📂Set Theory

Zermelo-Fraenkel Set Theory with the Axiom of Choice

Zermelo’s Axiomatic System

• [1] Axiom of Extensionality: $$\forall A \forall B ( \forall x ( x \in A \iff x \in B) )$$ It is said that two sets $A$, $B$ are equal if they have the same elements, and this is expressed as $A = B$.
• [2] Axiom of the Empty Set: $$\exists X \forall x \left( \lnot \left( x \in X \right) \right)$$ There exists a set $X$ that contains no elements, and this set $X$ is defined as the empty set.
• [3] Axiom of Pairing: $$\forall A \forall B \exists U ( a \in A \land b \in B )$$ For any two sets $A$, $B$, there exists a set $U$ that contains $A$ and $B$ as elements.
• [4] Axiom Schema of Specification: $$\forall X \exists A \forall a \left( a \in A \iff ( a \in X \land p(a)) \right)$$ For any set $X$, there exists a subset $A$ that consists of elements having property $p$.
• [5] Axiom of Union: $$\forall X \left( \exists U \left( \forall a \left( a \in x \land x \in X \implies a \in U \right) \right) \right)$$ For any set $X$, there exists a set $U$ that includes all elements of the elements of $X$.
• [6] Axiom of Power Set: $$\forall X \exists P \forall A ( A \subset X \implies A \in P)$$ For any set $X$, there exists a set $P$ that contains all the subsets of $X$.
• [7] Axiom of Infinity: $$\exists U \left( \emptyset \in U \land \forall X ( X \in U \implies S(X) \in U) \right)$$ There exists a set $U$ that contains the empty set $X$ and, if it contains $S(X)$, it also contains $S(X)$ as an element.

Zermelo’s axiomatic system was introduced to address the problems revealed in set theory by Russell’s paradox, comprising the seven axioms mentioned above. In contrast, Cantor’s set theory, which Cantor himself proclaimed, is also called Naive Set Theory. Zermelo’s axiomatic system, unlike Cantor’s set theory, includes axioms that clearly define many concepts defined in natural language through mathematical logic and solidify their existence. Concepts like the empty set, union, intersection, and power set were already treated as sufficiently natural, but in fact, those words alone were not enough.

Zermelo-Fraenkel Axiomatic System

• [8] Axiom of Regularity: $$\forall X \left( \exists x_{0} ( x_{0} \in X ) \implies \exists y ( y \in X \land \lnot \exists x ( x \in y \land x \in X )) \right)$$ Every set $X \ne \emptyset$ has an element that is disjoint from itself.
• [9] Axiom Schema of Replacement: $$\forall X \left( \forall x \in X \exists ! y \left( p(x,y) \right) \implies \exists Y \forall x \in X \exists y \in Y \left( p(x,y) \right) \right)$$ There exists a range for every function.

These additional two axioms form the Zermelo-Fraenkel axiomatic system, abbreviated as ZF Axiomatic System.

Zermelo-Fraenkel Axiomatic System with the Axiom of Choice

• [10] Axiom of Choice: $$\forall U \left( \emptyset \notin U \implies \exists f: U \to \bigcup_{X \in U \\ f(X) \in X } U \right)$$ For every set $U$ of non-empty sets, there exists a choice function $f$ that selects one element from each element of $U$.

Finally, the Zermelo-Fraenkel axiomatic system with the added Axiom of Choice is abbreviated as ZFC Axiomatic System. The Axiom of Choice seems obvious at first glance, but it is generally accepted because of the very useful results obtained when the Axiom of Choice is accepted. It is usually considered safe to refer to the axiomatic system of set theory in contemporary mainstream mathematics as the ZFC Axiomatic System. Whether the Axiom of Choice is actually true or false, its rejection does not cause any contradictions within ZF; however, such rejections are almost non-existent.