Axioms of ZFC - Core Definitions

The Zermelo-Fraenkel axioms with the Axiom of Choice (ZFC) form the standard foundation for modern mathematics. These axioms provide a rigorous framework for reasoning about sets and their properties, avoiding the paradoxes that plagued naive set theory.

DefinitionSet

A set is a collection of distinct objects, called elements or members. We write $x \in A$ to denote that $x$ is an element of set $A$ , and $x \notin A$ to denote that $x$ is not an element of $A$ . Two sets are equal if and only if they have the same elements:

A = B \iff \forall x(x \in A \leftrightarrow x \in B)

This is known as the axiom of extensionality.

The language of set theory is remarkably simple, consisting of a single binary relation symbol $\in$ (membership) and the usual logical symbols. All mathematical objects in ZFC are sets, and all relationships are ultimately expressed in terms of membership.

DefinitionSubset

A set $A$ is a subset of a set $B$ , written $A \subseteq B$ , if every element of $A$ is also an element of $B$ :

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

If $A \subseteq B$ and $A \neq B$ , we say $A$ is a proper subset of $B$ and write $A \subset B$ or $A \subsetneq B$ .

The axioms of ZFC can be divided into several categories. The axiom of extensionality establishes when sets are equal. The axiom of empty set guarantees the existence of a set with no elements, denoted $\emptyset$ or $\{\}$ . The axiom of pairing states that for any sets $a$ and $b$ , there exists a set $\{a, b\}$ containing exactly $a$ and $b$ .

ExampleBasic Set Constructions

Starting from the empty set $\emptyset$ , we can construct:

The set $\{\emptyset\}$ containing the empty set (by pairing)
The set $\{\emptyset, \{\emptyset\}\}$ (by pairing again)
More complex sets through iterated applications

These constructions form the basis for defining natural numbers in set theory, where we identify:

0 = \emptyset, \quad 1 = \{\emptyset\}, \quad 2 = \{\emptyset, \{\emptyset\}\}, \ldots

The axiom schema of comprehension (or separation) is crucial for avoiding paradoxes. It states that for any set $A$ and any property $P(x)$ definable in the language of set theory, there exists a set containing exactly those elements of $A$ satisfying $P$ :

\exists B \forall x(x \in B \leftrightarrow x \in A \land P(x))

This restricted form prevents Russell's paradox by requiring that we always separate from an existing set rather than forming arbitrary collections.

Remark

The axiom schema of comprehension differs from Frege's unrestricted comprehension principle, which led to Russell's paradox. By requiring that we separate from an existing set $A$ , we cannot form the "set of all sets not containing themselves," which would lead to contradiction.

The axiom of union guarantees that for any set $A$ , there exists a set $\bigcup A$ containing exactly those elements that belong to at least one member of $A$ . The axiom of power set ensures that for any set $A$ , there exists a set $\mathcal{P}(A)$ consisting of all subsets of $A$ .

These foundational axioms, together with the axiom of infinity, replacement, regularity (foundation), and choice, form the complete ZFC system that underpins modern mathematics.