. 3.

if $A\cap B=\varnothing,$
$\mathfrak{num}\left(A\right)=\mathfrak{num}\left(A^{\prime}\right),\
\mathfrak{num}\left(B\right)=\mathfrak{num}\left(B^{\prime}\right),$ then

$\mathfrak{num}\left(A\cup B\right)=\mathfrak{num}\left(A^{\prime}\cup
B^{\prime}\right)$

  4. 4.

if $\mathfrak{num}\left(A\right)=\mathfrak{num}\left(A^{\prime}\right),\
\mathfrak{num}\left(B\right)=\mathfrak{num}\left(B^{\prime}\right),$ then

$\mathfrak{num}\left(A\times B\right)=\mathfrak{num}\left(A^{\prime}\times
B^{\prime}\right)$

  5. 5.

if $A\in\Lambda_{S}$ and $b\in\Lambda,$ then
$\mathfrak{num}\left(A\times\left\\{b\right\\}\right)=\mathfrak{num}\left(A\right).$

If $F$ and $F^{\prime}$ are finite sets of the same cardinality, by 1.1 and
1.3, it follows that
$\mathfrak{num}\left(F\right)=\mathfrak{num}\left(F^{\prime}\right);$ then, by
1.5, the numerosities of finite sets can be identified with the natural
numbers $\mathbb{N}$. By 1.3, $\mathfrak{N}$ can be equipped with an
”addition” by setting

$\sigma+\tau=\mathfrak{num}\left(A\cup B\right)$

where $\sigma=\mathfrak{num}\left(A\right),$
$\tau=\mathfrak{num}\left(B\right)$ and $A\cap B=\varnothing$; similarly, by
1.4, we can define a ”multiplication”:

$\sigma\cdot\tau=\mathfrak{num}\left(A\times B\right)$

Clearly $0=\mathfrak{num}\left(\varnothing\right)$ is the neutral element with
respect to the addition and, by 1.5,
$\mathfrak{num}\left(\left\\{b\right\\}\right)=1$ is the neutral element with
respect to the multiplication for any $b\in\Lambda.$

###  2.2 The Euclidean Field Axiom

Our second axioms identifies the Euclidean (straight) line with field
$\mathbb{E}_{\kappa}\subseteq\mathbb{A}$. Usually the Euclidean line is
identified with the real line, however we think that this point of view is too
restrictive. In fact, the main intuitive peculiarities of the Euclidean line
are the following:

  * •

two oriented segments of the Euclidean line can be added or subtracted;

  * •

if we choose a unitary segment of the Euclidean line, two segments can be
multiplied or divided;

  * •

once we have chosen two distinguish points $O$ and $U$ on the line, every
magnitude can be posed in a biunivocal correspondence with a point (provided
that the unitary magnitude has been defined).

Then, if we take $0=O$ and $1=U,$ the Euclidean line gets the stucture of
ordered field and its points can be identified with numbers. Since the
numerosities can be considered magnitudes, the Euclidean line should be richer
than the real line. We can formalize these intuitive remarks by the following
axiom:

######  Axiom 2.

(Euclidean Field Axiom) There is an ordered field
$\mathbb{E}_{\kappa}\subset\mathbb{A}$ such that

  * •

$\mathbb{E}_{\kappa}$ contains the set numerosities $\mathfrak{N}$ and the
field operations $+,\cdot$ coincide with the numerosity operations;

  * •

for every $\xi\in\mathbb{E}_{\kappa},\ $there exists $E\in\Lambda,$ such that

$\left|\xi\right|<\mathfrak{num}\left(E\right).$ (3)

We will refer to $\mathbb{E}_{\kappa}$ as the Euclidean line or the Euclidean
field and its elements will be called Euclidean numbers. $\mathbb{E}_{\kappa}$
contains infinite numbers, then it is a non-Archimedean field. Now let us
recall some basic definitions relative to non-Archimedean fields. Since
$\mathbb{N\subset E}_{\kappa}$, the following definition makes sense:

######  Definition 2.

Let $\xi\in\mathbb{E}_{\kappa}$. We say that:

  * •

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

  * •

$\xi$ is finite (or bounded) if $\exists n\in\mathbb{N}$ such as $|\xi|<n$;

  * •

$\xi$ is infinite (or unbounded) if $\forall n\in\mathbb{N}$, $|\xi|>n.$

The following proposition establishes some relations among Euclidean numbers:

######  Proposition 3.

We have the following relations:

  * •

(i) If $\varepsilon$ e ${\delta}$ are infinitesimal, also
$\varepsilon+{\delta}$ ,$\mathbf{\;}\varepsilon-{\delta}$ and
$\varepsilon\cdot{\delta}$ are infinitesimal.

  * •

(ii) If $\xi$ e $\sigma$ are bounded also $\xi+\sigma$
,$\mathbf{\;}\xi-\sigma$ e $\xi\cdot\sigma$ are bounded.

  * •

(iii) If $\theta$ e $\tau$ are infinite, also $\theta\cdot\tau$ is infinite;
moreover if $\theta$ and $\tau$ are postive infinite (or negative), also
$\omega+\tau$ is postive infinite (or negative).

  * •

(iv) If $\varepsilon$ is infinitesimal and $\xi$ is bouded
$\varepsilon\cdot\xi$ is infinitesimal; moreover if $\varepsilon\neq 0$ and
$\xi$ is not infinitesimal, $\xi/\varepsilon$ is infinite.

Proof. The proof of this proposition is easy and it is left to the reder.

$\square$

######  Definition 4.

We say that two numbers $\xi,\zeta\in\mathbb{E}$κ 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. Then the following definition comes naturally

######  Definition 5.

If $\xi\in\mathbb{E}_{\kappa}$, the monad of $\xi$ is the set of all numbers
that are infinitely close to it:

$\mathfrak{m}\mathfrak{o}\mathfrak{n}(\xi)=\\{\zeta\in\mathbb{E}_{\kappa}\mathbb{\
}|\ \xi\sim\zeta\\},$

The galaxy of $\xi$ is the set of all numbers that are finitely close to it:

$\mathfrak{gal}(\xi)=\\{\zeta\in\mathbb{E}_{\kappa}\mathbb{\ }|\
\xi-\zeta\text{\ is a finite number}\\}.$

###  2.3 The Center Axiom

The notion of monad allows to state our last axiom:

######  Axiom 3.

Every monad $\mu$ has a distinguished point called center of $\mu$ and denoted
by $Ctr(\mu)$; the set $\mathfrak{C}$ of all the centers is an additive
subgroup of $\mathbb{E}_{\kappa}$ containing $\mathfrak{N.}$

For every $\xi\in\mathbb{E}_{\kappa}$, we will use the notation

$ctr\left(\xi\right)=Ctr(\mathfrak{mon}\left(\xi\right));$

so, every number $\xi\in\mathbb{E}_{\kappa}$ can be decomposed as follows:

$\xi=x+\varepsilon$ (4)

where $x=ctr\left(\xi\right)\ $and $\varepsilon\sim 0$. Then
$\mathbb{E}_{\kappa}$ can be decomposed as follows:

$\mathbb{E}_{\kappa}=\mathfrak{C}\times\mathfrak{mon}\left(0\right)$

and

$ctr:\mathbb{E}_{\kappa}\rightarrow\mathfrak{C}$

is a projection.

##  3 Structure of the Euclidean numbers

In this section we will examine some peculiarities of $\mathbb{E}_{\kappa}.$
In particular, we will see that $\mathbb{E}_{\kappa}$ contains the ordinal
numbers of accessible cardinalities and the real numbers. Moreover we will
introduce the notion $\Lambda$-limit which, in this context, is a very basic
tool.

###  3.1 The ordinal numerosities

In this subsection we introduce a set $\mathbf{Ord}\subset\mathfrak{N}$ that
is isomorphic (in a sense specified below) to the initial segment of length
$\kappa$ of ordinal numbers .

######  Definition 6.

The set $\mathbf{Ord}\subset\mathfrak{N}$ 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 recursive
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 $\mathbb{N}\subset\mathbf{Ord}$).

  * •

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 particular, $\omega=\mathfrak{num}\left(\mathbb{N}\right)$ is an ordinal.
This 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 in
$\mathbb{E}_{\kappa}$.

It is easy to see that $\mathbf{Ord}$ is well ordered and hence it is
isomorphic to the initial segment of length $\kappa$ of the full class of the
ordinals. Obviously, not all numerosities are ordinals: for example,
$\omega-1=\mathfrak{num}\left(\mathbb{N}^{+}\right)=\mathfrak{num}\left(\mathbb{N}\backslash\left\\{0\right\\}\right)$
is not an ordinal.

The most remarkable thing in this theory is that the numerosity operations $+$
and $\cdot,$ correspond to the natural (or Hessenberg) operations between
ordinals. We refer to [13] for an in-depth analysis of this topic.

######  Remark 7.

Notice that the existence of $\mathbf{Ord}$ depends only on Axiom 1.

###  3.2 The $\Lambda$-limit theorem

We set

$\mathfrak{L}=\left\\{\lambda\in\Lambda\ |\ \lambda\ \text{is a finite
set}\right\\}$

and

$\mathfrak{F}\left(\mathfrak{L},\mathbb{E}_{\kappa}\right)=\left\\{\varphi\in\mathbb{E}_{\kappa}^{\mathfrak{L}}\
|\ \exists A\in\Lambda,\forall\lambda\in\mathfrak{L},\
\varphi(\lambda)=\varphi(\lambda\cap A)\right\\}.$

Since $\left(\mathfrak{L},\subseteq\right)$ is a directed sets, the elements
of $\mathfrak{F}\left(\mathfrak{L},\mathbb{E}_{\kappa}\right)$ are nets. The
set $\mathfrak{F}\left(\mathfrak{L},\mathbb{E}_{\kappa}\right)$ is a partially
ordered commutative algebra over $\mathbb{E}_{\kappa}$ with the operations

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

######  Theorem 8.

($\mathbf{\Lambda}$-limit theorem) There is a unique ring homomorphism

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

such that,

$\forall A\in\Lambda,\ J(\psi_{A})=\mathfrak{num}\left(A\right)$

where

$\psi_{A}\left(\lambda\right)=\left|A\cap\lambda\right|.$ (5)

Proof: Let $\mathfrak{F}_{q}\left(\mathfrak{L},\mathbb{E}_{\kappa}\right)$ be
the $\mathbb{E}_{\kappa}$-subalgebra of
$\mathfrak{F}\left(\mathfrak{L},\mathbb{E}_{\kappa}\right)$ ”generated” by
$\left\\{\psi_{A}\ |\ A\in\Lambda\right\\}\ $namely the subset of the elements
$\varphi$ of $\mathfrak{F}\left(\mathfrak{L},\mathbb{E}_{\kappa}\right)$ which
can be written as follows:

$\varphi(\lambda)=\frac{\sum_{A\in\mathcal{A}}a_{A}\psi_{A}(\lambda)}{\sum_{B\in\mathcal{B}}b_{A}\psi_{B}(\lambda)}$

where $\mathcal{A},\mathcal{B}$ are finite subsets of $\Lambda$,
$a_{A},b_{A}\in\mathbb{E}_{\kappa}$, $\psi_{A},\psi_{B}$ are defined by (5)
and $\ \forall\lambda\in\mathfrak{L,\
}\sum_{B\in\mathcal{B}}b_{A}\psi_{B}(\lambda)\neq 0$. We define a field
homomorphism

$J_{q}:\mathfrak{F}_{q}\left(\mathfrak{L},\mathbb{E}_{\kappa}\right)\rightarrow\mathbb{E}_{\kappa}$

as follows:

$J_{q}(\varphi)=\frac{\sum_{A\in\mathcal{A}}a_{A}\cdot\mathfrak{num}\left(A\right)}{\sum_{B\in\mathcal{B}}b_{A}\cdot\mathfrak{num}\left(B\right)}$

Since $\mathop{\mathrm{I}m}\left(J_{q}\right)=\mathbb{E}_{\kappa}$ is a field,
$\ker\left(J_{q}\right)$ is a maximal ideal in
$\mathfrak{F}_{q}\left(\mathfrak{L},\mathbb{E}_{\kappa}\right);$ hence, the
set

$\mathcal{U}_{0}=\left\\{Q\in\mathfrak{L}\ |\
\exists\psi\in\ker\left(J_{q}\right),\ Q=\psi^{-1}(0)\right\\}$

is a filter over $\mathfrak{L}$. We denote by $\mathcal{U}$ an ultrafilter
such that $\mathcal{U}_{0}\subseteq\mathcal{U}$. Then also

$\mathfrak{I}:=\left\\{\psi\in\mathfrak{F}\left(\mathfrak{L},\mathbb{E}_{\kappa}\right)\
|\ \exists Q\in\mathcal{U},\ \forall\lambda\in Q,\ \psi(\lambda)=0\right\\}$

is a maximal ideal in
$\mathfrak{F}\left(\mathfrak{L},\mathbb{E}_{\kappa}\right)$ and hence

$\mathbb{F}:=\mathfrak{F}\left(\mathfrak{L},\mathbb{E}_{\kappa}\right)/\mathfrak{I}$

is a field. We will see that $\mathbb{F}$ is isomorphic to
$\mathbb{E}_{\kappa}.$ We denote by $\left[\varphi\right]$ a generic element
of $\mathbb{F}$ and we claim that

$\forall\left[\varphi\right]\in\mathbb{F},\ \exists\bar{\mu}\in\mathfrak{L},\
\exists Q\in\mathcal{U},\ \forall\lambda\in Q,\
\varphi\left(\lambda\right)=\varphi\left(\bar{\mu}\right),$

namely

$\left[\varphi\right]=\left[C_{\varphi\left(\bar{\mu}\right)}\right]$ (6)

where $\lambda\mapsto C_{\xi}\left(\lambda\right)$ denotes the net identically
equal to $\xi.$ In order to prove (6) we set

$R^{-}:=\left\\{\mu\in\mathfrak{L}\ |\
\left[C_{\varphi(\mu)}\right]<\left[\varphi\right]\right\\}$ (7)
$R^{0}:=\left\\{\mu\in\mathfrak{L}\ |\
\left[C_{\varphi(\mu)}\right]=\left[\varphi\right]\right\\}$ (8)
$R^{+}:=\left\\{\mu\in\mathfrak{L}\ |\
\left[C_{\varphi(\mu)}\right]>\left[\varphi\right]\right\\}$ (9)

By (7), if $R^{-}\neq\varnothing,\ \forall\mu\in R^{-},\ \exists
Q_{\mu}^{-}\in\mathcal{U}$ such that

$\forall\lambda\in Q_{\mu}^{-},\
C_{\varphi(\mu)}\left(\lambda\right)<\varphi\left(\lambda\right)$

then,

$\mu\in R^{-}\cap Q_{\mu}^{-}\Rightarrow\
C_{\varphi(\mu)}\left(\mu\right)<\varphi\left(\mu\right)$

and since, by definition $C_{\varphi(\mu)}\left(\mu\right)=\varphi(\mu),$ it
follows that

$\forall\mu\in R^{-},\ R^{-}\cap Q_{\mu}^{-}=\varnothing$

and hence,

$R^{-}\notin\mathcal{U}$

By (9), arguing in the same way, we have that

$R^{+}\notin\mathcal{U}$

Since $\left(R^{-}\cup R^{+}\right)\cup R^{0}=\mathfrak{L}$, it follows that
$R^{0}\in\mathcal{U}$ and hence $R^{0}\neq\varnothing$. Now, if you take
$\bar{\mu}$ in $R^{0}$, there is $Q^{0}\in\mathcal{U}$ such that

$\forall\lambda\in Q^{0},\
\varphi\left(\lambda\right)=C_{\varphi(\bar{\mu})}\left(\lambda\right)=\varphi(\bar{\mu}).$

namely (6) is satisfied. Now, we can extend $J_{q}$ to
$\mathfrak{F}\left(\mathfrak{L},\mathbb{E}_{\kappa}\right);$ given
$\varphi\in\mathfrak{F}\left(\mathfrak{L},\mathbb{E}_{\kappa}\right)$, we set

$J\left(\varphi\right)=\varphi(\bar{\mu}).$

So every function $\varphi$ in
$\mathfrak{F}\left(\mathfrak{L},\mathbb{E}_{\kappa}\right)$ is eventually
constant in the sense that

$\exists\xi\in\mathbb{E}_{\kappa},\exists Q\in\mathcal{U},\forall\lambda\in
Q,\ \varphi\left(\lambda\right)=\xi.$

Then $\mathbb{F}$ is isomorphic to $\mathbb{E}_{\kappa}.$

It remains to prove the uniqueness of $J.$ Let us assume that $J_{1}$ and
$J_{2}$ extend $J_{q}$ to all
$\mathfrak{F}\left(\mathfrak{L},\mathbb{E}_{\kappa}\right).$ We have to prove
that for every
$\varphi\in\mathfrak{F}\left(\mathfrak{L},\mathbb{E}_{\kappa}\right)$

$J_{1}\left(\varphi\right)=J_{2}\left(\varphi\right)$

We set

$c_{1}=J_{1}\left(\varphi\right);\ c_{2}=J_{2}\left(\varphi\right)$
$\displaystyle A_{\varphi}$ $\displaystyle=$
$\displaystyle\left\\{J_{q}\left(\psi\right)\ |\
\psi\in\mathfrak{F}_{q}\left(\mathfrak{L},\mathbb{E}_{\kappa}\right),\ \exists
Q\in\mathcal{U}_{0},\ \forall\lambda\in Q,\ \psi(\lambda),\
\psi\leq\varphi\right\\};\ $ $\displaystyle B_{\varphi}$ $\displaystyle=$
$\displaystyle\left\\{J_{q}\left(\psi\right)\
|\psi\in\mathfrak{F}_{q}\left(\mathfrak{L},\mathbb{E}_{\kappa}\right)\ \
\exists Q\in\mathcal{U}_{0},\ \forall\lambda\in Q,\ \psi(\lambda),\
\psi>\varphi\right\\}$

Clearly $\ \forall a\in A_{\varphi},\ \forall b\in B_{\varphi},\ $

$a\leq c_{1}\leq b\ \text{\ \ and\ \ }\ a\leq c_{2}\leq b.$ (10)

and hence, assuming that $c_{1}\leq c_{2}$

$\forall a\in A_{\varphi},\ \forall b\in B_{\varphi},\ 0\leq c_{2}-c_{1}\leq
b-a$

Since $A_{\varphi}\cup
B_{\varphi}=\mathfrak{F}_{q}\left(\mathfrak{L},\mathbb{E}_{\kappa}\right)$
contains $\frac{1}{\mathfrak{num}\left(E\right)}$ for any set
$E\in\Lambda_{S}\backslash\left\\{0\right\\}$, we have that

$\ 0\leq c_{2}-c_{1}\leq\frac{1}{\mathfrak{num}\left(E\right)}$

and so, by (3), $c_{2}=c_{1}.$

$\square$

######  Definition 9.

The number $J\left(\varphi\right)$ is called $\Lambda$-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 some of the properties of the usual limit over a net:

  * •

If eventually $\varphi(\lambda)\geq\psi(\lambda)$, then

$\lim_{\lambda\uparrow\Lambda}\
\varphi(\lambda)\geq\lim_{\lambda\uparrow\Lambda}~{}\psi(\lambda).$

  * •

If $\forall q\in\mathbb{Q}$, $C_{q}\left(\lambda\right)=q$, then

$\lim_{\lambda\uparrow\Lambda}\ C_{q}(\lambda)=q.$

  * •

For all
$\varphi,\psi\in\mathfrak{F}\left(\mathfrak{L},\mathbb{E}_{\kappa}\right)$

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

In this framework, $\Lambda$ can be regarded as the ”point at infinity” of
$\mathfrak{L}$. The $\Lambda$-limit is not a limit in a topological sense, in
fact there are also strong differences with a topological limit; we list some
of them:

  * •

Every net
$\varphi\in\mathfrak{F}\left(\mathfrak{L},\mathbb{E}_{\kappa}\right)$ has a
limit $L\in\mathbb{E}_{\kappa}$.

  * •

If $\forall\lambda\in\mathfrak{L}$, $\varphi(\lambda)\neq 0$, then
$\lim_{\lambda\uparrow\Lambda}\varphi(\lambda)\neq 0;$ in fact,

$\lim_{\lambda\uparrow\Lambda}\varphi(\lambda)\cdot\lim_{\lambda\uparrow\Lambda}\frac{1}{\varphi(\lambda)}=1$

and hence $\lim_{\lambda\uparrow\Lambda}\varphi(\lambda)\neq 0.$

  * •

If, $\xi\in\mathbb{E}_{\kappa}\backslash\mathbb{R}$,

$\lim_{\lambda\uparrow\Lambda}C_{\xi}\left(\lambda\right)\neq\xi$

For example, take

$\omega:=\lim_{\lambda\uparrow\Lambda}\left|\lambda\cap\mathbb{N}\right|;$

then $\forall\lambda\in\mathfrak{L,\
}\left|\lambda\cap\mathbb{N}\right|<\omega,$ so

$0>\lim_{\lambda\uparrow\Lambda}\left(\left|\lambda\cap\mathbb{N}\right|-\omega\right)=\lim_{\lambda\uparrow\Lambda}\left|\lambda\cap\mathbb{N}\right|-\lim_{\lambda\uparrow\Lambda}\omega=\omega-\lim_{\lambda\uparrow\Lambda}\omega.$

and hence

$\omega>\lim_{\lambda\uparrow\Lambda}\omega.$

The last statement suggests the following notation: for any
$\xi\in\mathbb{E}_{\kappa}$, we set

$\xi^{\ast}=\lim_{\lambda\uparrow\Lambda}\xi.$ (11)

###  3.3 The real numbers

We remark that the notion and the first properties of the $\Lambda$-limit do
not depend on Axiom 3. In this section we will see that also Axiom 3 is very
relevant.

######  Definition 10.

An Euclidean number is called standard if it is finite and it is the center of
a monad. The set of standard points will be denoted by $\mathbb{R}$, namely

$\mathbb{R}:=\mathfrak{C}\cap\mathfrak{gal}(0).$

If $\xi$ is a finite number, then $ctr(\xi)$ is called standard part of $x$
and will be denoted also by $st\left(\xi\right).$

First let us examine some (obvious) properties of the function $st(\cdot).$

######  Proposition 11.

Let $\xi$ and $\zeta$ be finite numbers, then

  1. 1.

$\xi\in\mathbb{R}\Leftrightarrow st\left(\xi\right)=\xi;$

  2. 2.

$\xi\leq\zeta\Rightarrow st\left(\xi\right)\leq st\left(\zeta\right);$

  3. 3.

$st\left(\xi+\zeta\right)=st\left(\xi\right)+st\left(\zeta\right);$

  4. 4.

$st\left(\xi\cdot\zeta\right)=st\left(\xi\right)\cdot st\left(\zeta\right);$

  5. 5.

if $st\left(\zeta\right)\neq 0,$ then
$st\left(\frac{\xi}{\zeta}\right)=\frac{st\left(\xi\right)}{st\left(\zeta\right)}.$

Proof: The first four statements trivially descend from (4) and Prop. 3. In
order to prove 11.4, we put

$\displaystyle\xi$ $\displaystyle=$ $\displaystyle r+\varepsilon$
$\displaystyle\zeta$ $\displaystyle=$ $\displaystyle s+\theta$

where $r,s\in\mathbb{R}$, $\varepsilon\sim\theta\sim 0.$ Then,

$st\left(\xi\cdot\zeta\right)=st\left[\left(r+\varepsilon\right)\left(s+\theta\right)\right]=st\left[rs+\left(\varepsilon
s+\theta r+\varepsilon\theta\right)\right]$

Since $\varepsilon s+\theta r+\varepsilon\theta\sim 0$, we have that
$st\left(\xi\cdot\zeta\right)=rs=st\left(\xi\right)\cdot
st\left(\zeta\right).$ Let us prove 11.5; by 11.4 we have that

$st\left(\zeta\right)\cdot
st\left(\frac{\xi}{\zeta}\right)=st\left(\zeta\cdot\frac{\xi}{\zeta}\right)=st\left(\xi\right);$

hence

$st\left(\frac{\xi}{\zeta}\right)=\frac{st\left(\xi\right)}{st\left(\zeta\right)}.$

$\square$

######  Theorem 12.

The set of standard numbers $\mathbb{R}$ is isomorphic to the set of real
numbers.

Proof. We will prove that every Cauchy sequence of rationals is convergent to
some $L\in\mathbb{R}$ with respect to the metric topology. Let $x_{n}$ be a
Cauchy sequence in $\mathbb{Q}$. We set

$\varphi(\lambda):=x_{\left|\mathbb{N}\cap\lambda\right|}$

and

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

Then, by Prop. 11.1, $L\in\mathbb{R}$. We have to prove that $L$ is the Cauchy
limit of $x_{n}$. We choose a number $\varepsilon\in\mathbb{Q}^{+}$; then,
there exists $n_{0}$ such that $\forall n,m\geq n_{0},$

$\left|x_{n}-x_{m}\right|<\varepsilon$

Now take $\lambda_{0}\in\mathfrak{L}$ such that
$\left|\mathbb{N}\cap\lambda_{0}\right|\geq n_{0};\ $thus,
$\forall\lambda\supset\lambda_{0},\ $we have that
$\left|\mathbb{N}\cap\lambda\right|\geq n_{0}.$ Then

$\left|\varphi(\lambda)-x_{m}\right|=\left|x_{\left|\mathbb{N}\cap\lambda\right|}-x_{m}\right|<\varepsilon$

and taking the $\Lambda$-limit, we get the conclusion:

$\varepsilon>\lim_{\lambda\uparrow\Lambda}\
\left|\varphi(\lambda)-x_{m}\right|=\left|\lim_{\lambda\uparrow\Lambda}\
\varphi(\lambda)-x_{m}\right|=\left|L-x_{m}\right|.$

$\square$

From now on, the set $\mathbb{R}$ of standard numbers will be identified with
the set of real numbers, namely the real number will be considered ”special”
points on the Euclidean line.

Given a net $\varphi:\Lambda\rightarrow\mathbb{R}$, since $\Lambda$ is a
directed set, also the Cauchy limit is well defined:

$L=\lim_{\lambda\rightarrow\Lambda}\varphi(\lambda)\Leftrightarrow\forall\varepsilon\in\mathbb{R}^{+},\exists\lambda_{0}\in\mathfrak{L},\forall\lambda\supset\lambda_{0},|\varphi(\lambda)-L|\
\leq\varepsilon$ (12)

Notice that in order to distinguish the Cauchy limit (12) from the
$\Lambda$-limit, we have used the symbols ”$\lambda\rightarrow\Lambda$” and
”$\lambda\uparrow\Lambda$” respectively.

The standard part of a number is related to the Cauchy notion of limit. If a
real net $x_{\lambda}$ admits the Cauchy limit, the relation with the
$\Lambda$-limit is given by the following identity:

$\lim_{\lambda\rightarrow\Lambda}\
x_{\lambda}=st\left(\lim_{\lambda\uparrow\Lambda}\ x_{\lambda}\right)$ (13)

Another important relation between the two limits is the following:

######  Proposition 13.

If

$\lim_{\lambda\uparrow\Lambda}\ x_{\lambda}=\xi\in\mathbb{E}_{\kappa}$

and $\xi$ is bounded, then there exist a sequence $\lambda_{n}\in\mathfrak{L}$
such that

$\lim_{n\rightarrow\infty}\ x_{\lambda_{n}}=st(\xi).$

Proof: Set $x_{0}=st(\xi)$ and for every $n\in\mathbb{N}$, take $\lambda_{n}$
such that $\ x_{\lambda_{n}}\in\left[x_{0}-1/n,\ x_{0}+1/n\right].$

$\square$

######  Remark 14.

As we have remarked in the intruduction, the Centrum Axiom is necessary to
prove Theorem 12. Actually it is not difficult to prove that the Centrum Axiom
is equivalent to the following:

$\mathbb{E_{\kappa}\ }\text{contains a subfield isomorphic to the field of
real numbers.}$

The notion of ”standard entity” can be extended from numbers (i.e. the real
numbers) to other elements of the universe by the following definition:

######  Definition 15.

An element $E\in V(\mathbb{R})$ is called standard and $V(\mathbb{R})$ is
called standard universe; $V(\mathbb{E}_{\kappa})$ is called Euclidean
universe.

Notice that

$V(\mathbb{R})\subset\Lambda\subset V(\mathbb{E}_{\kappa}).$

Also the second inclusion is strict since $V(\mathbb{E}_{\kappa})$ contains
sets of inaccessible cardinality such as $\mathbb{E}_{\kappa}$.

##  4 The Euclidean universe

In this section we will inestigate the structure of the Euclidean universe
$V(\mathbb{E}_{\kappa}).$

###  4.1 $\Lambda$-limit of sets

By Def. 9, the $\Lambda$-limit has been defined for every net
$\varphi\in\mathfrak{F}\left(\mathfrak{L},\mathbb{E}_{\kappa}\right);$ next we
will extend this notion to the nets of sets in

$\mathfrak{F}\left(\mathfrak{L},V_{n}(\mathbb{E}_{\kappa})\right)=\left\\{\Phi\in
V_{n}\left(\mathbb{E}_{\kappa}\right)^{\mathfrak{L}}\ |\ \exists
A\in\Lambda,\forall\lambda\in\mathfrak{L},\ \Phi(\lambda)=\Phi(\lambda\cap
A)\right\\}$

for every $n\in\mathbb{N}$. In the following, in order to simplify the
notation, a net of sets
$\Phi\in\mathfrak{F}\left(\mathfrak{L},V_{n}(\mathbb{E}_{\kappa})\right)$ will
be denoted by $\left\\{E_{\lambda}\right\\}$ where
$E_{\lambda}=\Phi\left(\lambda\right).$

We define the $\Lambda$-limit of sets by induction over $n.$ If $n=0,$
$\lim_{\lambda\uparrow\Lambda}$ $\Phi\left(\lambda\right)$ is a net of numbers
defined by Def. 9; if $n>0,$ we set

$E_{\Lambda}=\lim_{\lambda\uparrow\Lambda}\
E_{\lambda}:=\left\\{\lim_{\lambda\uparrow\Lambda}\ \Psi(\lambda)\ |\
\Psi\in\mathfrak{F}\left(\mathfrak{L},V_{n-1}(\mathbb{E}_{\kappa})\right),\forall\lambda\in\mathfrak{L}:\Psi(\lambda)\in
E_{\lambda}\right\\}.$ (14)

Clearly, by (1), $E_{\Lambda}\in V(\mathbb{E}_{\kappa}).$

######  Definition 16.

A set $E$ obtained as $\Lambda$-limit of a net of sets
$\left\\{E_{\lambda}\right\\}\in\mathfrak{F}\left(\mathfrak{L},V_{n}(\mathbb{E}_{\kappa})\right)$
is called internal. If not it, is called external.

For example the set $\mathbb{R}$ is external.

If $C_{A}\left(\lambda\right)=A\in\Lambda_{S}$ is a constant net, we set

$A^{\ast}:=\lim_{\lambda\uparrow\Lambda}\
C_{A}\left(\lambda\right)=\left\\{\lim_{\lambda\uparrow\Lambda}\
\Psi(\lambda)\ |\
\Psi\in\mathfrak{F}\left(\mathfrak{L},V_{n-1}(\mathbb{E}_{\kappa})\right),\forall\lambda\in\mathfrak{L}:\Psi(\lambda)\in
A\right\\};$ (15)

then, if $A\in V_{n}(\mathbb{E}_{\kappa}),$ also $A^{\ast}\in
V_{n}(\mathbb{E}_{\kappa}).$ This definition extends (11) to all the elements
of $\Lambda=\Lambda_{S}\cup\mathbb{E}_{\kappa}$. $A^{\ast}$ will be called the
$\ast$-transform of $A.$

The $\ast$-transform allows to build a family
$\left\\{\mathbb{E}_{j}\right\\}_{j\in\mathbf{Ord}}$ of subsets of
$\mathbb{E}_{\kappa}$ as follows:

  * •

$\mathbb{E}_{0}=\mathbb{R};$

  * •

$\mathbb{E}_{j+1}=\mathbb{E}_{j}^{\ast};$

  * •

if $j\leq\kappa$ is a limit ordinal, then
$\mathbb{E}_{j}=\mathop{\displaystyle\bigcup}\limits_{k<j}\mathbb{E}_{k}.$

###  4.2 $\Lambda$-limit of functions

Since in set theory a function $f$ can be identified with its graph
$\Gamma_{f},$

$f_{\Lambda}:=\lim_{\lambda\uparrow\Lambda}~{}f_{\lambda}$

is well defined. However, it is not immediate to see that $f_{\Lambda}$ is
function. For this reason, we will analyze this situation explicitly.

######  Theorem 17.

Given a net of functions $\left\\{f_{\lambda}\right\\}$

$f_{\lambda}:A_{\lambda}\rightarrow B_{\lambda},\ \ A_{\lambda},B_{\lambda}\in
V_{n}(\mathbb{E}_{\kappa}),$

then $f_{\Lambda}:A_{\Lambda}\rightarrow B_{\Lambda}$ defined by

$f_{\Lambda}\left(\lim_{\lambda\uparrow\Lambda}\
x_{\lambda}\right)=\lim_{\lambda\uparrow\Lambda}~{}f_{\lambda}\left(x_{\lambda}\right);$
(16)

is a function and we have that

$\Gamma_{f_{\Lambda}}=\left(\Gamma_{f}\right)_{\Lambda}$

Proof: First, we will prove that (16) is a good definition, namely that
$f_{\Lambda}(\xi)$ does not depend on the net $x_{\lambda}$ which defines
$\xi.$ We set

$\xi=\lim_{\lambda\uparrow\Lambda}\ x_{\lambda}=\lim_{\lambda\uparrow\Lambda}\
y_{\lambda}$

and we have to prove that

$\lim_{\lambda\uparrow\Lambda}\
f\left(x_{\lambda}\right)=\lim_{\lambda\uparrow\Lambda}\
f\left(y_{\lambda}\right).$

We take

$\chi\left(\lambda\right)=\left\\{\begin{array}[]{cc}1&if\ \
x_{\lambda}=y_{\lambda}\\\ 0&if\ \ x_{\lambda}\neq
y_{\lambda}\end{array}\right..$

Hence $\forall\lambda$,
$\chi\left(\lambda\right)+\left(x_{\lambda}-y_{\lambda}\right)\neq 0$ and so

$\lim_{\lambda\uparrow\Lambda}\
\left[\chi\left(\lambda\right)+\left(x_{\lambda}-y_{\lambda}\right)\right]\neq
0,$

then,

$\displaystyle\lim_{\lambda\uparrow\Lambda}\chi\left(\lambda\right)$
$\displaystyle=$
$\displaystyle\lim_{\lambda\uparrow\Lambda}\chi\left(\lambda\right)+\lim_{\lambda\uparrow\Lambda}\
x_{\lambda}-\lim_{\lambda\uparrow\Lambda}\ y_{\lambda}$ $\displaystyle=$
$\displaystyle\lim_{\lambda\uparrow\Lambda}\
\left[\chi\left(\lambda\right)+\left(x_{\lambda}-y_{\lambda}\right)\right]\neq
0.$

Moreover, we have that

$\forall\lambda,\
\chi\left(\lambda\right)\cdot\left[f_{\lambda}\left(x_{\lambda}\right)-\
f_{\lambda}\left(y_{\lambda}\right)\right]=0;$

then

$0=\lim_{\lambda\uparrow\Lambda}\left(\chi\left(\lambda\right)\cdot\left[f\left(x_{\lambda}\right)-\
f\left(y_{\lambda}\right)\right]\right)=\lim_{\lambda\uparrow\Lambda}\chi\left(\lambda\right)\cdot\lim_{\lambda\uparrow\Lambda}\left[f_{\lambda}\left(x_{\lambda}\right)-f_{\lambda}\left(y_{\lambda}\right)\right].$

Since $\lim_{\lambda\uparrow\Lambda}\chi\left(\lambda\right)\neq 0,$ we have
that

$0=\lim_{\lambda\uparrow\Lambda}\left[f_{\lambda}\left(x_{\lambda}\right)-\
f_{\lambda}\left(y_{\lambda}\right)\right]=\lim_{\lambda\uparrow\Lambda}\
f_{\lambda}\left(x_{\lambda}\right)-\lim_{\lambda\uparrow\Lambda}\
f_{\lambda}\left(y_{\lambda}\right).$

Finally, it is immediate to check that $f^{\ast}$ is the graph of the function
(16), in fact

$\displaystyle\left(\Gamma_{f}\right)_{\Lambda}$ $\displaystyle=$
$\displaystyle\left\\{\lim_{\lambda\uparrow\Lambda}\left(x_{\lambda},f_{\lambda}\left(x_{\lambda}\right)\right)\
|\ \forall\lambda,\
\left(x_{\lambda},f_{\lambda}\left(x_{\lambda}\right)\right)\in\Gamma_{f_{\lambda}}\right\\}$
$\displaystyle=$ $\displaystyle\left\\{\left(\lim_{\lambda\uparrow\Lambda}\
x_{\lambda},\lim_{\lambda\uparrow\Lambda}f_{\lambda}\left(x_{\lambda}\right)\right)\
|\ \forall\lambda,\ x_{\lambda}=f_{\lambda}\left(x_{\lambda}\right)\right\\}$
$\displaystyle=$ $\displaystyle\left\\{\left(\xi,\
f_{\Lambda}\left(\xi\right)\right)\ |\
\xi=f_{\Lambda}\left(\xi\right)\right\\}=\Gamma_{f_{\Lambda}}.$

$\square$

######  Definition 18.

A function $f$ obtained as $\Lambda$-limit of a net of functions $f_{\lambda}$
is called internal. Otherwise is called external.

If $\left\\{f\right\\}$ is a constant net, we set

$f^{\ast}=\lim_{\lambda\uparrow\Lambda}\ f;$ (17)

then, if $f:A\rightarrow B,$ then $f^{\ast}:A^{\ast}\rightarrow B^{\ast}$ and
$f^{\ast}$ will be called the $\ast$-transform of $f.$

###  4.3 Hyperfinite sets

Another fundamental notion in Euclidean calculus is the following:

######  Definition 19.

We say that a set $F\in V\left(\mathbb{E}_{\kappa}\right)$ is hyperfinite if
there is a net $\left\\{F_{\lambda}\right\\}_{\lambda\in\Lambda}$ of finite
sets such that

$F=\lim_{\lambda\uparrow\Lambda}~{}F_{\lambda}=\left\\{\lim_{\lambda\uparrow\Lambda}\
x_{\lambda}\ |\ x_{\lambda}\in F_{\lambda}\right\\}$

The hyperfinite sets share many properties of finite sets. For example, a
hyperfinite set $F\subset\mathbb{E}_{\kappa}$ has a maximum $x_{M}$ and a
minimum $x_{m}$ respectively given by

$x_{M}=\lim_{\lambda\uparrow\Lambda}\max F_{\lambda};\ \
x_{m}=\lim_{\lambda\uparrow\Lambda}\min F_{\lambda}$

Moreover, it is possible to ”add” the elements of an hyperfinite set of
numbers. If $F$ is an hyperfinite set of numbers, the hyperfinite sum of the
elements of $F$ is defined as follows:

$\sum_{x\in F}x=\ \lim_{\lambda\uparrow\Lambda}\sum_{x\in F_{\lambda}}x.$

One peculiarity of Euclidean analysis the possibility to associate a unique
hyperfinite set $E^{\circledcirc}$ to any set $E\in V(\mathbb{E}_{\kappa})$
according to the following definition:

######  Definition 20.

Given a set $E\in\Lambda_{S},$ the set

$E^{\circledcirc}:=\lim_{\lambda\uparrow\Lambda}\ \left(E\cap\lambda\right)$

is called hyperfinite extension of $E.$

If $F=\lim_{\lambda\uparrow\Lambda}~{}F_{\lambda}$ is a hyperfinite set, its
hypercardinality is given by

$\left|F\right|^{\ast}=\lim_{\lambda\uparrow\Lambda}~{}\left|F_{\lambda}\right|$

were $\left|\cdot\right|^{\ast}$ is the $\ast$-tranform of the fuction
”cardinality” defined on finite sets.

Notice that, by vitue of (5), the hypercardinality of $E^{\circledcirc},$
given by

$\left|E^{\circledcirc}\right|^{\ast}=\lim_{\lambda\uparrow\Lambda}\
\left|E\cap\lambda\right|,$

is the numerosity of $E$ as it has been defined by Axiom 1.

If we put

$E^{\sigma}=\left\\{x^{\ast}\ |\ x\in E\right\\},$

we can associate the sets $E^{\sigma},$ $E^{\circledcirc}$ and $E^{\ast}$ to
any set $E\in\Lambda.$ They are ordered as follows:

$E^{\sigma}\subseteq E^{\circledcirc}\subseteq E^{\ast};$

in particular, if $E\subseteq\mathbb{R}$, $E^{\sigma}=E.$ The hyperfinite
analysis is very relevant in the applications and the operator
”$\circledcirc$” plays a special role. You can see some examples of this fact
in section 6.3.

##  5 Nonstandard Analysis

Even if some notions and definitions of Nonstandard Analysis have already been
introduced in the previous sections, now we will treat this topic in details.
In this section, we assume that the reader is familiar with the basic notions
on NSA.

###  5.1 Nonstandard theories

In this subsection, we recall the basic notion of Nonstandard Analysis how
have been developed in the ”superstructure” approach. Following Keisler (see
[17]), we give the following definition:

######  Definition 21.

A nonstandard theory is a triple
$\left(V\left(\mathbb{R}\right),V\left(\mathbb{R}^{\mathbb{\bullet}}\right),\bullet\right)$
such that333To be precise, Keisler calls
$\left(V\left(\mathbb{R}\right),V\left(\mathbb{R}^{\mathbb{\bullet}}\right),\bullet\right)$
nonstandard universe while we use the espression nonstandard universe to
denote the set $V\left(\mathbb{R}^{\mathbb{\bullet}}\right)$.

  * •

$V\left(\mathbb{R}\right)$ is a superstructure over $\mathbb{R}$ called
standard universe;

  * •

$\mathbb{R}^{\mathbb{\bullet}}$ is a set such that $\mathbb{R\subset
R}^{\mathbb{\bullet}}$ which is called field of the ($\bullet$)-hyperreal
numbers;

  * •

$V\left(\mathbb{R}^{\mathbb{\bullet}}\right)$ is a superstructure over
$\mathbb{R}^{\mathbb{\bullet}}$ called nonstandard universe;

  * •

the map

$\bullet~{}:V\left(\mathbb{R}\right)\rightarrow
V\left(\mathbb{R}^{\mathbb{\bullet}}\right)$

satisfies the Leibniz principle and

$\forall r\in\mathbb{R},r=r^{\mathbb{\bullet}}.$ (18)

We recall the notion of Leibniz (or transfer) Principle. It is well known that
the map $\bullet$ transforms any elementary sentence
$P(a_{1},a_{2},...,a_{n})$ to a elementary sentence
$P(a_{1}^{\mathbb{\bullet}},a_{2}^{\mathbb{\bullet}},...,a_{n}^{\mathbb{\bullet}})$
in $V\left(\mathbb{R}^{\mathbb{\bullet}}\right)$ where $a_{1},a_{2},...,a_{n}$
are constants in $V\left(\mathbb{R}\right).$ The adjective elementary refers
to the fact that the quantifiers in elementary sentences are of the form
($\forall x\in y$) or ($\exists x\in y$) where $x$ is a variable and $y$ is a
constant or a variable. The Leibniz principle states that
$P(a_{1},a_{2},...,a_{n})$ is true if an only if
$P(a_{1}^{\mathbb{\bullet}},a_{2}^{\mathbb{\bullet}},...,a_{n}^{\mathbb{\bullet}})$
is true. For details, see e.g. [8] or [17].

######  Definition 22.

Given two sets $\mathbb{A}$ and $\mathbb{S}$, a superstructure embedding is a
triple
$\left(V\left(\mathbb{A}\right),V\left(\mathbb{S}\right),\bullet\right)$
where$\ \bullet:V\left(\mathbb{A}\right)\rightarrow V\left(\mathbb{S}\right)$
is a injective map such that

$\mathbb{A^{\mathbb{\bullet}}=S}$ $\forall x,y\in V\left(\mathbb{A}\right),\
x\in y\Leftrightarrow x^{\mathbb{\bullet}}\in y^{\mathbb{\bullet}}$

The following fact is well known:

######  Theorem 23.

If $\left(V\left(\mathbb{A}\right),V\left(\mathbb{S}\right),\bullet\right)$ is
superstructure embedding then the map $\bullet$ satisfies the Leibniz
principle.

Proof: This result can be proved by induction over the complexity of the
sentences; see e.g. [8] Th. 5.8. or [17].

$\square$

By the above theorem, we get the following result:

######  Corollary 24.

If $\left(V\left(\mathbb{R}\right),V\left(\mathbb{S}\right),\bullet\right)$ is
a superstructure embedding such that $\mathbb{R\neq S}$, then,
$\left(V\left(\mathbb{R}\right),V\left(\mathbb{R^{\mathbb{\bullet}}}\right),\bullet\right)$
is a nonstandard theory with $\mathbb{R^{\mathbb{\bullet}}=S}$.

An isomorphism beetwen nonstandard theories is defined as follows:

######  Definition 25.

Two nonstandard theories
$\left(V\left(\mathbb{R}\right),V\left(\mathbb{R}^{\mathbb{\bullet}}\right),\bullet\right)$
and
$\left(V\left(\mathbb{R}\right),V\left(\mathbb{R}^{\star}\right),\star\right)$
are isomorphic if there is a map

$h:V\left(\mathbb{R}^{\mathbb{\bullet}}\right)\rightarrow
V\left(\mathbb{R}^{\star}\right)$

such that

  1. 1.

$\forall r\in\mathbb{R},$ $h(r)=r;$

  2. 2.

$h$ maps $\mathbb{R}^{\mathbb{\bullet}}$ one to one onto $\mathbb{R}^{\star};$

  3. 3.

for each $A\in
V\left(\mathbb{R}^{\mathbb{\bullet}}\right)\backslash\mathbb{R}^{\mathbb{\bullet}},$

$h(A)=\left\\{h(a)\ |\ a\in A\right\\}$

  4. 4.

for each $A\in V\left(\mathbb{R}\right),$ $h(A^{\mathbb{\bullet}})=A^{\star}.$

######  Definition 26.

A nonstandard theory
$\left(V\left(\mathbb{R}\right),V\left(\mathbb{R}^{\mathbb{\bullet}}\right),\bullet\right)$
is called saturated if any family of sets $\mathfrak{S}\in
V_{n}\left(\mathbb{R}\right)^{\mathbb{\bullet}}$ with cardinality smaller than
$\mathbb{R}^{\mathbb{\bullet}}$ and with the finite intersection property has
non empty intersection; namely if

$S_{1}\cap...\cap S_{n}\neq\varnothing,\ \ S_{i}\in\mathfrak{S};\
\left|\mathfrak{S}\right|<\left|\mathbb{R}^{\mathbb{\bullet}}\right|$

then

$\mathop{\displaystyle\bigcap}\mathfrak{S}\neq\varnothing.$

Among all the nonstandard theories there is a privileged one which is unique
up to isomorphisms.

######  Theorem 27.

A saturated nonstandard theory
$\left(V\left(\mathbb{R}\right),V\left(\mathbb{R}^{\bullet}\right),\bullet\right)$
with

$\left|\mathbb{R}^{\mathbb{\bullet}}\right|=\kappa$

is unique up to isomorphism. In this case,
$V\left(\mathbb{R}^{\bullet}\right)$ will be called Keisler universe.

Proof: See [17].

$\square$

###  5.2 The Normal Universe

According to the theory of the previous section we give following

######  Definition 28.

If ”$\ast$” is the map defined by (15), $V\left(\mathbb{R}^{\ast}\right)$ will
be called normal universe; $\mathbb{R}^{\ast}$ will be called normal Euclidean
field and from now on, it will be simply denoted by $\mathbb{E}$.

######  Theorem 29.

The triple
$\left(V\left(\mathbb{R}\right),V\left(\mathbb{E}\right),\ast\right)$ is a
nonstandard theory and $V\left(\mathbb{E}\right)$ is a Keisler universe.

Proof: By the definition of $V\left(\mathbb{E}\right),$ it follows that
$\left(V\left(\mathbb{R}\right),V\left(\mathbb{E}\right),\ast\right)$ is a
superstructure embedding. Then by Cor. 24, we have to prove that
$\left|\mathbb{E}\right|=\kappa$ and that
$\left(V\left(\mathbb{R}\right),V\left(\mathbb{E}\right),\ast\right)$ is
saturated. Since

$\mathbb{E=R}^{\ast}=\left\\{\lim_{\lambda\rightarrow\Lambda}\ x_{\lambda}\ |\
x_{\lambda}\in\mathfrak{F}\left(\mathfrak{L},\mathbb{R}\right)\right\\}$

we have that

$\left|\mathbb{E}\right|\leq\left|\mathfrak{F}\left(\mathfrak{L},\mathbb{R}\right)\right|$

Moreover

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

and hence

$\left|\mathfrak{F}\left(\mathfrak{L},\mathbb{R}\right)\right|=\left|\mathop{\displaystyle\bigcup}\limits_{A\in\Lambda}\mathbb{R}^{A}\right|$

and since $\left|\mathbb{R}^{A}\right|<\kappa$ and
$\left|\Lambda\right|=\kappa$, we have that

$\left|\mathbb{E}\right|\leq\left|\mathop{\displaystyle\bigcup}\limits_{A\in\Lambda}\mathbb{R}^{A}\right|=\kappa$

Also we have that

$\left|\mathbb{E}\right|\geq\left|\mathfrak{N}\right|\geq\left|\mathbf{Ord}\right|=\kappa.$

Then $\left|\mathbb{E}\right|=\kappa$. It remains to prove that it is
saturated.

If $\mathfrak{S}\in V_{n}\left(\mathbb{R}\right)^{\mathbb{\ast}},$ then

$\mathfrak{S}=\left\\{E_{\mu}\ |\ \mu\in H\right\\}$

where $H$ is a set of indices with $\left|H\right|<\kappa$. Since
$\left|H\right|<\kappa,$ it is not restrictive to assume that
$H\subset\mathfrak{L}$. For every $\mu\in H$, let $\varphi_{\mu}(\lambda)$ be
a net such that

$\lim_{\lambda\uparrow\Lambda}\ \varphi_{\mu}(\lambda)=E_{\mu}$

For any fixed $\lambda$, pick an element

$\psi(\lambda)\in\mathop{\displaystyle\bigcap}_{\mu\subseteq\lambda}\varphi_{\mu}(\lambda)$

if this intersection is nonempty. Otherwise, pick

$\psi(\lambda)\in\mathop{\displaystyle\bigcap}_{\mu\subseteq\lambda\backslash\left\\{x_{1}\right\\}}\varphi_{\mu}(\lambda);\
x_{1}\in\lambda$

if this intersection is nonempty, and continue in this manner until the
element $\psi(\lambda)$ is defined. In case that this intersection is always
empty, we set $\psi(\lambda)=\varnothing.$ As a consequence of this
definition, the following property holds:

$\mathop{\displaystyle\bigcap}_{\mu\subseteq\tau}\varphi_{\mu}(\lambda)\neq\varnothing\Rightarrow\forall\lambda\supseteq\tau,\
\psi(\lambda)\in\mathop{\displaystyle\bigcap}_{\mu\subseteq\tau}\varphi_{\mu}(\lambda)$
(19)

Now let $\tau\in H$ be fixed. By the finite intersection property,

$\varnothing\neq\mathop{\displaystyle\bigcap}_{\mu\subseteq\tau}E_{\mu}=\mathop{\displaystyle\bigcap}_{\mu\subseteq\tau}\lim_{\lambda\uparrow\Lambda}\
\varphi_{\mu}(\lambda)=\lim_{\lambda\uparrow\Lambda}\left(\mathop{\displaystyle\bigcap}_{\mu\subseteq\tau}\varphi_{\mu}(\lambda)\right)$

Then, there exists a qualified set $Q\in\mathcal{U}$ ($\mathcal{U}$ is defined
in the proof of Th. 8) such that,

$\forall\lambda\in Q,\
\mathop{\displaystyle\bigcap}_{\mu\subseteq\tau}\varphi_{\mu}(\lambda)\neq\varnothing;$

and hence, by (19), it follows that

$\forall\lambda\in Q,\
\psi(\lambda)\in\mathop{\displaystyle\bigcap}_{\mu\subseteq\tau}\varphi_{\mu}(\lambda)\neq\varnothing$

and taking the $\Lambda$-limit

$\lim_{\lambda\uparrow\Lambda}\
\psi(\lambda)\in\mathop{\displaystyle\bigcap}_{\mu\subseteq\tau}\lim_{\lambda\uparrow\Lambda}\
\varphi_{\mu}(\lambda)=\mathop{\displaystyle\bigcap}_{\mu\subseteq\tau}E_{\mu}$

and in particular,

$\lim_{\lambda\uparrow\Lambda}\ \psi(\lambda)\in E_{\tau}$

As this holds for every $\tau\in H$, we conclude that

$\lim_{\lambda\uparrow\Lambda}\
\psi(\lambda)\in\mathop{\displaystyle\bigcap}_{\tau\in
H}E_{\tau}=\mathop{\displaystyle\bigcap}\mathfrak{S.}$

$\square$

$V\left(\mathbb{E}\right)$ is not the only Keisler universe contained in
$V\left(\mathbb{E}_{\kappa}\right)$. For example, using the notations at the
end of section 4.1, we have that also

$\left(V\left(\mathbb{R}\right),V\left(\mathbb{E}_{2}\right),\ast\ast\right),\
\left(V\left(\mathbb{R}\right),V\left(\mathbb{E}_{3}\right),\ast\ast\ast\right),\
\ etc.$

are saturated nonstandard theories and hence we have infinite Keisler
universes included in $V\left(\mathbb{E}_{\kappa}\right)$. More in general,
for $0<j\leq\kappa$, it is possible to define a Keisler universe

$\left(V\left(\mathbb{R}\right),V\left(\mathbb{E}_{j}\right),\ast^{j}\right)$

as follows:

  * •

if $j=1,$ $E^{\ast^{1}}=E^{\ast}$

  * •

if $j=k+1,$ $E^{\ast^{j+1}}=\left(E^{\ast^{j}}\right)^{\ast}$

  * •

if $k\ $is a limit ordinal, and $E\in V\left(\mathbb{E}_{k}\right),$ then
$E\in V\left(\mathbb{E}_{j}\right)$ for some $j<k$ and hence
$E^{\ast^{k}}=E^{\ast^{j}}.$

In particular we have that all the fields $\mathbb{E}_{j},$ $j\leq\kappa$ are
isomorphic; for $j<\kappa,$ the map

$\ast~{}:\mathbb{E}_{j}\rightarrow\mathbb{E}_{j+1}$

is a field homomorphism and if $k$ is a limit ordinal, the map

$\ast~{}:\mathbb{E}_{k}\rightarrow\mathbb{E}_{k}$

is a field isomorphism. In any case, the only fixed points of $\ast$ are the
real numbers. The spaces $\mathbb{E}_{j}$’s differ from each other by the way
they are embedded in $\mathbb{A}$.

Moreover in a Euclidean universe there are other interesting superstructure
embeddings which can be useful in some application. For example,

$\left(V\left(\mathbb{E}_{j}\right),V\left(\mathbb{E}_{j+1}\right),\ast\right)$

is a superstructure embedding; however is not a nonstandard theory, since
$\mathbb{E}_{j}\neq\mathbb{R}$; moreover
$\left(V\left(\mathbb{E}_{j}\right),V\left(\mathbb{E}_{j+1}\right),\ast\right)$
violates an other important request of Def. 21, namely

$\exists\xi\in\mathbb{E}_{j},\ \xi\neq\xi^{\ast}$

in contrast with (18). Nevertheless, by Th. 23,
$\left(V\left(\mathbb{E}_{j}\right),V\left(\mathbb{E}_{j+1}\right),\ast\right)$
satisfies the Leibniz principle. Then the following fact follows
straightforwardly:

######  Theorem 30.

$\left(V\left(\mathbb{E}_{\kappa}\right),V\left(\mathbb{E}_{\kappa}\right),\ast\right)$
is a superstructure embedding that satisfies the Leibniz principle.

###  5.3 The $\alpha$-theory

The $\alpha$-theory has been introduced in [7] and it represents an elementary
approach to nonstandard analysis particularly suitable for some application;
see e.g [8], [12], [5] and references. Actually, the Euclidean universe
contains many non-standard universes which can be easily defined and,
therefore, are more suitable for the elementary applications of practitioners.
The $\alpha$-theory is one of them and it can be constructed using the notion
of $\alpha$-limit where $\alpha=\mathfrak{num}\left(\mathbb{N}^{+}\right)$,
$\mathbb{N}^{+}=\mathbb{N}\backslash\left\\{0\right\\}$.

######  Definition 31.

Given a sequence $\varphi:$ $\mathbb{N}\rightarrow
V_{n}(\mathbb{E}_{\kappa}),$ we set

$\lim_{n\uparrow\alpha}\ \varphi(n):=\lim_{\lambda\uparrow\Lambda}\
\varphi\left(\left|\lambda\cap\mathbb{N}^{+}\right|\right).$

If $C_{b}(n)$ is the constant sequence with value $b\in
V(\mathbb{E}_{\kappa}),$ we set

$b^{\ast_{\alpha}}=\lim_{n\uparrow\alpha}\ C_{b}(n)$

Then, by Th. 23, it follows that

$\left(V\left(\mathbb{R}\right),V\left(\mathbb{R}^{\ast_{\alpha}}\right),\ast_{\alpha}\right)$

is a nonstandard theory.

######  Definition 32.

The nonstandard theory
$\left(V\left(\mathbb{R}\right),V\left(\mathbb{R}^{\ast_{\alpha}}\right),\ast_{\alpha}\right)$
is called $\alpha$-theory.

If

$i:\mathbb{N}\rightarrow\mathbb{E}_{\kappa};\ i(n)=n$

then taking the $\alpha$-limit we get that

$\lim_{n\uparrow\alpha}\ i(n)=\lim_{\lambda\uparrow\Lambda}\
i\left(\left|\lambda\cap\mathbb{N}^{+}\right|\right)=\mathfrak{num}(\mathbb{N}^{+})=\alpha$

hence the set of ($\ast_{\alpha}$)-hyperreal numbers
$\mathbb{R}^{\ast_{\alpha}}$ can be characterized as follows:

$\mathbb{R}^{\ast_{\alpha}}=\left\\{\lim_{n\uparrow\alpha}\ \varphi(n)\ |\
\varphi\,:\mathbb{N}\rightarrow\mathbb{R}\right\\}$

namely, every ($\ast_{\alpha}$)-hyperreal numbers is the $\alpha$-limit of
real sequence. For example, we have that $\omega$ is a
($\ast_{\alpha}$)-hyperreal number since

$\displaystyle\omega$ $\displaystyle=$
$\displaystyle\mathfrak{num}(\mathbb{N})=\mathfrak{num}(\mathbb{N}^{+})+\mathfrak{num}(\left\\{0\right\\})$
$\displaystyle=$ $\displaystyle\alpha+1=\lim_{n\uparrow\alpha}\ (n+1).$

As we will see in section 7.1, the construction of a model of the theory,
there is an ultrafilter $\mathcal{U}$ which plays a central role (see
definition (22)). By choosing $\mathcal{U}$ in a suitable way, then we get the
following result:

######  Theorem 33.

It is compatible with axioms 1-3 that the number $\alpha$ satisfies the
following properties:

  * •

Divisibility Property : For every $k\in\mathbb{N}$, the number $\alpha$ is a
multiple of $k$ and the numerosity of the set of multiples of k:

$\mathfrak{num}(\\{k,2k,3k,...,nk,...\\})=\frac{\alpha}{k}$

  * •

Root Property: For every $k\in\mathbb{N}$, the number $\alpha$ is a k-th power
and the numerosity of the set of k-th powers:

$\mathfrak{num}(\\{1^{k},2^{k},3^{k},...,n^{k},...\\})=\sqrt[k]{\alpha}$

  * •

Power Property: If we set $\wp_{fin}(A)=\\{F\in\wp(A)\ |\ F\ $is a finite
set$\\},$ then

$\mathfrak{num}(\wp_{fin}(\mathbb{N}^{+}))=2^{\alpha}$

  * •

Integer numbers Property:

$\mathfrak{num}(\mathbb{Z})=2\alpha+1$

  * •

Rational numbers Property: For every $q\in\mathbb{Q}$,

$\mathfrak{num}(\left(q,q+1\right]\cap\mathbb{Q})=\mathfrak{num}(\left(0,1\right]\cap\mathbb{Q})=\alpha$

and

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

Proof: See [8], Sections 16.6 and 16.7.

$\square$

###  5.4 About the idea of continuum

The idea of (linear) continuum is described or modeled by the geometric line.
In classical Euclidean geometry, lines and segments are not considered sets of
points; on the contrary, in the last two centuries the reductionist attitude
of modern mathematics has described Euclidean geometry through a set
interpretation. In the last century, the geometric continuum has been
identified with the Dedekind continuum and the geometric line has been
identified with the set of real numbers (once the origin $O$ and a unitary
segment $OU$ have been fixed). Even is this identification, today, is almost
universally accepted, we have seen that also the Euclidean line, as defined by
axiom 2, has some right to represent the geometric continuum. In this section
we will compare $\mathbb{R}$ and $\mathbb{E\ }$($\mathbb{\cong E}_{\kappa}$)
with respect to the idea of geometric continuum.

In our naive intuition, we think of a linear continuum as a linearly ordered
set without interruptions, that is, without holes between one part and the
other. Let’s make this definition rigorous. Contrary to our intuition, a set X
satisfying the following property

$\forall a,b\in X,\ a<b,\ \exists c\in X,\ a<c<b$

it cannot be considered a continuum: this notion, satisfied for example by the
set of rational numbers, is not a good candidate for a continuum as the set of
rationals is full of holes represented by irrational numbers.

So we are led to discuss the notion of saturation and to the Eudoxus
Principle:

######  Definition 34.

A linearly ordered set $X$ is called saturated if it satisfies following
property: given two sets $A,B\subset X,$ such that

$\left|A\right|,\left|B\right|<\left|X\right|$ (20) $\forall a\in A,\ \forall
b\in B,\ a<b$

then $\exists c\in X,$

$\forall a\in A,\ \forall b\in B,\ a\leq c\leq b.$

Clearly the rational numbers are a saturated field but they cannot be
considered a continuum. On the contrary the real numbers are not saturated.

######  Definition 35.

(Eudoxus Principle) A linearly ordered Abelian group $\mathbb{F}$ satisfies
the Eudoxus Principle if given two sets $A,B\subset\mathbb{F}$ such that

$\forall a\in A,\ \forall b\in B,\ a<b$ $\forall\varepsilon\in\mathbb{F}^{+},\
\exists a\in A,\ \exists b\in B,\ b-a<c.$

then $\exists c\in X,$

$\forall a\in A,\ \forall b\in B,\ a\leq c\leq b.$

Using these notions, we can characterize $\mathbb{R}$ and $\mathbb{E}$ as
follows:

######  Theorem 36.

The field of the real numbers $\mathbb{R}$ is the only field that:

  * •

(i) satisfies the Eudoxus Principle,

  * •

(ii) satisfies the Archimedes’ Axiom, namely :

$\forall a,b\in\mathbb{F}^{+},\ \exists c\in\mathbb{N},\ na>b$

Proof: Well known.

$\square$

The request (ii) is necessary; in fact, for example, the field of rational
functions with a suitable order structure444For example the field of rational
fuction $\mathbb{F}$ can be equipped with an order structure by setting
$\mathbb{F}^{+}=\left\\{\frac{r_{n}x^{n}+r_{n-1}x^{n-1}+....+r_{0}}{w_{m}x^{m}+w_{m-1}x^{m-1}+....+w_{0}}\
|\ \frac{z_{n}}{w_{m}}>0\right\\}.$ satisfy (i) but not (ii).

######  Theorem 37.

The field of the Euclidean numbers $\mathbb{E}$ is the smallest field that:

  * •

(i) is saturated,

  * •

(ii) is a real closed field, namely every polynomial of odd degree has at
least one root.

Proof: $\mathbb{Q}$ is the smallest saturated field but it is not real closed.
It is well known that a linearly ordered set $X$ is strongly saturated iff
$\left|X\right|$ is strongly inaccessible. Then $\left|\mathbb{F}\right|$ must
be inaccessible. The conclusion follows by the fact that
$\left|\mathbb{E}\right|=\kappa$ and it is saturated by Th. 29.

$\square$

Notice that the request (ii) is necessary; in fact $\mathbb{Q}^{\ast}$ is a
saturated field, but it does not satisfy (i) since the equation $x^{3}=2$ does
not have any solution in $\mathbb{Q}^{\ast}$. We observe that the request (ii)
fits well the idea of continuity, in fact, a polynomial of odd degree must
take positive and negative values and hence, by continuity, it must have some
$0$’s.

The above discussion suggests the following definitions of continuum:

######  Definition 38.

A linearly ordered Abelian group $\mathbb{F}$ is a Dedekind continuum if it
satisfies the following property: given two sets $A,B\subset X$ such that

$A,B\neq\varnothing$ (21) $\forall a\in A,\ \forall b\in B,\ a<b$

then $\exists c\in X,$

$\forall a\in A,\ \forall b\in B,\ a\leq c\leq b;$

A linearly ordered ordered field $\mathbb{F}$ is an absolute continuum555This
notion of absolute contiuum has been introduced by Ehrlich in [15]. However in
his definition $X$ is a class in the sense of Von Neumann–Bernays–Gödel set
theory. if it is saturated and real closed.

With these notions of continuity $\mathbb{R}$ and $\mathbb{E}$ have the
following characterization: $\mathbb{R}$ is the only Dedekind continuum field;
$\mathbb{E}$ is the smallest absolutely continuum field.

##  6 Euclidean Calculus

In this section we pretend to not know the classical calculus and we will
define the basic notion of calculus, derivative and integral, in the most
natural way provided that you are equipped with infinitesimal and infinite
numbers. Hence, these definitions are very similar to those of the XVIII
century. With these definitions, we will discover that every function is both
integrable and it has a left and right derivative. Of course, if a function is
differentiable, the Euclidean derivative corresponds to the usual derivative
and if it is Lebesgue-integrable, the Euclidean integral corresponds to the
usual Lebesgue integral. We limit this game to the normal functions as defined
below.

Of course this game could be extended to other notions and to a larger class
of functions and it might have some interest for the foundations and the
philosophy of Mathematics.

The idea to work directly in a nonstandard unverse is not new; we recall [19],
[1] and [16]. This section can be considered an other experiment in that
direction.

###  6.1 Normal functions and sets

In most application, the space $\mathbb{E}_{\kappa}$ and the Euclidean
universe $V\left(\mathbb{E}_{\kappa}\right)$ are too large and hence might
imply useless technicalities. It is more convenient to work in the normal
Euclidean field $\mathbb{E}=\mathbb{R}^{\ast}$ and in the normal universe
$V\left(\mathbb{E}\right)$. So we are lead to the following definition:

######  Definition 39.

A function $f:\mathbb{E\rightarrow E}$ is called normal if

$f=h^{\ast}$

where $h\ $is a standard real function, i.e. $f\in\mathbb{R}^{E},\
E\subseteq\mathbb{R}.$

If $f$ is normal then $\forall x\in\mathbb{R}$,
$f\left(x\right)\in\mathbb{R}$.

######  Definition 40.

A subset $N\subset\mathbb{E}$ is called normal if $N=A^{\ast}$ for some set
$A\subset\mathbb{R}$.

######  Remark 41.

In the nonstandard analysis community there is the habit to call standard both
functions and sets of the form $f^{\ast},A^{\ast}$ and functions and sets in
$V(\mathbb{R}).$ Here we call standard the elements of $V(\mathbb{R})$ and
normal their counterpart defined as above.

The usual functions used in the applications of mathematics can be regarded as
normal functions and not as standard functions. The advantadge of this point
of view is that the main notions of infinitesimal analysis can be defined
using the ”actual infinitesimal” in a natural way and hence they assume a
different meaning.

###  6.2 The notion of derivative

Since the normal functions are in a biunivocal correspondence with the real
functions, sometimes we will denote both with the same symbol. The same we
will do with the intervals.

Now let us introduce some notions of Euclidean calculus. In order to intoduce
a ”Euclidean derivative”, we will take the advantage to have a distinguished
infinite number, namely $\alpha;$ then we can define a distinguished
infinitesimal number as follows:

$\eta:=\frac{1}{\alpha}.$

######  Definition 42.

The right derivative of a normal function
$f:\left(a,b\right)\rightarrow\mathbb{E}$ in a standard point $x_{0}\in(a,b)$,
in the sense of Euclidean Calculus, is defined as follows:

$D^{+}f(x_{0})=ctr\left(\frac{f(x_{0}+\eta)-f(x_{0})}{\eta}\right);$

similarly the left $\mathbb{E}$-derivative is defined as follows:

$D^{-}f(x_{0})=ctr\left(\frac{f(x_{0})-f(x_{0}-\eta)}{\eta}\right);$

the mean $\mathbb{E}$-derivative is defined as follows:

$Df(x_{0})=\frac{1}{2}\left[D^{+}f(x_{0})+D^{-}f(x_{0})\right]=ctr\left(\frac{f(x_{0}+\eta)-f(x_{0}-\eta)}{2\eta}\right).$

We say that a function is derivable in a point $x_{0}\in(a,b)$ if
$Df(x_{0})=D^{+}f(x_{0})$ and $Df(x_{0})\in\mathbb{R}$. In this case,

$Df(x_{0})=st\left(\frac{f(x_{0}+\eta)-f(x_{0})}{\eta}\right)$

is called generalized derivative in the sense of Euclidean Calculus or simply
$\mathbb{E}$-derivative.

It is easy to check that given a real function $f$ differentiable in a point
$x_{0}\in\left(a,b\right)\cap\mathbb{R},$ then

$Df(x_{0})=f^{\prime}(x_{0})$

but the converse is not true; for example the function
$f(x)=x\sin\frac{1}{x^{2}}$ is not differentiable for $x=0,$ but

$D^{+}f(0)=ctr\left(\sin\frac{1}{\eta^{2}}\right)=ctr\left(\sin\frac{1}{\left(-\eta\right)^{2}}\right)=D^{-}f(0)$

and hence the $\mathbb{E}$-derivative, given by

$\left[D\left(x\sin\frac{1}{x^{2}}\right)\right]_{x=0}=st\left(\sin\frac{1}{\eta^{2}}\right),$

is well defined.

The notion of $\mathbb{E}$-derivative is more general and consequently the
fact that if $f\ $is $\mathbb{E}$-derivable does not imply that $f$ (resticted
to $\mathbb{R}$) is continuous. For example the Dirichlet function

$f_{D}(x):=\left\\{\begin{array}[]{cc}1&if\ x\in\mathbb{Q}^{\ast}\\\ &\\\
0&if\ x\in\mathbb{R}^{\ast}\backslash\mathbb{Q}^{\ast}\end{array}\right.$

is $\mathbb{E}$-derivable in every point with $Df_{D}=0$ (remember that by Th.
33, $\eta\in\mathbb{Q}^{\ast}$), but it is not continuous.

As far, we have defined the derivative of a normal function in a standard
point. The following definition extends the notion of derivative to every
Euclidean point, namely it defines the function ”derivative” in all the points
of $\left(a,b\right)$ while Def. 42 defines it only for
$x\in\left(a,b\right)\cap\mathbb{R}$.

######  Definition 43.

The $\mathbb{E}$-derivative of a normal derivable function
$f:\left(a,b\right)\rightarrow\mathbb{E}$ is defined by

$Df:=\left(Df|_{\left(a,b\right)\cap\mathbb{R}}\right)^{\ast}.$

Example: take $f(x)=x\sin\frac{1}{x^{2}};$ then

$D\left(x\sin\frac{1}{x^{2}}\right)=\left\\{\begin{array}[]{cc}\sin\frac{1}{x^{2}}-\frac{2}{x^{2}}\cos\frac{1}{x^{2}}&if\
\ x\in\mathbb{E}\backslash\left\\{0\right\\}\\\ &\\\
st\left(\sin\frac{1}{\eta^{2}}\right)&if\ \ x=0\end{array}\right..$

Obviously, the $\mathbb{E}$-derivability does not imply the differentiability
defined as follows:

######  Definition 44.

A normal function $f:\left(a,b\right)\rightarrow\mathbb{E}$ is said to be
differentiable in a point $x_{0}\in(a,b)$ if there exists a linear function
$t\mapsto df\left(x_{0}\right)\left[t\right]$ such that, for every
infinitesimal $\varepsilon$,

$f(x_{0}+\varepsilon)=f(x_{0})+df\left(x_{0}\right)\left[\varepsilon\right]+\varepsilon\varepsilon_{1}$

where $\varepsilon_{1}$ is an infinitesimal (which might depend on
$\varepsilon$).

It is immediate to check that a function is differentiable in $x_{0}$ if and
and only if

$\forall\varepsilon\in\mathfrak{mon}\left(0\right)\backslash\left\\{0\right\\},\
\
Df\left(x_{0}\right)=st\left(\frac{f(x_{0}+\varepsilon)-f(x_{0})}{\varepsilon}\right)$

Then, if a function is differentiable in $x_{0}\in(a,b)\cap\mathbb{R}$, it has
the $\mathbb{E}$-derivative in that point and

$df\left(x_{0}\right)\left[t\right]=Df(x_{0})\cdot t.$

but the converse is not true. The derivability of a function does not coincide
with the differentiability; it is well known that in ”classic calculus” this
phenomenon occurs only in dimension $\geq 2.$

######  Remark 45.

With the above definitions, the two classical problems of the ”istantaneous
velocity” and of the ”tangent” get different solutions given by the
$\mathbb{E}$-derivative and the differentaial respectively. They coincide only
for continuous functions.

Even if the $\mathbb{E}$-derivative is weaker than the usual one, it is quite
surprising that the main theorems of calculus remain true. For example, let us
consider the Fermat theorem:

######  Theorem 46.

(Fermat theorem) If a normal function
$f:\left(a,b\right)\rightarrow\mathbb{E}$ achieves a local maximum (or
minimum) in a point $x_{0}\in(a+\eta,b-\eta)$ and it has the
$\mathbb{E}$-derivative in that point, then

$Df(x_{0})=0$

Proof: We have that $D^{+}f(x_{0})\leq 0,\ $and $D^{-}f(x_{0})\geq 0.$ Since
$D^{+}f(x_{0})=$ $D^{-}f(x_{0}),$ it follows that

$Df(x_{0})=D^{+}f(x_{0})=0.$

$\square$

Following the usual procedure, we can prove Rolle theorem, the Lagrange
intermediate value theorem and most of the theorems of real calculus for a
class of function that are not necessarely differentiable, but have only the
$\mathbb{E}$-derivative. We will sketch this fact (see [3] for details). First
of all we recall some well known fact in NSA:

######  Definition 47.

A function

$f:D\rightarrow\mathbb{E},\ D\subset\mathbb{E}$

is called continuous in a point $\xi\in D$ iff

$\xi\sim x\Rightarrow f\left(\xi\right)\sim f\left(x\right);$

It is called continuous in $D$ if it is normal and it is continuous in every
point $\xi\in D\cap\mathbb{R}$. It is called uniformly continuous in $D$ if it
is normal and it is continuous in every point $\xi\in D.$

######  Theorem 48.

(Weierstrass) Let $f$ be a continuous fuction on an interval
$\left[a,b\right].$ Then $f$ has a maximum point in $\left[a,b\right]\ $and it
is a standard point.

Proof: Since the set $\left[a,b\right]\cap\mathbb{R}^{\circledcirc}$ is
hyperfinite, $f$ restricted to $\left[a,b\right]\cap\mathbb{R}^{\circledcirc}$
has a maximum point $\xi.$ We calim that $c=st\left(\xi\right)$ is the maximum
in $\left[a,b\right];$ in fact, since
$\left[a,b\right]\cap\mathbb{R}\subset\left[a,b\right]\cap\mathbb{R}^{\circledcirc},$
$\forall x\in\left[a,b\right]\cap\mathbb{R}$

$f\left(\xi\right)\geq f\left(x\right)$

and hence by the continuity of $f,$ $\forall
x\in\left[a,b\right]\cap\mathbb{R}$

$f\left(c\right)=st\left[f\left(\xi\right)\right]\geq
st\left[f\left(x\right)\right]=f(x).$

The inequality above, can be extended to every
$\zeta=\lim_{\lambda\uparrow\Lambda}x_{\lambda}\in\left[a,b\right],$
$x_{\lambda}\in\left[a,b\right]\cap\mathbb{R}$. In fact, since
$f\left(c\right)\geq f(x_{\lambda}),$

$f\left(c\right)=\lim_{\lambda\uparrow\Lambda}f\left(c\right)=\lim_{\lambda\uparrow\Lambda}f\left(x_{\lambda}\right)=f\left(\zeta\right).$

$\square$

So we have the following result involving the $\mathbb{E}$-derivative:

######  Lemma 49.

(Rolle) Let $f$ be a continuous fuction on an interval $\left[a,b\right]$ such
that $f(a)=$ $f(b);$ then if $f$ is $\mathbb{E}$-derivable in
$\left(a,b\right)$, there is a point $c\in\left(a,b\right)$ such that

$Df\left(c\right)=0$

Proof: By Fermat’s and Weierstrass’ theorems, the proof is equal to the usual
one.

$\square$

######  Theorem 50.

(Lagrange) Let $f$ be a continuous fuction on an interval $\left[a,b\right]$
and $\mathbb{E}$-derivable in $\left(a,b\right)$, there is a point
$c\in\left(a,b\right)$ such that

$Df\left(c\right)=\frac{f(b)-f(a)}{b-a}.$

Proof: By Rolle’s lemma, the proof is equal to the usual one.

$\square$

These results show that even if a $\mathbb{E}$-derivable function can be quite
wild (think of the Dirichlet function), the continuous $\mathbb{E}$-derivable
functions behave quite well. For example, the space of the solutions of the
equation

$Df=0$

in general, is not finite-dimensional. However, by the Lagrange theorem it
follows that the only continuous functions which solve the above equations are
the constants. Among the other consequences of the Lagrange’s teorem, we get
the following result:

######  Theorem 51.

A sufficient condition for a function $f$ to be differentiable in
$x_{0}\in(a,b)\cap\mathbb{R}$ is that both $f$ and $Df$ be continuous in
$x_{0}.$

Proof: See [3].

$\square$

This discussion shows that the notion of $\mathbb{E}$-derivability, even if it
is essentially irrelevant for the applications, it seems interesting for the
foundation of the notion of derivative and its relation with the
differentiability.

######  Remark 52.

In the framework of Euclidean calculus there are several other notions of
”generalzed derivative” which make sense. For example we can define the right
grid derivative as follows:

$D^{+}f(x_{0})=ctr\left(\frac{f(x^{+})-f(x)}{x^{+}-x}\right)$

where

$x^{+}=\min\ \left\\{y\in\mathbb{R}^{\circledcirc}\ |\ y>x\right\\}$

and similarly the left grid derivative etc. An other notion of generalized
derivative useful for the applications can be found in [4]. In this paper the
notion of grid derivative is combined with the notion of weak derivative in
such a way to include the derivative of distributions (identified with
suitable internal functions).

###  6.3 The integral

Also the definition of the integral takes advantage of a peculiarity of
Euclidean analysis, namely of the operator ”$\circledcirc$” introduced by Def.
20.

######  Definition 53.

Given a normal function $f:\left[a,b\right]\rightarrow\mathbb{E}$, we define
the $\mathbb{E}$-integral as follows:

$\int_{a}^{b}f(x)dx:=ctr\left(\sum_{x\in\left[a,b\right]^{\circledcirc}}f(x)\left(x^{+}-x\right)\right)$

where

$x^{+}=\min\ \left\\{y\in\mathbb{R}^{\circledcirc}\ |\ y>x\right\\}.$

Clearly, if $f$ is Riemann integrable, the $\mathbb{E}$-integral coincides
with the Riemann integral. Moreover the $\mathbb{E}$-integral is well defined
for every normal function even when $\left[a,b\right]=\mathbb{R}$ or/and $f$
is unbounded. However, the most interesting property of the
$\mathbb{E}$-integral is given by the following theorem:

######  Theorem 54.

Let $f$ be a bounded Lebesgue integrable function, then the
$\mathbb{E}$-integral is equal to the Lebesgue integral.

Proof: Assume that $f$ is a bounded Lebesgue integrable function in
$\left[a,b\right]$ and set

$f_{\lambda}(x):=\sum_{z\in\left[a,b\right]\cap\lambda}f(z)\chi_{\left[z,z_{\lambda}^{+}\right)}(x)$

where $\chi_{\left[z,z_{\lambda}^{+}\right)}$ is the characteristic function
of $\left[z,z_{\lambda}^{+}\right)$ and

$z_{\lambda}^{+}=\min\ \left\\{y\in\mathbb{\lambda}\ |\ y>z\right\\}.$

Now let us denote by $\int_{L}$ the Lebesgue integral; then

$\int_{L}f_{\lambda}(x)dx=\int f_{\lambda}(x)dx$

The net of functions $\left\\{f_{\lambda}\right\\}$ converges to $f$ in every
point $x\in\left[a,b\right]\cap\mathbb{R}$ since
$\left[a,b\right]\cap\mathbb{R}\subset\left[a,b\right]^{\circledcirc};$ in
fact, eventually we have that $\forall x\in\left[a,b\right]\cap\mathbb{R}$

$f_{\lambda}(x)=f(x)$

and hence by (13), $\forall x\in\left[a,b\right]\cap\mathbb{R}$,

$\lim_{\lambda\rightarrow\Lambda}\ f_{\lambda}(x)=st\left[f(x)\right]=f(x)$

where $\lim_{\lambda\rightarrow\Lambda}$ is the usual Cauchy limit. By the
Dominated Convergence Theorem,

$\lim_{\lambda\rightarrow\Lambda}\int_{L}f_{\lambda}(x)dx=\int_{L}\lim_{\lambda\rightarrow\Lambda}\
f_{\lambda}(x)dx=\int_{L}f(x)dx$

On the other hand,

$\displaystyle\lim_{\lambda\uparrow\Lambda}\ \int f_{\lambda}(x)dx$
$\displaystyle=$ $\displaystyle\lim_{\lambda\uparrow\Lambda}\
\left[\sum_{z\in\left[a,b\right]\cap\lambda}f(z)\chi_{\left[z,z_{\lambda}^{+}\right)}(x)\right]$
$\displaystyle=$
$\displaystyle\sum_{x\in\left[a,b\right]^{\circledcirc}}f(x)\left(x^{+}-x\right)$

Then, by (13),

$\int_{L}f(x)dx\sim\sum_{x\in\left[a,b\right]^{\circledcirc}}f(x)\left(x^{+}-x\right)$

and hence

$\int_{L}f(x)dx=st\left(\sum_{x\in\left[a,b\right]^{\circledcirc}}f(x)\left(x^{+}-x\right)\right)=\int
f(x)dx.$

$\square$

######  Remark 55.

All the normal functions are $\mathbb{E}$-integrable; however the functions
which are not Lebesgue-integrable might have a pathological behavior; for
example their integral is not invariant for translations. Nevertheless
$\mathcal{L}^{1}$, the space of the Lebesgue-integrable functions, can be
easily characterized as the closure of the continuous functions with compact
support with respect to the norm

$\left\|f\right\|=\int\left|f(x)\right|dx.$

##  7 Consistency of the axioms

In this sections, we will prove the consistency of the three axioms introduced
in section 2 by building a model in ZFC+{Axiom of Inaccessibility}.

######  Remark 56.

In many models on nonstandard analysis, the axiom of regularity of ZFC may
fail for external sets (see e.g. [14, 10]). However, in this paper, both, the
standard universe $V\left(\mathbb{R}\right)$ and the Euclidean universe
$V\left(\mathbb{E}_{\kappa}\right)$ contain only sets of finite rank and this
peculiarity allows to have a model in ZFC. We are forced to work with sets of
finite rank by axiom 1. In fact, assuming that $\Lambda$ contains a set $A$ of
infinite rank, we would get a contradiction. Take for instance a set a defined
as follows:

$A=\left\\{b_{n}\ |\ n\in\mathbb{N}\right\\}$

where $b_{0}=a$ and

$b_{n+1}=\left(b_{n},a\right)$

namely

$A=\left\\{a,\left(a,a\right),\left(a,a,a\right),\left(a,a,a,a\right),,....\right\\}.$

Then, setting

$B=A\times\left\\{a\right\\}=\left\\{\left(a,a\right),\left(a,a,a\right),\left(a,a,a,a\right),....\right\\}=\left\\{b_{n}\
|\ n\in\mathbb{N}^{+}\right\\}$

we have that $B\subset A$ and hence, by Axiom 1.2

$\mathfrak{num}\left(B\right)<\mathfrak{num}\left(A\right),$

while, by Axiom 1.5

$\mathfrak{num}\left(B\right)=\mathfrak{num}\left(A\times\left\\{a\right\\}\right)=\mathfrak{num}\left(A\right)\cdot\mathfrak{num}\left(\left\\{a\right\\}\right)=\mathfrak{num}\left(A\right)\cdot
1=\mathfrak{num}\left(A\right).$

Contradiction!

###  7.1 The construction of the field $\mathbb{E}$

We assume that $\mathbb{A}$ is a set of atoms having cardinality
$\left|\mathbb{A}\right|=\kappa$ and that it contains a set $\mathbb{R}$
isomorphic to the real numbers. Moreover, we assume that $\Lambda_{S}$,
$\Lambda$ and $\mathfrak{L}$ be sets as defined in section 2.1.

If $n_{0}\in\mathbb{N},\ $and $\lambda_{0}\in\mathfrak{L}$, we set

$Q\left(n_{0},\lambda_{0}\right)=\left\\{V_{n}\left(\lambda\right)\in\mathfrak{L}\
|\ n\in\mathbb{N},\ n\geq n_{0};\ \lambda\in\mathfrak{L},\
\lambda\supseteq\lambda_{0}\right\\}$

We have that $Q\left(n_{0},\lambda_{0}\right)\subset\mathfrak{L}$ and

$Q\left(n_{0},\lambda_{0}\right)\cap
Q\left(m_{0},\mu_{0}\right)=Q\left(\max\left\\{n_{0},m_{0}\right\\},\
\lambda_{0}\cup\mu_{0}\right)$

Hence there exists an ultrafilter $\mathcal{U}$ over $\mathfrak{L}$ such that
$\forall n\in\mathbb{N}$, $\forall\lambda\in\mathfrak{L}$,

$Q\left(n,\lambda\right)\in\mathcal{U}.$ (22)

Now, for any $j<\kappa$, we define by transfinite induction a sequence of
ordered fields $\mathbb{K}_{j}\subset\mathbb{A}$ such that
$\left|\mathbb{K}_{j}\right|<\kappa$ and

$\mathbb{K}_{j}\subset\mathbb{K}_{k}\ \ \ \ \,\text{if}\ \ j<k.$

For $j=0,$ we set

$\mathbb{K}_{0}=\mathbb{R}$

and for every ordinal $j<\kappa,$ we set

$\mathfrak{F}_{j+1}:=\mathfrak{F}\left(\mathfrak{L},\mathbb{K}_{j}\right)/I_{j}$
(23)

where

$\mathfrak{F}_{j}\left(\mathfrak{L},\mathbb{K}_{j}\right)=\left\\{\varphi\in\left(\mathbb{K}_{j}\right)^{\mathfrak{L}}\
|\ \forall\lambda\in\mathfrak{L},\
\varphi(\lambda)=\varphi(\lambda\cap\mathbb{K}_{j})\right\\}$

and $I_{j}\subset\mathfrak{F}_{j}\left(\mathfrak{L},\mathbb{K}_{j}\right)$ is
the maximal ideal defined as follows:

$I_{j}:=\left\\{\varphi\in\mathfrak{F}_{j}\left(\mathfrak{L},\mathbb{K}_{j}\right)\
|\ \exists Q\in\mathcal{U},\ \forall\lambda\in Q,\
\varphi(\lambda)=0\right\\}$

Then $\mathfrak{F}_{j+1}$ is a field and the projection

$\Pi_{j}:\mathfrak{F}_{j}\left(\mathfrak{L},\mathbb{K}_{j}\right)\rightarrow\mathfrak{F}_{j+1}$

defined by

$\Pi_{j}\left(\varphi\right)=\left[\varphi\right]_{\mathcal{U}}:=\varphi+I_{j}$

is a surjective ring homomorphism. Now, since
$\left|\mathbb{K}_{j}\right|<\kappa,$ also
$\left|\mathfrak{F}_{j+1}\right|<\kappa;$ then we can define an injetive map

$\Theta_{j}:\mathfrak{F}_{j+1}\rightarrow\mathbb{A}$

such that $\forall\xi\in\mathbb{K}_{j}$,

$\Theta_{j}\left(\left[C_{\xi}\right]_{\mathcal{U}}\right)=\xi$

where $\forall\lambda\in\mathfrak{L,\
}C_{\xi}\left(\lambda\right):=\xi\in\mathbb{K}_{j}$. The set
$\mathbb{K}_{j+1}:=\mathop{\mathrm{I}m}\Theta_{j}$ can be equipped with the
structure of ordered field by setting, $\forall\xi,\zeta\in\mathbb{K}_{j+1}$

$\displaystyle\xi+\zeta$ $\displaystyle=$
$\displaystyle\Theta_{j}\left(\Theta_{j}^{-1}\left(\xi\right)+\Theta_{j}^{-1}\left(\zeta\right)\right)$
$\displaystyle\xi\cdot\zeta$ $\displaystyle=$
$\displaystyle\Theta_{j}\left(\Theta_{j}^{-1}\left(\xi\right)\cdot\Theta_{j}^{-1}\left(\zeta\right)\right).$

Then $\mathbb{K}_{j+1}$ is a field which contains the real numbers. Now set

$J_{j}=\Theta_{j}\circ\Pi_{j}:\mathfrak{F}\left(\mathfrak{L},\mathbb{K}_{j}\right)\rightarrow\mathbb{K}_{j+1}$

By this construction, $J_{j}$ is a surjective ring homomorphism.

If $k\leq\kappa$ is a limit ordinal, we set

$\mathbb{K}_{k}=\mathop{\displaystyle\bigcup}\limits_{j<k}\mathbb{K}_{j}$

and

$J_{k}=\ \underset{\longrightarrow}{\lim}J_{j}$

namely, $J_{k}\left(\varphi\right)=J_{j}\left(\varphi\right)$ for every $j$
such that $\varphi\in\mathfrak{F}\left(\mathfrak{L},\mathbb{K}_{j}\right).$
Then, also

$J_{k}:\mathfrak{F}\left(\mathfrak{L},\mathbb{K}_{k}\right)\rightarrow\mathbb{K}_{k}$

is a surjective ring homomorphism.

In particular, if $j=\kappa$, we have that

$J:=J_{\kappa}:\mathfrak{F}\left(\mathfrak{L},\mathbb{K}_{\kappa}\right)\rightarrow\mathbb{K}_{\kappa}\subset\mathbb{A}$
(24)

is a surjective ring homomorphism and
$\left|\mathbb{E}_{\kappa}\right|=\kappa$.

If $A\in\Lambda$, by the definition of $\Lambda$ (see (1)), we have that
$\left|A\right|<\kappa,$ then the net $\left\\{\lambda\mapsto\left|\lambda\cap
A\right|\right\\}\in\mathfrak{F}\left(\mathfrak{L},\mathbb{K}_{\kappa}\right).$
Then, we can define the numerosity of $A$ as follows:

$\mathfrak{num}\left(A\right)=J\left(\left|\lambda\cap A\right|\right)$ (25)

where, with some abuse of notation, $\left|\lambda\cap A\right|$ denotes the
net $\left\\{\lambda\mapsto\left|\lambda\cap A\right|\right\\}.$

###  7.2 Proof of the consistency of Axioms 1-3

Now we can prove the consistency of our axioms.

######  Theorem 57.

The numerosity function defined by (25) satisfies the request of Axiom 1.

Proof : 1.1 and 1.2 follows directly by the definition (25) of
$\mathfrak{num}$.

1.3 \- We have that

$\displaystyle\mathfrak{num}\left(A\cup B\right)$ $\displaystyle=$
$\displaystyle\ J\left(\left|\lambda\cap\left(A\cup
B\right)\right|\right)=J\left(\left|\left(\lambda\cap
A\right)\cup\left(\lambda\cap B\right)\right|\right)$ $\displaystyle=$
$\displaystyle J\left(\left|\left(\lambda\cap
A\right)\right|+\left|\left(\lambda\cap
B\right)\right|\right)=J\left(\left|\left(\lambda\cap
A\right)\right|\right)+J\left(\left|\left(\lambda\cap A\right)\right|\right)$
$\displaystyle=$
$\displaystyle\mathfrak{num}\left(A\right)+\mathfrak{num}\left(B\right).$

1.4 \- Let $n_{0}$ be so large that $A,B\in
V_{n_{0}}\left(\mathbb{E}_{\kappa}\right),$ then $A\times B\in
V_{n_{0}+2}\left(\mathbb{E}_{\kappa}\right);$ if we take
$V_{n}\left(\lambda\right)\in Q\left(n_{0}+2,\lambda\right),$ we have that

$\displaystyle\left|V_{n}\left(\lambda\right)\cap\left(A\times
B\right)\right|$ $\displaystyle=$
$\displaystyle\left|\left(V_{n}\left(\lambda\right)\cap
A\right)\times\left(V_{n}\left(\lambda\right)\cap B\right)\right|$ (26)
$\displaystyle=$ $\displaystyle\left|V_{n}\left(\lambda\right)\cap
A\right|\cdot\left|V_{n}\left(\lambda\right)\cap B\right|$

If we set

$\varphi_{E}\left(\lambda\right)=\left|V_{n}\left(\lambda\right)\cap E\right|$

by (26), we have that $\forall\lambda\in Q\left(n_{0}+2,\lambda_{0}\right),$

$\varphi_{A\times
B}\left(\lambda\right)=\varphi_{A}\left(\lambda\right)\cdot\varphi_{B}\left(\lambda\right)$

and since $Q\left(n_{0}+2,\lambda_{0}\right)\in\mathcal{U}$,

$J\left(\varphi_{A\times B}\right)=J\left(\varphi_{A}\right)\cdot
J\left(\varphi_{B}\right)$

Then

$\mathfrak{num}\left(A\times B\right)=J\left(\varphi_{A\times
B}\right)=J\left(\varphi_{A}\right)\cdot
J\left(\varphi_{B}\right)=\mathfrak{num}\left(A\right)\cdot\mathfrak{num}\left(B\right)$
(27)

1.5 \- It follows immediately from (27).

$\square$

######  Theorem 58.

The set $\mathbb{E}_{\kappa}$ satisfies the request of Axiom 2.

Proof : Trivial by our construction.

$\square$

We define the set of the Euclidean integers as follows:

$\mathfrak{Z:=}\mathop{\displaystyle\bigcup}\limits_{j<\kappa}\mathfrak{Z}_{j}$

where

$\mathfrak{Z}_{0}=\mathbb{Z}$

and, for $j<\kappa$

$\mathfrak{Z}_{j}:=J\left(\mathop{\displaystyle\bigcup}\limits_{k<j}\mathfrak{Z}_{k}\right)$
(28)

Before proceeding we need the following

######  Lemma 59.

Every number $\xi\in\mathbb{E}_{\kappa}$ can be decomposed as follows:

$\xi=\zeta+\mu$ (29)

where $\zeta\in\mathfrak{Z}$ and $0\leq\mu<1$.

Proof: Let $\xi\in\mathbb{E}_{j}.$ We argue by transfinite induction over $j.$
If $j=0,$ (29) holds with $\zeta\in\mathbb{Z}$. If $j>0,$ then,

$\xi:=J\left(\left\\{\xi_{\lambda}\right\\}\right)$

with $\xi_{\lambda}\in\mathbb{E}_{k}$ for some $k<j.$ By our inductive
assumption,

$\xi_{\lambda}=\zeta_{\lambda}+\mu_{\lambda},\ \
\zeta_{\lambda}\in\mathfrak{Z\ \ }\text{and\ \ }0\leq\mu_{\lambda}<1.$

Then

$\xi:=J\left(\left\\{\zeta_{\lambda}+\mu_{\lambda}\right\\}\right)=J\left(\left\\{\zeta_{\lambda}\right\\}\right)+J\left(\left\\{\mu_{\lambda}\right\\}\right)$

By (28),
$\zeta:=J\left(\left\\{\zeta_{\lambda}\right\\}\right)\in\mathfrak{Z}$ and if
we set $\mu:=J\left(\left\\{\mu_{\lambda}\right\\}\right),$ we have that
$0\leq\mu<1.\ $Then (29) is satisfied.

$\square$

We now set

$\mathfrak{C:=}\left\\{\zeta+r\ |\ \zeta\in\mathfrak{Z},\
r\in\mathbb{R}\right\\}.$ (30)

Clearly $\mathfrak{Z}$ and $\mathbb{R}$ are additive subgroups of
$\mathbb{E}_{\kappa}$ and hence also $\mathfrak{C}$ is an additive group.
$\mathfrak{C}$ is the set of the centers.

######  Theorem 60.

The set $\mathfrak{C}$ defined by (30) satisfies the requests of Axiom 3.

Proof. Since $\mathbb{E}_{\kappa}$ contains the real numbers, if
$\theta\in\mathbb{E}_{\kappa}$ is bounded, $st(\theta)$ is well defined. If
$\xi\in\mathbb{E}_{\kappa}$, by Lemma 59, we can write $\xi=\zeta+\mu,\ $with
$\zeta\in\mathfrak{Z\ }$and $0\leq\mu<1;$ then we set

$ctr(\zeta+\mu)=\zeta+st(\mu).$

Then $ctr(\zeta+\mu)\in\mathfrak{C}$.

$\square$

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