The Mathematical Universe

The present unit is part of the following walks

Introduction

In the following you will find a short summary of this unit. For detailed information please see the full text or download the pdf document at the end of this page.


The present unit is the first unit of the walk The Axioms of Zermelo and Fraenkel.


We will explain the first axioms of Zermelo and Fraenkel, namely

The other axioms are explained in the remaining units of the walk The Axioms of Zermelo and Fraenkel.


In addition you will learn the meaning of the following terms:


The main result of this unit is



The Mathematical Universe

The basic axiom explains what the mathematical universe and what a set is:


Axiom. (ZFC-0: Basic Axiom) (a) A mathematical universe ${\cal U}$ consists of sets.

(b) There is the following relation between the sets of the universe ${\cal U}$: For any two sets $A$ and $B$ of the universe ${\cal U}$, either the set $A$ is an element of the set $B$ or the set $A$ is no element of the set $B$.

(c) If the set $A$ is an element of the set $B$, then we also say that the set $A$ is contained in the set $B$ or, equivalently that the element $A$ is contained in the set $B$. (Note that the expression $A$ is contained in the set $B$ means $A \in B$ and not $A \subseteq B$ ($A$ is a subset of $B$) as defined in the definition of a subset.)

(d) If the set $A$ is an element of the set $B$, then we write $A \in B$. If the the set $A$ is no element of the set $B$, then we write $A \notin B$.

French / German. Universe = Univers = Universum. Set = Ensemble = Menge. Element = Élément = Element



Subsets and the Axiom of Extension

The axiom of extension explains when two sets are equal:


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

(a) The set $A$ is called a subset of the set $B$ if every element of the set $A$ is also an element of the set $B$. If the set $A$ is a subset of the set $B$, we write $A \subseteq B$.

(b) If the set $A$ is a subset of the set $B$ and if the set $B$ contains an element $b$ not contained in the set $A$, then the set $A$ is called a proper subset of the set $B$. In this case we write $A \subset B$.

French / German. Subset = Sous-ensemble = Teilmenge; Proper subset = Sous-ensemble propre = Echte Teilmenge.


Axiom. (ZFC-1: Axiom of Extension) (a) Two sets $A$ and $B$ are equal if and only if the set $A$ is a subset of the set $B$ and if the set $B$ is a subset of the set $A$.

(b) If the sets $A$ and $B$ are equal, then we write $A = B$. If the sets $A$ and $B$ are not equal, then we write $A \neq B$. Hence, we have

$$
A = B \mbox{ if and only if } A \subseteq B \mbox{ and } B \subseteq A.
$$

French / German. Axiom of extension = Axiome d’extensionalité = Extensionalitätsaxiom.


Definition. A set which does not contain any element is called empty. The empty set is denoted by $\emptyset$.

French / German. Empty set = Ensemble vide = Leere Menge


Axiom. (ZFC-2: Axiom of Existence) There exists an empty set.

French / German. Axiom of existence = Axiome d’existence = Existenzaxiom.



Sentences

In order to formulate mathematical definitions and theorems, we need a mathematical language. We will choose a set-theoretical language whose main building blocks are sentences and elementary sentences:


Definition. (a) There are four elementary sentences:

(i) The set $A$ is an element of the set $B$, or, equivalently, $A \in B$.

(ii) The set $A$ is no element of the set $B$, or, equivalently, $A \notin B$.

(iii) The sets $A$ and $B$ are equal, or, equivalently, $A = B$.

(iv) The sets $A$ and $B$ are distinct, or, equivalently, $A \neq B$.

(b) The variables appearing in an elementary sentence $\varphi$ are called the variables of the elementary sentence $\varphi$. We also say that a variable $x$ is contained in the elementary sentence $\varphi$.

French / German. Elementary Sentence = Terme élémentaire = Elementare Aussage.


Definition. (a) A sentence is defined by the following recursive rules:

(i) Every elementary sentence is a sentence.

(ii) If $\varphi$ is a sentence, then the expression negation of $\varphi$ is a sentence abbreviated by $\neg \varphi$.

The sentence $\neg \varphi$ is defined to be true if and only if the sentence $\varphi$ is false.

(iii) If $\varphi$ and $\psi$ are two sentences, then the expression $\varphi$ and $\psi$ is a sentence abbreviated by $\varphi \land \psi$.

The sentence $\varphi \land \psi$ is defined to be true if and only if the sentences $\varphi$ and $\psi$ are true.

(iv) If $\varphi$ and $\psi$ are two sentences, then the expression $\varphi$ or $\psi$ is a sentence abbreviated by $\varphi \lor \psi$.

The sentence $\varphi \lor \psi$ is defined to be true if and only if at least one of the sentences $\varphi$ and $\psi$ is true.

(v) If $\varphi$ and $\psi$ are two sentences, then the expression If $\varphi$, then $\psi$ is a sentence abbreviated by $\varphi \rightarrow \psi$.

The sentence $\varphi \rightarrow \psi$ is defined to be true if and only if the sentence $\varphi$ is false or if the sentences $\varphi$ and $\psi$ are both true.

(vi) If $\varphi$ and $\psi$ are two sentences, then the expression $\varphi$ if and only if $\psi$ is a sentence abbreviated by $\varphi \leftrightarrow \psi$.

The sentence $\varphi \leftrightarrow \psi$ is defined to be true if and only if the the sentences $\varphi$ and $\psi$ are both true or both false.

(vii) If $\varphi$ is a sentence, then the expression There exists a set $X$ such that $\varphi$ is a sentence abbreviated by $\exists \: X \: \varphi$.

The sentence $\exists \: X \: \varphi$ is defined to be true if the sentence $\varphi$ is true for at least one set $X$.

(viii) If $\varphi$ is a sentence, then the expression For all sets $X$, (we have) $\varphi$ is a sentence abbreviated by $\forall \: X \: \varphi$.

The sentence $\forall \: X \: \varphi$ is defined to be true if the sentence $\varphi$ is true for all sets $X$.

(ix) Let $\varphi$ be a sentence, and let $x$ be a variable of the sentence $\varphi$. If the variable $x$ appears in an expression of the form for all $x$ ($\forall \: x$) or of the form there exists an element $x$ ($\exists \: x$), the variable $x$ is called a bounded variable of the sentence $\varphi$.

If $x$ and $y$ are two bounded variables of a sentence $\varphi$, then the variables $x$ and $y$ have to be different. This means that the names $x$ and $y$ of the variables have to be different. However, it is possible that $x = y$, that is, the different variables $x$ and $y$ may represent the same set $A$.

(b) The sentence $\exists \: x \: \big(( x \in X) \land \varphi \big)$ is abbreviated by $\exists \: x \in X \: \varphi$. Analogously, the sentence $\forall \: x \: \big(( x \in X) \land \varphi \big)$ is abbreviated by $\forall \: x \in X \: \varphi$.

(c) The variables appearing in a sentence $\varphi$ are called the variables of the sentence $\varphi$. We also say that a variable $x$ is contained in the sentence $\varphi$.

(d) Let $\varphi$ be a sentence, and let $x$ be a variable of the sentence $\varphi$. If the variable $x$ is not bounded, then it is called a free variable or, equivalently, a parameter of the sentence $\varphi$.

(e) Formulas without brackets are executed in the following order: $\neg$, $\land$, $\lor$, $\rightarrow$, $\leftrightarrow$.

French / German. Sentence = Terme = Aussage. Bounded variable = Variable liée = Gebundene Variable. Free variable = Variable libre = Freie Variable.



The Axiom of Specification

The axiom of specification is a powerful tool to guarantee the existence of subsets of a given set.


Axiom. (ZFC-3: Axiom of Specification) Let $A$ be a set, and let $\varphi = \varphi(x)$ be a sentence containing the free variable $x$ (and possibly more variables).

Then there exists a subset $B$ of the set $A$ consisting of all elements $x$ of the set $A$ such that the sentence $\varphi = \varphi(x)$ is true. The set $B$ is denoted by

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

French / German. Axiom of specification = Axiome de compréhension or Axiome de séparation = Aussonderungsaxiom.


Definition. Let $A$ be a set, let $\varphi$ be a sentence, and let $x$ be a free variable of the sentence $\varphi$.

We say that an element $a$ of the set $A$ fulfills the condition $\varphi = \varphi(x)$ if the sentence $\varphi(a)$ is true or, equivalently, if the element $a$ is contained in the set $\{ x \in A \mid \varphi(x) \}$.


Theorem. (a) Let $A$ be an arbitrary set. Then there exists a set $B$ which is no element of the set $A$, that is, $B \notin A$.

(b) There is no set of all sets, that is, there is no set $A$ such that we have $X \in A$ for all sets $X$.



Notes and References

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


Do you want to learn more? The next axioms of Zermelo and Fraenkel are explained in Unit Unions and Intersections of Sets.



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The Mathematical Universe

The pdf document is the full text including the proofs.

Current Version: 1.0.4 from October 2020.