The Mathematical Universe

Introduction

Please note that the text on this website gives a summary of the Unit The Mathematical Universe. 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:

ZFC-0:	Basic Axiom
ZFC-1:	Axiom of Extension
ZFC-2:	Axiom of Existence
ZFC-3:	Axiom of Specification
ZFC-4:	Axiom of Foundation
ZFC-5:	Axiom of Pairing
ZFC-6:	Axiom of Unions
ZFC-7:	Axiom of Powers
ZFC-8:	Axiom of Substitution
ZFC-9:	Axiom of Choice
ZFC-10:	Axiom of Infinity

The present unit explains the axioms ZFC-0 to ZFC-4. Axioms ZFC-5 to ZFC-7 are explained in the Unit [Unions and Intersections of Sets]. Axioms ZFC-8 and ZFC-9 are explained in the Unit [Families]. Finally, Axiom ZFC-10 is explained in the Unit [Successor Sets].

The mathematical universe:

We will call the totality of mathematics a mathematical universe or just a universe. Note that we speak of a mathematical universe and not of the mathematical universe. There may and do exist different mathematical universes with similar properties. You may think of the definition of a group. All groups have the same (defining) properties, however there exist different groups. Our mathematical universes will all obey the same axioms, namely the axioms of Zermelo and Fraenkel listed above, but there do exist different universes fulfilling these axioms.

In principle, we should have to start every definition and every theorem by the words “Let ${\cal U}$ be a mathematical universe (fulfilling the axioms of Zermelo and Fraenkel), and …“. However, the mathematical tradition is to omit this expression.

The first main idea is to postulate that a mathematical universe consists of a collection of sets. In other words, every mathematical object is a set. This approach sounds contra-intuitive since we are used to distinguish between sets and their elements like the set $\Bbb{N}_0$ of the natural numbers and its elements $0, 1, 2, \ldots$.

Looking at the simple set $A := \big\{ \{x\}, \{y\} \big\}$ it becomes obvious that there is no strict distinction between sets and elements. The sets $\{x\}$ and $\{y\}$ are at the same time elements of the set $A$. So we make one step further and say that every element is also a set. For example, the natural numbers will be defined as specific sets, namely

$$
0 := \emptyset \mbox{ (empty set)}, 1 := \{0\}, \ldots, n+1 := \{0, 1, 2, \ldots, n\}, \ldots
$$

and so on. For more detailed information about natural numbers see the Unit [Natural Numbers].

The second main idea is to postulate that the sets of a universe have the property that for each two sets $A$ and $B$ of the universe we either have

$$
A \in B \mbox{ or } A \notin B.
$$

In other words, we assume that for each two sets $A$ and $B$ of a universe ${\cal U}$, exactly one of the two relations $A \in B$ or $A \notin B$ holds. However, it may be difficult to decide which of the two possibilities is correct for two given sets $A$ and $B$.

This is the main content of the basic axiom.

Subsets and the axiom of extension:

The basic axiom allows us to define a subset of a set in the usual way: A set $A$ is a subset of a set $B$ if every element of the set $A$ is also an element of the set $B$. In this case we write $A \subseteq B$.

Our next task is to postulate when two sets $A$ and $B$ are equal: The axiom of extension answers this question as follows:

Two sets $A$ and $B$ are equal if and only if we have

$$
A \subseteq B \mbox{ and } B \subseteq A.
$$

In this case we write $A = B$. Otherwise, we write $A \neq B$.

We want to exclude the case that a mathematical universe does not contain any sets. Therefore, we will require that each universe contains at least one set, namely the empty set. This is the content of the axiom of existence.

Sentences:

So far we know what a mathematical universe ${\cal U}$ is, namely a collection of sets with the property that for each two sets $A$ and $B$ of the universe ${\cal U}$ we either have

$$
A \in B \mbox{ or } A \notin B.
$$

The next question that we want to answer is: How do we formulate a mathematical sentence? Or, equivalently: What is the mathematical language? Given two sets $A$ and $B$ of a universe ${\cal U}$, we have introduced the following four possible relations between these two sets:

$$
A \in B, \: A \notin B, \: A = B \mbox{ and } A \neq B.
$$

These four expressions are called elementary sentences. They form the elementary part of the mathematical language. Mathematical sentences are then recursively defined, for example by $\varphi \land \psi$ ($\varphi$ and $\psi$) or by $\varphi \lor \psi$ ($\varphi$ or $\psi$) if $\varphi$ and $\psi$ are mathematical sentences. A complete definition of sentences will be given later.

The elementary sentences and the sentences already constitute the mathematical language. Every definition and every theorem can be expressed as a sentence. A definition like the definition of a subset introduced above is nothing else than an abbreviation for a specific sentence.

The axiom of specification:

We are used to define subsets of a given set by specifying some interesting properties. For example, the set $E$ of the even integers is defined as follows:

$$
E := \{ x \in \Bbb{Z} \mid x \mbox{ even} \} = \{ x \in \Bbb{Z} \mid \exists \: z \in \Bbb{Z} : x = 2z \}.
$$

Note that the expression $\exists \: z \in \Bbb{Z} : x = 2z$ is a sentence. The axiom of specification guarantees the existence of sets like the set $E$: Given a set $A$ and a mathematical property (sentence) $\varphi = \varphi(x)$ depending on a variable $x$, the axiom of specification guarantees the existence of the set

$$
\{ x \in A \mid \varphi(x) \}.
$$

In the above example, $\varphi(x)$ is the sentence $\exists \: z \in \Bbb{Z} : x = 2z$.

However, this set may be empty. An important conclusion of the axiom of specification is Theorem [#nst1-th-no-universe] stating that for each set $A$ of a universe ${\cal U}$, there exists a set $B$ of the universe ${\cal U}$ which is no element of the set $A$. In particular, there does not exist a set $A$ containing all sets of the universe.

The axiom of foundation:

The axiom of foundation excludes some strange constellations like

$$
A \in A \mbox{ or } (A \in B \mbox{ and } B \in A).
$$

The Mathematical Universe

The Basic Axiom and the Definition of a Universe:

Examples of Universes:

Historical Note:

At the end of the 19th century and in the beginning of the 20th century was a growing interest into set theory. This interest had two main sources:

Richard Dedekind (1831 – 1916) investigated systematically the nature of the natural numbers (see [Dedekind-1888]) and of the real numbers (see [Dedekind-1872]). Georg Cantor (1845 – 1918) started a systematic research on sets and the cardinalities of sets. He started this research with the observation that the cardinality of the set of the real numbers is strictly greater than the cardinality of the set of the natural numbers (see [Cantor-1874]).

On the other hand David Hilbert (1862 – 1943) emphasized the importance of the axiomatization of mathematics and published in 1899 his Grundlagen der Geometrie where he provided a modern axiomatics of a 3-dimensional affine space.

In 1908 Ernst Zermelo published a fundamental paper containing an axiomatics of mathematics based on set theory where he refers explicitly to the former work of Cantor and Dedekind. This paper contains already most of the axioms of Zermelo and Fraenkel listed in the introduction.

The full set of axioms as described in the introduction is only contained in the paper [Zermelo-1930] of Zermelo.

The basic axiom reads in [Zermelo-1908b] as follows:

1. Die Mengenlehre hat zu tun mit einem Bereich ${\cal B}$ von Objekten, die wir einfach als Dinge bezeichnen wollen, unter denen die Mengen einen Teil bilden. … Von einem Dinge $a$ sagen wir, es existiere, wenn es dem Bereich ${\cal B}$ angehört; …

2. Zwischen den Dingen des Bereiches ${\cal B}$ bestehen gewisse Grundbeziehungen der Form $a \: \varepsilon \: b$. Gilt für zwei Dinge $a$, $b$ die Beziehung $a \: \varepsilon \: b$, so sagen wir “$a$ sei Element der Menge $b$” oder $b$ enthalte $a$ als Element oder besitze das Element $a$. Ein Ding $b$, welches ein anderes $a$ als Element enthält, kann immer als eine Menge bezeichnet werden, aber auch nur dann – mit einer einzigen Ausnahme (Axiom II).

(See [Zermelo-1908b], p. 262.)

Set theory is concerned with a domain ${\cal B}$ of individuals, which we shall call simply objects and among which are the sets. … We say of an object $a$ that it exists if it belongs to the domain ${\cal B}$; …

2. Certain fundamental relations of the form $a \: \varepsilon \: b$ obtain between the objects of the domain ${\cal B}$. If for two objects $a$ and $b$ the relation $a \: \varepsilon \: b$ holds, we say $a$ is an element of the set $b$, $b$ contains $a$ as an element, or $b$ possesses the element $a$. An object $b$ may be called a set if and – with a single exception (Axiom II) – only if it contains another object, $a$, as an element.

(See [Zermelo-1908b-en], p. 201.)

It is remarkable that Zermelo does not define what a set is, but that he restricts himself to describe the properties of a set. This approach follows the geometric axiomatics of Hilbert in [Hilbert-1899] where points, lines and planes are not defined, but their properties are described in the corresponding axioms.

Note that Zermelo makes a difference between a set and an object insofar as a set has to contain at least one element (with one exception which is of course the empty set). So the objects of Zermelo are the sets of the basic axiom.

An earlier definition of a set is due to Richard Dedekind who gives the following definition:

Es kommt sehr häufig vor, dass verschiedene Dinge $a, b, c, \ldots$ … im Geiste zusammengestellt werden, und man sagt dann, dass sie ein System $S$ bilden; man nennt die Dinge $a, b, c, \ldots$ die Elemente des Systems $S$ …; umgekehrt besteht $S$ aus diesen Elementen. Ein solches System $S$ … ist als Gegenstand unseres Denkens ebenfalls ein Ding …

(See [Dedekind-1888] or [Dedekind-CW], vol. 3, p. 344 – 345.)

It happens quite often that different things $a, b, c \ldots$ … are summarized in one’s mind’s eye; in this case one says that they form a system $S$; the things $a, b, c, \ldots$ are called the elements of the system $S$ …; conversely, the system $S$ consists of these elements. Such a system $S$ … as an object of our thinking is again a thing …

(Translation by the author.)

Dedekind distinguishes between elements (things) and sets (systems) and points out that every set (system) is also an element (thing). In our terminology, the sets are the things (and of course the systems) of Dedekind.

The symbol $\in$ has been introduced by Giuseppe Peano (1858 – 1932) as the Greek letter $\varepsilon$:

Signo $K$ significatur classis, sive entium aggregatio. Signum $\varepsilon$ significat est. Ita $a \: \varepsilon \: b$ legitur $a$ est quoddam $b$; $a \: \varepsilon \: K$ significat $a$ est quaedam classis; $a \: \varepsilon \: P$ significat $a$ est quaedam propositio.

(See [Peano-1899], p. x.)

The sign $K$ means class, or aggregate of objects. The sign $\varepsilon$ means is. Thus $a \: \varepsilon \: b$ is read $a$ is a $b$; $a \: \varepsilon \: K$ means $a$ is a class; $a \: \varepsilon \: P$ means $a$ is a proposition.

(See [Peano-1899-en], p. 89.)

Subsets and the Axiom of Extension

Definition of a Subset:

The Axiom of Extension:

Elementary Properties of Sets:

When is $A \not\subseteq B$?

The Empty Set and the Axiom of Existence:

Historical Note

An early definition of subsets is given by Richard Dedekind:

Ein System $A$ heißt Teil eines Systems $S$, wenn jedes Element von $A$ auch Element von $S$ ist.

(See [Dedekind-1888] or [Dedekind-CW], p.345.)

A aystem $A$ is called part of a system $S$ if every element of $A$ is also an element of $S$.

(Translation by the author.)

Dedekind uses the word system for sets. The symbol $\subset$ has been introduced by Ernst Schröder (1841 – 1902):

Das andere Zeichen $\subset$ lese man: untergeordnet, auch, wenn man will: subordinirt. … Die Kopula ist wird bald die eine, bald die andere der beiden Beziehungen ausdrücken, die wir mittels der Zeichen $\subset$ und $=$ dargestellt haben. Zu Ihrer Darstellung wird sich darum ein aus den beiden letzten zusammengesetztes Zeichen € … empfehlen. Ausführlichst wird dieses Zeichen als untergeordnet oder gleich zu lesen sein.

(See [Schroeder-1890], p. 129 and p. 132.)

The other sign $\subset$ is read inferior, or, if you want subordinate. … The copula is will express the one or the other of the two relations that we represent by the signs $\subset$ and $=$. To its representation the sign € built from these two signs commends itself. … In detail this sign has to be read as subordinate or equal.

(Translation by the author.)

Schröder introduced these signs in the context of logic.

The sign $\subseteq$ instead of € has been used later on by Felix Hausdorff (1868 – 1942) (see [Hausdorff-1914], p. 3).

The axiom of extension has already been introduced by Dedekind:

Das System $S$ ist daher dasselbe wie das System $T$, in Zeichen $S = T$, wenn jedes Element von $S$ auch Element von $T$ und jedes Element von $T$ auch Element von $S$ ist.

Footnote: Auf welche Weise diese Bestimmtheit zustande kommt, und ob wir einen Weg kennen, um hierüber zu entscheiden, ist für alle Folgende gänzlich gleichgültig;

(See [Dedekind-1888] or [Dedekind-CW], p. 345.)

The system $S$ is therefore the same as the system $T$, notation $S = T$, if every element of $S$ is also an element of $T$ and if every element of $T$ is also an element of $S$.

Footnote: It is completely indifferent for the following how this relation is achieved or whether we know a way how to decide about this relation.

(Translation by the author.)

Dedekind wants to express in the footnote that the way how an element is constructed does not matter for the decision whether it is element of a set. For example, we have

$$ \{2\} = \{ \sqrt{4} \} = \{ \lim_{n \rightarrow \infty} 2 + \frac{1}{n} \}.$$

We can also speak of the set

$$ A := \{(a, b, c) \in \mathbb{N} \times \mathbb{N} \times \mathbb{N} \mid a^n + b^n = c^n \mbox{ for some } n \geq 3 \}. $$

Today, we know that $A = \emptyset$. Before the proof of Andrew Wiles and others we did not know a way how to decide about the relation $x \in A$.

This is also the reason why this axiom is called the axiom of extension in contrary to an axiom of intension.

The first almost complete list of axioms is contained in [Zermelo-1908b]: The axiom of extension is called the Axiom der Bestimmtheit:

Axiom I. Ist jedes Element einer Menge $M$ gleichzeitig Element von $N$ und umgekehrt, ist also gleichzeitig $M$ € $N$ und $N$ € $M$, so ist immer $M = N$. Oder kürzer: jede Menge ist durch ihre Elemente bestimmt.

(See [Zermelo-1908b], p. 263.)

Axiom I. (Axiom of extensionality.) If every element of a set $M$ is also an element of $N$ and vice versa, if, therefore, both $M$ € $N$ and $N$ € $M$, then always $M = N$; Or more briefly: Every set is determined by its elements.

(See [Zermelo-1908b-en], p. 201.)

The axiom of existence is also contained in [Zermelo-1908b]: It is part of the so-called Axiom der Elementarmengen:

Axiom II. Es gibt eine (uneigentliche) Menge, die Nullmenge $0$, welche gar keine Elemente enthält. …

(See [Zermelo-1908b], p. 263.)

Axiom II. (Axiom of elementary sets.) There exists a (fictitious) set, the null set, $0$, that contains no element at all. …

(See [Zermelo-1908b-en], p. 202.)

For a long time the empty set has been abbreviated by the number $0$. This fits very well to the definition of the natural numbers: One defines $0 := \emptyset$. The symbol $\emptyset$ for the empty set has been introduced by Bourbaki. See for example [Bourbaki-2006], page E.II,6.

Sentences

In order to formulate mathematical definitions and theorems, we need a mathematical language. We will choose a set-theoretical language. Its main building blocks are sentences. We start with the so-called elementary sentences:

Definition of Elementary Sentences:

Elementary Sentence	Negation
The set $A$ is an element of the set $B$. Equivalently, $A \in B$.	The set $A$ is no element of the set $B$. Equivalently, $A \notin B$.
The set $A$ is no element of the set $B$. Equivalently, $A \notin B$.	The set $A$ is an element of the set $B$. Equivalently, $A \in B$.
The sets $A$ and $B$ are equal. Equivalently, $A = B$.	The sets $A$ and $B$ are distinct. Equivalently, $A \neq B$.
The sets $A$ and $B$ are distinct. Equivalently, $A \neq B$.	The sets $A$ and $B$ are equal. Equivalently, $A = B$.

(b) The negation of an elementary sentence is again an elementary sentence. For each elementary sentence $\varphi$, there is exactly one negation $\neg \varphi$.

Definition of Sentences:

Negations of Sentences:

Sentence	Negation
$\varphi \land \psi$.	$\neg \varphi \lor \neg \psi$.
$\varphi \lor \psi$.	$\neg \varphi \land \neg \psi$.
$\varphi \rightarrow \psi$.	$\varphi \land (\neg \psi)$.
$\varphi \leftrightarrow \psi$.	$\varphi \leftrightarrow \neg \psi$.
$\exists \: X \: \varphi$.	$\forall \: X \: \neg \varphi$.
$\forall \: X \: \varphi$.	$\exists \: X \: \neg \varphi$.

Historical Note

Zermelo noticed in [Zermelo-1908b] that a mathematical expression needs to fulfill some formal requirements. He introduced for that purpose the notion definite:

4. Eine Frage oder Aussage ${\cal F}$, über deren Gültigkeit oder Ungültigkeit die Grundbeziehungen des Bereiches vermöge der Axiome und der allgemeingültigen logischen Gesetze ohne Willkür entscheiden, heißt definit.

(See [Zermelo-1908b], p. 263.)

4. A question or assertion ${\cal F}$ is said to be definite if the fundamental relations of the domain, by means of the axioms and the universally valid laws of logic, determine without arbitrariness whether it holds or not.

(See [Zermelo-1908b-en], p. 201.)

The current definition of sentences stems from Thoralf Skolem (1887 – 1963):

… – erwähne ich hier die 5 Grundoperationen der mathematischen Logik. wobei ich die Bezeichnungen E. Schröders [Schroeder-1890] benutze:

(1x) Die Konjunktion, …
(1+) Die Disjunktion, …
(2) Die Negation, …
(3x) In jedem Falle Gültigkeit, …
(3+) In mindestens einem Falle Gültigkeit, …

…

Unter einer definiten Aussage kann man jetzt einen endlichen Ausdruck verstehen, der von Elementaraussagen der Form $a \in b$ oder $a = b$ mit Hilfe der 5 genannten Operationen aufgebaut ist.

(See [Skolem-1923].)

I mention the five basic operations of mathematical logic here, using Schröder’s notation [Schroeder-1890]:

(1x) Conjunction, …
(1+) Disjunction, …
(2) Negation, …
(3x) Universal quantification, …
(3+) Existential quantification, …

…

By a definite proposition we now mean a finite expression constructed from elementary propositions of the form $a \in b$ or $a = b$ by means of the five operations mentioned.

(See [Skolem-1923-en], p. 292 – 293.)

The Axiom of Specification

The Axiom of Specification:

Elementary Applications of the Axiom of Specification:

The Non-existence of the Set of all Sets:

Historical Note:

Georg Cantor defined a set as follows:

Unter einer Menge verstehen wir jede Zusammenfassung $M$ von bestimmten wohlunterschiedenen mathematischen Objekten $m$ unserer Anschauung oder unseres Denkens (welche die Elemente von $M$ genannt werden) zu einem Ganzen. In Zeichen drücken wir dies so aus:

$$ M = \{ m \}. $$

(See [Cantor-1895], p. 481.)

By a set we understand any collection $M$, gathered into a whole, of certain well-distinguished objects $m$ of our perception or our thought (which are called the elements of $M$). We express this by the following notation:

$$ M = \{ m \}. $$

(Translation by the author based on [Zermelo-1908b-en].)

Comparing this definition with Remark [#nst1-rem-comprehension-no-set] we can say that Cantor’s\ definition of a set is very close to the definition of a set $A$ as

$$ A := \{ x \mid \varphi(x) \}. $$

It turned out that this definition of sets results in some contradictions which have also been called paradoxes or antimonies. One of the most famous antimonies is the paradox of Bertrand Russell (1872 – 1970) published in the Principles of Mathematics:

Thus we must conclude again that the classes which as ones are not members of themselves as many do not form a class.

(See [Russell-1903], No. 102, p. 152.)

A class is a set in our terminology. This paradox is the antimony that we explained in Remark [#nst1-rem-comprehension-no-set].

By the way, in a footnote Zermelo states that he has found this antimony independently from B. Russell before 1903 and that he had communicated it to David Hilbert:

Indessen hatte ich selbst diese Antinomie unabhängig von Russell gefunden und sie schon vor 1903 u.a. Herrn Prof. Hilbert mitgeteilt.

(See [Zermelo-1908a], p. 118 – 119.)

I had , however, discovered this antinomy myself, independently of Russell, and had communicated it prior to 1903 to Professor Hilbert among others.

(See [Zermelo-1908a-en], p. 191.)

The axiom of specification has been introduced by Zermelo as the Axiom der Aussonderung:

Axiom III. Ist die Klassenaussage ${\cal F}(x)$ definit für alle Elemente einer Menge $M$, so besitzt $M$ immer eine Untermenge $M_{{\cal F}}$, welche alle diejenigen Elemente $x$ von $M$, für welche ${\cal F}(x)$ wahr ist, und nur solche als Elemente enthält.

(See [Zermelo-1908b], p. 263.)

Axiom III. (Axiom of separation.) Whenever the propositional function ${\cal F}(x)$ is definite for all elements of a set $M$, $M$ possesses a subset $M_{{\cal F}}$ containing as elements precisely those elements $x$ of $M$ for which ${\cal F}(x)$ is true.

(See [Zermelo-1908b-en], p. 202.)

The condition that the propositional function ${\cal F}(x)$ is definite means that ${\cal F}(x)$ is a sentence.

The antinomy of Remark [#nst1-rem-comprehension-no-set] becomes the positive existence theorem (Theorem [#nst1-th-no-universe]) stating that for each set $A$ there exists an element (set) $b$ such that $b \notin A$. This conclusion is also already contained in the foundation paper of Zermelo:

10. Theorem. Jede Menge $M$ besitzt mindestens eine Untermenge $M_0$, welche nicht Element von $M$ ist.

(See [Zermelo-1908b], p. 264.)

10. Theorem. Every set $M$ possesses at least one subset $M_0$ that is not an element of $M$.

(See [Zermelo-1908b-en], p. 203.)

The Axiom of Foundation

See also Example [#nst1-ex-specification]. Intersections of sets are discussed in more detail in Unit [Unions and Intersecions of Sets].

Condition (ii) of the following proposition explains why these elements are called element minimal:

The for us most interesting conclusions of the axiom of foundation can be found in Unit [Unions and Intersecions of Sets] after the introduction of the axiom of pairing.

Historical Note:

Ernst Zermelo has updated his list of axioms of 1908 ([Zermelo-1908b-en]) in 1930 and has included the axiom of foundation:

F) Axiom der Fundierung: Jede (rückschreitende) Kette von Elementen, in welcher jedes Glied Element des vorangehenden ist, bricht mit endlichem Index ab bei einem Urelement. Oder, was gleichbedeutend ist: Jeder Teilbereich $T$ enthält wenigstens ein Element $t_0$, das kein Element $t$ in $T$ hat.

(See [Zermelo-1930], p. 31.)

F) Axiom of Foundation: Each (regressive) chain of elements, in which every link is an element of the previous element, terminates with finite index at an ur-element. Or, equivalently: Each set $T$ contains at least one element $t_0$ that has no element $t$ in $T$.

(Translation by the author.)

Zermelo points out that his axiom of foundation relies on earlier work of Abraham Fraenkel and John von Neumann (1903 – 1957):

Axiom der Beschränktheit. Außer den durch die Axiome II bis VII (bzw. VIII) geforderten Mengen existieren keine weiteren Mengen.

(See [Fraenkel-1928], p. 355.)

Axiom of boundedness. There do not exist any sets except the sets required by Axioms II to VII (resp. VIII).

(Translation by the author.)

There are the following correspondences: Axiom II – ZFC-5 (axiom of pairing); Axiom III – ZFC-6 (axiom of unions); Axiom IV – ZFC-7 (axiom of powers); Axiom V – ZFC-3 (axiom of specification); Axiom VI – ZFC-9 (axiom of choice); Axiom VII – ZFC-10 (axiom of infinity).

The axiom of boundedness is already mentioned in [Fraenkel-1922], p. 234, but it is not really explained there.

Es gibt kein II. Ding $a$ mit der folgenden Eigenschaft: Es ist für jede endliche Ordnungszahl (d.h. ganze Zahl) $n$: $[a, n+1] \in [a, n]$.

(See [von-Neumann-1925], p. 239.)

There is no II. thing $a$ with the following property: For each finite ordinal number (that is, natural number) $n$: $[a, n+1] \in [a, n]$.

(Translation by the author.)

A II. thing $a$ stands for a function $a$. $[a, n]$ stands for the expression $a(n)$.

Notes and References

The first book I read about set theory was Naive Set Theory by Paul R. Halmos (1916 – 2006) [Halmos-1960}. I still find that it is one of the best introductions into set theory. The units of the mathematical garden about set theory are very much in its spirit.

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

References:

This book is a reprint of the edition of 1970. The four chapters of [Bourbaki 2006] have been first published separately in the years 1954 (Description de la mathématique formelle) 1939 and 1954 (Théorie des ensembles), 1956 (Ensembles ordonnés, cardinaux, nombres entiers) and 1956 (Structures).

Translation of [Bourbaki 2006] into English.

The first edition appeared in 1919.

This article is a translation of [Peano-1899] into English.

Note that this article is often cited as [Skolem 1922].

The first edition appeared in 1967.

This article is a translation of [Zermelo-1908a] into English.

This article is a translation of [Zermelo-1908b] into English.

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