Set Theory

Preface:

I have been learning axiomatic set theory lately and would like to use this blog as a way to document the notes as I learn them. If you are not a student of mathematics who somehow stumbled upon this post for learning just the basic amount of set theory for learning other subjects in applied mathematics then you shouldn't fret. Although this document is not about naive set theory, we will develop and mention all the ideas of set theory you may need in as simple a way as possible. If you are a mathematics student, then you can use this post as some kind of lecture notes if you ever attended lectures.

Basic Definitions:

We will separate the notion of classes and sets. All objects are classes and those classes that can be contained in other classes will be referred to as elements or sets. We will consider the

\in

to be a binary, the membership relation along with the

=

the equality predicate. We are now ready to make our first definition that will use formal notation.

Definition:

A=B: Two classes $A$ and $B$ are said to be equal if every class that has $A$ as an element also has the class $B$ as an element and vice versa.

(A = B) \iff (\forall X)[A\in X \iff B\in X]

We will also make the following definition for subsets.

Definition:

$A\subseteq B$ : $A$ is said to be a subset of $B$ if every element of $A$ is also an element of $B$ .

A\subseteq B \iff \forall t[t\in A \implies t \in B]

ZFC Axioms:

So, we will begin by mentioning the axioms of Zermelo and Fraenkel as mentioned in common textbooks on set theory like Jech (2003), Enderton (1977).

Axioms:

Axiom of Extensionality: If $A$ and $B$ have the same elements, then they are equal.

\forall A \forall B [\forall x (x\in A \iff x \in B) \implies A = B]

or alternatively

\forall A \forall B [\forall x (x\in A \iff x\in B) \implies \forall X (A\in X \iff B \in X)]

Axiom of Empty Set: There exists a set having no elements.

\exists A (\forall x, x\notin A)

Axiom of Pairing: For any $A$ and $B$ , there exists a set that contains exactly $A$ and $B$ .

\forall A \forall B \exists C [\forall x (x \in C \iff x = A \lor x = B)]

Axiom of Power Set: For any set A, there exists a set whose elements are exactly the subsets of $A$ .

\forall A \exists B \forall x (x\subseteq A \iff x\in B)

Axiom Schema of Separation: Let $\varphi$ be any formula with all free variables among $x, c, t_{1}, \dots, t_{n}$ ( $B$ is not free in $\varphi$ ). Then,

\forall t_{1}\forall t_{2}\dots\forall t_{n}\forall c \exists B \forall x[x\in B \iff (x\in c) \land \varphi(x,c, t_{1}, \dots, t_{n})]

Axiom of Union: For any $A$ , there exists a set whose elements are exactly the members of members of $A$ .

\forall A \exists B \forall x [x\in B \iff (\exists a\in A) x\in a]

Axiom of Infinity: There exists an infinite set.
Axiom of Replacement:
Axiom of Regularity:
Axiom of Choice:

The axioms

A-1 \text{ to } A-9

are the

ZF

axioms which becomes

ZFC

with the addition of the axiom

A-10

(AC).

Set Theory

Preface:

Basic Definitions:

ZFC Axioms:

Bibliography: