Following the discovery of considerable paradoxes in Cantor's naive set theory, a movement grew among mathematicians and logicians to find a more rigorous basis for Mathematics. In 1908 Ernest Zermelo proposed the first axiomatic system, and later Abraham Fraenkel formulated amendments for it. Together their axiomatic system is often refered to as Zermelo-Fraenkel, and is denoted by $ZF$.

In the beginning there was math. And math was without form, and structure; and uncetainty was upon the face the Congress. And the spirit of Cantor moved upon the discarded drafts.
And Zermelo said, Let there be the empty set, and there was an empty set, and it was unique.

This is not the start, but it's a start: Let there be $a$, and let $\varphi(a)$ be a statement about $a$, then if $\varphi(a)$ is true, $\varphi$ is called a property of $a$, or a formula. The collection of all $a$, such that the statement $\varphi(a)$ is true, is called a class and is denoted by $\{a : \varphi(a)\}$. If we have $A:=\{a : \varphi(a)\}$, we can denote that $\varphi(a_{0})$ is true by $a_{0} \in A$, or false by $a_{0} \notin A$.

Now let $\mathcal{M}$ be a collection of classes, then $\mathcal{M}$ is called a set theory model if the following axioms hold:

1. The Axiom of Empty Set: Exists $E \in \mathcal{M}$ such that for each $x \in \mathcal{M}$ holds that $x \notin E$. Such class is called empty.

• Notation: Let $A \in \mathcal{M}, B \in \mathcal{M}$, if for each $a \in \mathcal{M}$ that holds that $a\in A$ holds that $a \in B$, denote $A \subseteq B$, if additionally exists $b \in \mathcal{M}$ such that $b \in B$ but $b \notin A$, denote $A \subset B$.
• Corollary: Let $A \in \mathcal{M}, B \in \mathcal{M}, C \in \mathcal{M}$, then:
  1. For each empty $E \in \mathcal{M}$ holds that $E \subseteq A$.
  2. $A \subseteq A$.
  3. If $A \subseteq B$ and $B \subseteq C$, then $A \subseteq C$, and can also be denoted $A \subseteq B \subseteq C$.

2. The Axiom of Extensionality: Let $A \in \mathcal{M}, B \in \mathcal{M}$, then $A=B$ if and only if $A \subseteq B$ and $B \subseteq A$.

• Corollary (the empty set is unique): Let $E \in \mathcal{M}$ and $E' \in \mathcal{M}$ be empty, then $E = E'$, it is denoted by $\emptyset$ and it is called the empty set.

3. The Axiom of Pairing: Let $A \in \mathcal{M}, B \in \mathcal{M}$, then exists $C \in \mathcal{M}$ such that $A \in C$ and $B \in C$.

• Notation: Let $A$ be a class, and let $\varphi$ be a property, then: $\{a \in A : \varphi(a)\}:=\{a : a \in A \text{ and } \varphi(a)\}$.

4. The Axiom Schema of Separation: Let $A \in \mathcal{M}$ and let $\varphi$ be a property, then: $\{a \in A : \varphi(a)\} \in \mathcal{M}$.

• Notations: Let $a,b$ be, we denote:
  1. $\{a\}:=\{x:x=a\}$
  2. $\{a,b\}:=\{x:x=a \text{ or } a=b\}$
• Corollary: Let $A \in \mathcal{M}, B \in \mathcal{M}$ then:
  1. $\{A,B\} \in \mathcal{M}$
  2. $\{A\} \in \mathcal{M}$
  3. $\{A,B\} = \{B,A\}$
  4. $\{A,A\} = \{A\}$
• Notation: Let there $\mathcal{A} \in \mathcal{M}$, then:
  1. $\bigcup \mathcal{A} := \{a : \text{ exists } A \in \mathcal{A} \text{ s.t. } a \in A\}$, also denoted by $\bigcup\limits _{A \in \mathcal{A}} A$.
  2. $\bigcap \mathcal{A} := \{a : \text{ for all } A \in \mathcal{A} \text{ h.t. } a \in A\}$, also denoted by $\bigcap\limits _{A \in \mathcal{A}} A$.

5. The Axiom of Union: Let $A \in \mathcal{M}$, then $\bigcup A \in \mathcal{M}$.

• Notations: Let $A \in \mathcal{M}$, then:
  1. $A \cap B := \bigcap \{A,B\}$
  2. $A \cup B := \bigcup \{A,B\}$, and if additionally $A \cap B = \emptyset$, denote $A \uplus B$ and call $A$ and $B$ disjoint.
  3. $A \setminus B := \{a \in A : a \notin B\}$
  4. $A \,\triangle\, B := (A \setminus B) \cup (B \setminus A)$
• Corollary: Let $A \in \mathcal{M}, B \in \mathcal{M}$, then:
  1. $\bigcap A \in \mathcal{M}$
  2. $A \cup B \in \mathcal{M}$
  3. $A \cap B \in \mathcal{M}$
  4. $A \setminus B \in \mathcal{M}$
  5. $A \,\triangle\, B \in \mathcal{M}$
  6. $A \cup \emptyset = A$
  7. $A \cap \emptyset = \emptyset$
• Corollary: Let $A \in \mathcal{M}, B \in \mathcal{M}, C \in \mathcal{M}$, then:
  1. Commutativity:
     1. $A \cup B = B \cup A$
     2. $A \cap B = B \cap A$
     3. $A \,\triangle\, B = B \,\triangle\, A$
  2. Associativity:
     1. $(A \cup B) \cup C = A \cup (B \cup C)$, also denoted by $A \cup B \cup C$.
     2. $(A \cap B) \cap C = A \cap (B \cap C)$, also denoted by $A \cap B \cap C$.
     3. $(A \,\triangle\, B) \,\triangle\, C = A \,\triangle\, (B \,\triangle\, C)$, also denoted by $A \,\triangle\, B \,\triangle\, C$.
  3. Distributivity:
     1. $A \cap (B \cup C) = (A \cap B) \cup(A \cap C)$
     2. $A \cup (B \cap C) = (A \cup B) \cap(A \cup C)$
  4. De-Morgan:
     1. $C \setminus (A \cap B) = (C \setminus A) \cup (C \setminus B)$
     2. $C \setminus (A \cup B) = (C \setminus A) \cap (C \setminus B)$

6. The Axiom of the Power Set Let $A \in \mathcal{M}$, then exists $C \in \mathcal{M}$ such that for each $B \in \mathcal{M}$ holds that $B \in C$ if and only if $B \subseteq A$, it is denoted by $P(A)$ and is called the power set of $A$.

• Corollary: Let $A \in \mathcal{M}$, then:
  1. $\emptyset \in P(A)$
  2. $A \in P(A)$
  3. $\{\emptyset, A\} \subseteq P(A)$
  4. $\{\emptyset, \{A\}\} = P(\{A\})$

7. The Axiom of Regularity: Let $A \in \mathcal{M}$ not empty, then exists $B \in A$ such that $A \cap B = \emptyset$.

• Claim: Let $A \in \mathcal{M}, B \in \mathcal{M}$, then:
  1. $A \notin A$
  2. Either $A \notin B$ or $B \notin A$
• Proof:
  1. $A \in \mathcal{M}$ and therefore $\{A\} \in \mathcal{M}$, and from the axiom of regularity $A \cap \{A\} = \emptyset$, and $A \notin A$.
  2. $A \in \mathcal{M}, B \in \mathcal{M}$, and therefore $\{A,B\} \in \mathcal{M}$, and from the axiom of regularity holds that either $A \cap \{A ,B\} = \emptyset$ or $B \cap \{A,B\} = \emptyset$, and therefore either $A \notin B$ or $B \notin A$.

8. The Axiom of Infinity: Exists $I \in \mathcal{M}$ such that $\emptyset \in I$ and for each $x \in I$ holds that $x \cup \{x\} \in I$.

• I will discuss the implications of this axiom in the next post.

9. Axiom Schema of Replacement: Let $A \in \mathcal{M}$, then for each property $\varphi$ such that for each $a \in A$ exists and is unique $b \in \mathcal{M}$ such that $\varphi(a, b)$ holds true, holds that $\{b : a \in A \text{ and } \varphi(a,b)\} \in \mathcal{M}$.

• I will discuss the implications of this axiom in a post about functions.

Definition: Let $\mathcal{M}$ be a set theory model, and let $A \in \mathcal{M}$, then $A$ is called a set or a family. A class that is not a set is often called a proper class.

Definition: Let $A$ be a set and Let $B \subseteq A$, then $B$ is called a subset of $A$, and if $B \subset A$ then B is called a proper subset of $A$.