# Successor Sets and the Axioms of Peano

The present unit is part of the following walks

## Introduction

Please note that the text on this web page gives a summary of the Unit Successor Sets and the Axioms of Peano. A full text including the proofs is available as a separate pdf-document at the end of this page.

The most prominent system of axioms for mathematics is the axiomatics of Zermelo and Fraenkel abbreviated by ZFC where the letter C stands for the axiom of choice. It consists of the following axioms:

Axioms ZFC-0 to ZFC-4 are explained in Unit The Mathematical Universe. Axioms ZFC-5 to ZFC-7 are explained in Unit Unions and Intersections of Sets. Axioms ZFC-8 and ZFC-9 are explained in Unit Families and the Axiom of Choice. Finally, Axiom ZFC-10 will be explained in the present unit.

I think it is really amazing, if not a small miracle, that ten axioms are a sufficient basis for (almost) the complete mathematical theory.

### Successor Sets and the Axiom of Infinity:

Within the axiomatics of Zermelo and Fraenkel the natural numbers are defined as special sets. The main idea is as follows: One sets

\begin{eqnarray*}
0 & := & \emptyset \\
1 & := & \{ 0 \} = \{ \emptyset \} = \emptyset \cup \{ \emptyset \} = 0 \cup \{ 0 \} \\
2 & := & \{ 0, 1 \} = 1 \cup \{ 1 \} \\
3 & := & \{ 0, 1, 2 \} = 2 \cup \{ 2 \} \\
& \vdots & \\
n + 1 & := & \{ 0, 1, \ldots, n \} = n \cup \{ n \}.
\end{eqnarray*}

Then one defines the set $\mathbb{N}_0$ of the natural numbers as the set

$$\mathbb{N}_0 := \{0, 1, \ldots, n, \ldots \}.$$

The axiom of infinity will be needed to guarantee the existence of the set $\mathbb{N}_0$ of all natural numbers.

These definitions will only take place in Unit Natural Numbers and the Principle of Induction introducing the natural numbers. The present unit is devoted to the necessary preparations consisting of the successor sets and the axioms of Peano.

Since the natural numbers will be defined recursively by

$$0 := \emptyset \mbox{ and } n + 1 := n \cup \{ n \},$$

the first step in this direction is the investigation of sets of the form

$$A \cup \{ A \}.$$

Such a set is called a successor set, and it is denoted by

$$A^+ := A \cup \{ A \}.$$

In Unit Natural Numbers and the Principle of Induction we will define

$$n+1 := n ^+ = n \cup \{ n \},$$

that is, the number $n + 1$ will be defined to be the successor of the number $n$. Successor sets have the following important properties:

(i) We have $A^+ \neq \emptyset$ (Proposition [#nst-prop-successor-non-empty]).

(ii) We have $A \in A^+$ and $A \subseteq A^+$ (Proposition [#nst-prop-successor-element-subset]).

(iii) We have $A^+ \neq A$ (Proposition [#nst-prop-A-different-A-plus]).

The next question is how to construct the set

$$\mathbb{N}_0 = \{0, 1, 2, \ldots, n, n + 1, \ldots \}$$

of all natural numbers: For its construction we need a new axiom, namely the axiom of infinity guaranteeing the existence of a set ${\cal A}$ fulfilling the following conditions:

(i) The empty set $\emptyset$ is an element of the set ${\cal A}$.

(ii) If the set $A$ is an element of the set ${\cal A}$, then its successor $A^+$ is also an element of the set ${\cal A}$.

A set ${\cal A}$ fulfilling Conditions (i) and (ii) is called a successor set.

The most important conclusion of the axiom of infinity is the existence of a “minimal” successor set $\omega$ (see Theorem [#nst-th-existence-omega]). More precisely, Theorem [#nst-th-existence-omega] states that there exists exactly one successor set $\omega$ with the following property: If ${\cal S}$ is a successor set, then the set $\omega$ is a subset of the set ${\cal S}$.

### The Minimal Successor Set:

As explained above the axiom of infinity will allow to define the set $\mathbb{N}_0$ of the natural numbers (see Unit Natural Numbers and the Principle of Induction). The set $\mathbb{N}_0$ will be defined as the minimal successors set $\omega$.

It follows from the definition of the minimal successor set $\omega$ that we have

$$0 = \emptyset \in \omega, 1 = 0^+ \in \omega, 2 = 1^+ \in \omega, \ldots, n+1 = n^+ \in \omega, \ldots$$

which motivates the definition $\mathbb{N}_0 := \omega$. Of course, we will define $n + 1 := n^+$ for each natural number $n$.

The first important properties of the minimal successor set $\omega$ are as follows:

(i) Given a non-empty set $A$ of the minimal successor set $\omega$ there exists a set $B$ of the set $\omega$ such that $A = B^+$ (Theorem [#nst-th-minimal-successor-set-existence-predecessor]). This means: Given a natural number $n \neq 0$ there exists a natural number $m$ such that $n = m + 1$.

(ii) Given two elements $A$ and $B$ of the minimal successor set $\omega$ such that $A^+ = B^+$ we have $A = B$ (Theorem [#nst-th-minimal-successor-set-plus-injective]). This means: Given two natural numbers $m$ and $n$ such that $m + 1 = n + 1$ we have $n = m$.

A technically important tool is the observation that the minimal successor set is a so-called transitive set (see Proposition [#nst-prop-minimal-successor-set-collection]).

### The Axioms of Peano:

The axioms of Peano have been introduced by Giuseppe Peano as an axiomatic basis for the definition of he natural numbers.

Within the axiomatics of Zermelo and Fraenkel we go one step further: We first give an axiomatic basis for the theory of sets or even for mathematics in general and then deduce the axioms of Peano from the axioms of Zermelo and Fraenkel.

The five axioms of Peano and their meaning are as follows:

There exists a set $\omega$ such that

(P1) The set $\omega$ has a distinguished element $0$. This means that the set $\mathbb{N}_0$ contains the number $0$.

(P2) There exists a function $+ : \omega \rightarrow \omega$, $x \mapsto x^+$ from the set $\omega$ into itself. This means that the function $n \mapsto n + 1$ is a function from the set of the natural numbers into itself.

(P3) We have $x^+ \neq 0$ for all elements $x$ of the set $\omega$. This means that we have $n + 1 \neq 0$ for all natural numbers $n$.

(P4) If $x$ and $y$ are two elements of the set $\omega$ such that $x^+ = y^+$, then we have $x = y$. This means that $m + 1 = n + 1$ implies $m = n$ for all natural numbers $m$ and $n$.

(P5) If $A$ is a subset of the set $\omega$ such that

$$0 \in A \mbox{ and } x^+ \in A \mbox{ for all } x \in A,$$

then we have $A = \omega$. This means that if $A$ is a subset of the set $\mathbb{N}_0$ such that

$$0 \in A \mbox{ and } n + 1 \in A \mbox{ for all } x \in A,$$

then we have $A = \mathbb{N}_0$.

A set $X$ fulfilling the axioms of Peano is called a Peano set. Axiom (P5) guarantees that the set $\omega$ is not too big, and it allows to introduce the principle of induction. In the present unit we restrict ourselves to prove the fact that the minimal successor set fulfills the axioms of Peano (Theorem [#nst-th-minimal-successor-set-axiom-peano]). Further conclusions like the principle of induction are subject of Unit Natural Numbers and the Principle of Induction).

### The Recursion Theorem:

Let us look at the recursive definition of the addition of two natural numbers $m$ and $n$. We set

$$m + 0 := m \mbox{ and } m + (n + 1) := (m + n) + 1 \mbox{ for all } n \in \mathbb{N}_0.$$

In other words, we are looking for a function $\alpha_m : \mathbb{N}_0 \rightarrow \mathbb{N}_0$ such that

$$\alpha_m(0) = m \mbox{ and } \alpha_m(n + 1) = \alpha_m(n) + 1 \mbox{ for all } n \in \mathbb{N}_0.$$

Then we can define $m + n := \alpha_m(n)$.

More generally, let $X$ be a set, let $f : X \rightarrow X$ be a function, and let $x_0$ be an element of the set $X$. Then we are looking for a function $\alpha : \mathbb{N}_0 \rightarrow X$ such that

$$\alpha(0) = x_0 \mbox{ and } \alpha(n + 1) = f( \alpha(n) ) \mbox{ for all } n \in \mathbb{N}_0.$$

In our example we have $X = \mathbb{N}_0$, $x_0 = m$ and $f(x) := x + 1$ for all elements $x$ of the set $X = \mathbb{N}_0$. The existence of such a function $\alpha : \mathbb{N}_0 \rightarrow X$ is the content of the recursion theorem (Theorem [#nst-th-minimal-successor-set-recursion-theorem]).

### Isomorphisms of Peano Sets:

As explained above one of the main results of this unit is the fact that the minimal successor set is a Peano set. In fact, up to isomorphisms, the minimal successor set is the only Peano set. More precisely, given a Peano set $A$ there exists one (and only one) isomorphism $\alpha : \omega \rightarrow A$ from the minimal successor set $\omega$ onto the set $A$ (Theorem [#nst-th-axiom-peano-bijection]).

## Successor Sets and the Axiom of Infinity

### Definition of a Successor:

Definition. Let $A$ be a set. Then the set $A^+ := A \cup \{ A \}$ is called the successor of the set $A$.

French / German. Successor = Successeur = Nachfolger.

Examples. (a) Let $A := \emptyset$ be the empty set. Then we have

$$A^+ = A \cup \{ A \} = \emptyset \cup \{ \emptyset \} = \{ \emptyset \}.$$

(b) Let $A := \{a, b\}$ for two elements $a$ and $b$.

Then we have

$$A^+ = A \cup \{ A \} = \{a, b\} \cup \{ \{a, b\} \} = \{a, b, \{a, b\} \}.$$

### Elementary Properties of Successors:

Proposition. Let $A$ be a set, and let $A^+$ be its successor. Then the successor is non-empty.

Proposition. Let $A$ be a set, and let $A^+$ be its successor. Then we have

$$A \in A^+ \mbox{ and } A \subseteq A^+,$$

that is, the set $A$ is at the same time an element and a subset of the set $A^+$.

In the proof of Proposition [#nst-prop-A-different-A-plus] we will need the following result of Unit Unions and Intersections of Sets:

Theorem. For each set $A$ we have $A \notin A$.

Proposition. Let $A$ be a set, and let $A^+$ be its successor. Then we have $A^+ \neq A$.

### The Axiom of Infinity:

Axiom. (ZFC-10: Axiom of Infinity) There exists a set ${\cal A}$ fulfilling the following conditions:

(i) The empty set $\emptyset$ is an element of the set ${\cal A}$.

(ii) If $A$ is an element of the set ${\cal A}$, then its successor $A^+$ is also an element of the set ${\cal A}$.

### Definition of a Successor Set:

Definition. Let ${\cal A}$ be a set fulfilling the following conditions:

(i) The empty set $\emptyset$ is an element of the set ${\cal A}$.

(ii) If $A$ is an element of the set ${\cal A}$, then its successor $A^+$ is also an element of the set ${\cal A}$.

Then the set ${\cal A}$ is called a successor set.

Note that the axiom of infinity guarantees that there exists at least one successor set.

### Elementary Properties of Successor Sets:

Proposition. Let ${\cal C}$ be a non-empty set of successor sets. Then the intersection

$${\cal I} := \bigcap_{{\cal A} \in {\cal C}} {\cal A}$$

is also a successor set.

Theorem. There exists exactly one successor set $\omega$ with the following property: If ${\cal S}$ is a successor set, then the set $\omega$ is a subset of the set ${\cal S}$.

## The Minimal Successor Set

### Definition of the Minimal Successor Set:

Definition. Let $\omega$ be the successor set with the following property: If ${\cal S}$ is a successor set, then the set $\omega$ is a subset of the set ${\cal S}$ (Theorem [#nst-th-existence-omega]).

The successor set $\omega$ is called the minimal successor set. It is denoted by $\omega$.

Remark. In Unit Natural Numbers and the Principle of Induction we will set $\mathbb{N}_0 := \omega$, that is, we will define that the set of the natural numbers is the minimal successor set $\omega$. In addition, we will set $n + 1 := n^+$.

### Elementary Properties of the Minimal Successor Set:

Theorem. Let $\omega$ be the minimal successor set, and let ${\cal S}$ be a successor set which is a subset of the set $\omega$. Then we have ${\cal S} = \omega$.

Theorem. Let $\omega$ be the minimal successor set, and let $\emptyset \neq A$ be an element of the set $\omega$. Then there exists an element $B$ of the set $\omega$ such that $A = B^+$.

Remark. Theorem [#nst-th-minimal-successor-set-existence-predecessor] is an important technical tool in the study of successor sets. In view of the definition $\mathbb{N} = \omega$ it just means that for each natural number $n \neq 0$ there exists a natural number $m$ such that $n = m +1$.

Definition. A set $A$ is called a transitive set if every element of the set $A$ is at the same time a subset of the set $A$.

French / German. Transitive set = Ensemble transitif = Transitive Menge.

Proposition [#nst-prop-minimal-successor-set-collection] will be used in the proof of Theorem[#nst-th-minimal-successor-set-plus-injective].

Proposition. Let $\omega$ be the minimal successor set.

(a) The minimal successor set $\omega$ is a transitive set.

(b) Let $A$ be an element of the set $\omega$. Then the set $A$ is a transitive set.

Theorem. Let $\omega$ be the minimal successor set, and let $A$ and $B$ be two elements of the set $\omega$. If $A^+ = B^+$, then we have $A = B$.

### Historical Notes:

The idea of an infinite set is quite old. Early definitions have probably been of the following form: A set $A$ is called infinite if we have

$$|A| > n \mbox{ for all } n \in \mathbb{N}.$$

A good example for this approach is the wording of Euclid expressing that there exist infinitely many prime numbers:

Proposition 20. Prime numbers are more than any assigned multitude of prime numbers.

See [Heath-1908b], Book IX, no. 20, p. 412.

In the context of the axiomatization of mathematics it was a new idea that one has to give an argument why infinite sets, in particular the set $\mathbb{N}_0$ of the natural numbers, exist.

The first breakthrough was the paper Was sind und was sollen die Zahlen? of 1888 by Richard Dedekind. In this paper he gives a formal introduction of the natural numbers and gives a formal definition of an infinite set:

64. Erklärung. Ein System $S$ heißt unendlich, wenn es einem echten Teile seiner selbst ähnlich ist […]; im entgegengesetzten Fall heißt $S$ ein endliches System.

See [Dedekind-CW], Vol. 1, p. 356.

64. Explanation. A system $S$ is called infinite if it is similar to a proper subset of the system $S$ ) […]; in the opposite case $S$ is called a finite system.

(Translation by the author.)

Dedekind uses the word system for sets. Two sets $A$ and $B$ are called similar if there exists a bijective mapping from the set $A$ onto the set $B$.

Dedekind also postulated the existence of an infinite set:

66. Satz. Es gibt unendliche Systeme.

See [Dedekind-CW], Vol. 1, p. 357.

64. Theorem. There exist infinite systems.

(Translation by the author.)

Dedekind’s proof is of a more philosophical nature and cannot be seen as a rigorous mathematical proof. If you want, you can understand his “proof” as a (philosophical) argument that makes the postulation of the existence of infinite sets more plausible. With this interpretation one may say that Dedekind was the first to introduce the axiom of infinity.

By the way, the further procedure of Dedekind is quite similar to the procedure explained above: The axiom of infinity provides the existence of at least one successor set. As a next step the minimal successor set $\omega$ is constructed followed by the definition $\mathbb{N}_0 := \omega$.

Dedekind also postulates the existence of an infinite set and then constructs a specific infinite set $N$ followed by the definition $\mathbb{N}_0 := N$.

Ernst Zermelo presents his first version of the axiom of infinity in his first axiomatization paper of 1908 as follows:

Axiom VII (Axiom des Unendlichen). Der Bereich enthält mindestens eine Menge $Z$, welche die Nullmenge als Element enthält und so beschaffen ist, dass jedem ihrer Elemente $a$ ein weiteres Element der Form $\{a\}$ entspricht, oder welche mit jedem ihrer Elemente $a$ auch die entsprechende Menge $\{ a \}$ enthält.

See [Zermelo-1908b], pp. 266-267.

Axiom VII (Axiom of Infinity). There exists in the domain at least one set $Z$ that contains the null set as an element and is so constituted that to each of its elements $a$ there corresponds a further element of the form $\{a\}$, in other words, that with each of its elements $a$ it also contains the corresponding set $\{ a \}$ as an element.

See [Zermelo-1908b-en], p. 204.

The domain is the mathematical universe (see Unit The Mathematical Universe). This axiom is very close to our axiom of infinity, but there is an important difference:

Zermelo has in mind to construct the natural numbers as follows:

$$0 := \emptyset, \: 1 := \{ 0 \} = \{ \emptyset \}, \: 2 := \{ 1 \} = \{ \{ \emptyset \} \}, \: 3 := \{ 2 \} = \{ \{ \{ \emptyset \} \} \}, \ldots$$

During the investigation of ordinal numbers (see Unit [Ordinal Numbers – in preparation]), John von Neumann introduced the following way to define natural numbers:

$$0 := \emptyset, \: 1 := 0 \cup \{ 0 \} = \{ 0 \} = \{ \emptyset \}, \: 2 := 1 \cup \{ 1 \} = \{ 0, 1 \} = \{ \emptyset, \{ \emptyset \} \},$$

$$3 := 2 \cup \{ 2 \} = \{ 0, 1, 2 \} = \{ \emptyset, \{ \emptyset \}, \{ \emptyset, \{ \emptyset \} \} \}, \ldots$$

In his article Zur Einführung der transfiniten Zahlen from 1922 this reads as follows:

\begin{eqnarray*}
0 & := & 0, \\
1 & := & (0), \\
2 & := & (0, (0)), \\
3 & := & (0, (0), (0, (0))).
\end{eqnarray*}

See [von-Neumann-1922], p. 199.

## The Axioms of Peano

### The Axioms of Peano:

Definition. Let $A$ be a set.

(a) The set $A$ fulfills the axioms of Peano if it fulfills the following conditions:

(P1) The set $A$ contains a distinguished element $0$. In particular, the set $A$ is not empty.

(P2) There exists a function $^+ : A \rightarrow A$, $x \mapsto x^+$ from the set $A$ into itself.

(P3) We have $x^+ \neq 0$ for all elements $x$ of the set $A$.

(P4) If $x$ and $y$ are two elements of the set $A$ such that $x^+ = y^+$, then we have $x = y$, that is, the function $^+ : A \rightarrow A$ is injective.

(P5) If $B$ is a subset of the set $A$ such that

$$0 \in B \mbox{ and } x^+ \in B \mbox{ for all } x \in B,$$

then we have $B = A$.

(b) Let $A$ be a set fulfilling the axioms of Peano. Then the set $A$ is called a Peano set.

French / German. Peano set = Ensemble de Peano = Peano-Menge.

Theorem. The minimal successor set $\omega$ is a Peano set where $0 := \emptyset$ and $A^+ := A \cup \{A\}$ for all elements $A$ of the set $\omega$.

### Elementary Properties of Peano Sets:

Proposition. Let $A$ be a set fulfilling the axioms of Peano, and let $0 \neq x$ be an element of the set $A$. Then there exists an element $y$ of the set $A$ such that $x = y^+$.

Proposition. Let $A$ be a set fulfilling the axioms of Peano. Then the function $\alpha : A \rightarrow A \setminus \{ 0 \}$, $\alpha : x \mapsto x^+$ is bijective.

### Historical Notes:

The axioms of Peano have been introduced by Giuseppe Peano:

Axiomata

1. $1 \in N$.
2. $a \in N \Rightarrow a = a$.
3. $a, b, c \in N \Rightarrow ( a = b \Leftrightarrow b = a )$.
4. $a, b \in N \Rightarrow \big( (a = b) \land (b = c) \Rightarrow (a = c) \big)$.
5. $(a = b) \land (b \in N) \Rightarrow a \in N$.
6. $a \in N \Rightarrow a + 1 \in N$.
7. $a, b \in N \Rightarrow \big( (a = b) \Leftrightarrow (a + 1 = b + 1) \big)$.
8. $a \in N \Rightarrow a + 1 \neq 1$.
9. $(1 \in K) \land \big( ( x \in K) \Rightarrow ( x + 1 \in K) \big) \Rightarrow N \subseteq K$.

Definitiones

10. $2 = 1 + 1$; $3 = 2 + 1$; $4 = 3 + 1$; etc.

See [Peano-1889], p. 1.

We have transformed the original text of Peano into modern notations. Condition (1) corresponds to Condition (P1) of our definition of a Peano set. Peano starts with the number 1, whereas we have started with the number 0. Condition (6) corresponds to Condition (P2). Condition (8) corresponds to Condition (P3). Condition (7) corresponds to Condition (P4). Finally, Condition (9) corresponds to Condition (P5).

## The Recursion Theorem

The recursion theorem allows the recursive definition of functions. We will apply it in Theorem [#nst-th-minimal-successor-set-axiom-peano-bijection].

Theorem. (Recursion Theorem) Let $X$ be a non-empty set, let $f : X \rightarrow X$ be a function from the set $X$ into itself, and let $a$ be an element of the set $X$.

Then there exists exactly one function $\alpha : \omega \rightarrow X$ from the minimal successor set $\omega$ into the set $X$ fulfilling the following conditions:

(i) We have $\alpha(\emptyset) = a$.

(ii) We have $\alpha(A^+) = f( \alpha(A) )$ for each element $A$ of the set $\omega$.

French / German. Recursion theorem = Thórème de récursivité = Rekursionssatz.

### Historical Notes:

The method of the recursive definition of a sequence is quite old. For example, Blaise Pascal uses a recursive definition in his Traité du Triangle Arithmétique from 1665 in order to define what is today called the binomial coefficients:

Le nombre de chaque cellule est égal à celui de la cellule qui la précède dans son rang perpendiculaire, plus à celui qui la précède dans son rang parallèle.

See [Pascal-1665], Traité du Triangle, p. 1.

The number in each cell is equal to the number in the cell which precedes it in its perpendicular rang plus the number in the cell which precedes it in its parallel rang.

(Translation by the author.)

In modern terminology Pascal defines the binomial coefficients by the equation

$${n \choose k} = {n – 1 \choose k – 1} + {n – 1 \choose k}.$$

For more details about binomial coefficients and the Triangle of Pascal see Unit [The Binomial Coefficients and the Triangle of Pascal – in preparation].

The formal definition and proof of the recursion theorem has been given by Richard Dedekind in 1888:

126. Satz. Ist eine beliebige (ähnliche oder unähnliche) Abbildung $\theta$ eines Systems $\Omega$ in sich selbst und außerdem ein bestimmtes Element $\omega$ aus $\Omega$ gegeben, so gibt es eine und nur eine Abbildung $\psi$ der Zahlenreihe $N$, welche den Bedingungen

I. $\psi(N) \subseteq \Omega$,

II. $\psi(1) = \omega$,

III. $\psi(n’) = \theta \psi(n)$ genügt, wo $n$ jede Zahl bedeutet.

See [Dedekind-CW], Volume 1, p. 371.

126. Theorem. If $\theta$ is a (similar or non-similar) mapping from a system $\Omega$ into itself and if there is an element $\omega$ of $\Omega$, then there exists one and only one mapping $\psi$ from the number line $N$ which fulfills the conditions

I. $\psi(N) \subseteq \Omega$,

II. $\psi(1) = \omega$,

III. $\psi(n’) = \theta \psi(n)$ where $n$ is any number.

(Translation by the author.)

Similar means bijective. The number line $N$ is of course the set $\mathbb{N}_0$ of the natural numbers. The expression $\theta \psi(n)$ means $\theta(\psi(n))$. Furthermore, we have $n’ = n^+ = n + 1$.

## Isomorphisms of Peano Sets

Definition. Let $A$ and $B$ be two Peano sets.

(a) A function $\alpha : A \rightarrow B$ is called an isomorphism from the Peano set $A$ onto the Peano set $B$ if the following conditions are fulfilled:

(i) The function $\alpha : A \rightarrow B$ is bijective. Let $\beta := \alpha^{-1} : B \rightarrow A$.

(ii) We have $\alpha(0_A) = 0_B$.

(iii) We have $\alpha(x^+) = \alpha(x)^+$ for all elements $x$ of the set $A$.

(iv) We have $\beta(0_B) = 0_A$.

(v) We have $\beta(y^+) = \beta(y)^+$ for all elements $y$ of the set $B$.

(b) If $A$ and $B$ are two Peano sets such that there exists an isomorphism $\alpha: A \rightarrow B$ from the Peano set $A$ onto the Peano set $B$, then the Peano sets $A$ and $B$ are called isomorphic. In this case we write $A \cong B$.

French / German. Isomorphic Peano sets = Ensembles de Peano isomorphes = Isomorphe Peano-Mengen.

Proposition. Let $A$ and $B$ be two Peano sets.

(a) Let $\alpha : A \rightarrow B$ be an isomorphism from the Peano set $A$ onto the Peano set $B$, and let $\beta := \alpha^{-1} : B \rightarrow A$. Then the function $\beta : B \rightarrow A$ is an isomorphism from the Peano set $B$ onto the Peano set $A$.

(b) Let $\alpha : A \rightarrow B$ be a bijective function from the Peano set $A$ onto the Peano set $B$ such that

$$\alpha(0_A) = 0_B \mbox{ and } \alpha(x^+) = \alpha(x)^+ \mbox{ for all } x \in A.$$

Then the function $\alpha : A \rightarrow B$ is an isomorphism from the Peano set $A$ onto the Peano set $B$.

Proposition. Let $A$, $B$ and $C$ be three Peano sets, and let $\alpha : A \rightarrow B$ and $\beta : B \rightarrow C$ be two isomorphisms from the Peano set $A$ onto the Peano set $B$ and from the Peano set $B$ onto the Peano set $C$, respectively.

Then the composite $\gamma := \beta \circ \alpha : A \rightarrow C$ is an isomorphism from the Peano set $A$ onto the Peano set $C$.

Theorem. Let $A$ be a Peano set, and let $\omega$ be the minimal successor set. Then the Peano sets $A$ and $\omega$ are isomorphic.

Theorem. Let $A$ and $B$ be two Peano sets.

(a) The Peano sets $A$ and $B$ are isomorphic.

(b) There exists exactly one isomorphism $\alpha : A \rightarrow B$ from the Peano set $A$ onto the Peano set $B$.

## Notes and References

A list of textbooks about set theory is contained in Unit [Literature about Set Theory].

[Dedekind-1888] Dedekind, Richard (1888). Was sind und was sollen die Zahlen? Braunschweig: Vieweg.

[Dedekind-CW] — (1932). Gesammelte mathematische Werke.
Ed. by Robert Fricke, Emmy Noether, and Öystein Ore.
Braunschweig: Vieweg.
There are three volumes: Volume 1: (1930), Volume 2: (1931), Volume 3: (1932).

[Heath-1908b] Heath, Thomas, ed. (1908). Euclid: The Thirteen Books of the Elements.
Vol. 2. Cambridge: Cambridge University Press.
The second volume contains books III to IX.

[Pascal-1665] Pascal, Blaise (1665). Traité du Triangle Arithmétique avec quelques autres petits Traités sur la même Matière.
Paris: Guillaume Desprez.

[Peano-1889] Peano, Giuseppe (1889). Arithmetices Prinicipia – Nova Methodo Exposita.
Romae and Florentiae: Augustae Taurinorum.
This book is published under the name Ioseph Peano.

[von-Neumann-1922] von Neumann, John (1922). “Zur Einführung der transfiniten Zahlen”.
In: Acta litterarum ac scientiarum Regiae Universitatis Hungaricae Francisco-Josephinae: Sectio scientiarum mathematicarum 1, pp. 199–208.

[Zermelo-1908b] Zermelo, Ernst (1908). “Untersuchungen über die Grundlagen der Mengenlehre I”.
In: Mathematische Annalen 65, pp. 261–281.

[Zermelo-1967] — (1967). “Investigations in the Foundations of Set Theory I”.
In: From Frege to Gödel. A Source Book in Mathematical Logic, 1879 – 1931.
Ed. by van Heijenoort, pp. 199–215.