ZFC

Overview

Set theory begins with two primitive notions of sets and membership. Other axioms are defined relative to these concepts. The Axiom of Choice is elaborated on separately.

In general, set theory prefers working with hereditary sets, also known as pure sets. These are sets whose elements are all hereditary sets; that is, all elements of the sets are themselves sets, as are all elements of the elements, and so on.

Sets are often denoted using roster notation in which members are specified explicitly in a comma-delimited list surrounded by curly braces. Alternatively, abstraction (or set-builder notation) defines sets using an entrance requirement. Examples of the set of prime numbers less than $10$:

• Roster notation: $\{2,3,5,7\}$
• Set-builder notation: $\{x\mid x<10\wedge x\text{ is prime}\}$

Extensionality

If two sets have exactly the same members, then they are equal:

Empty Set Axiom

There exists a set having no members:

Pairing Axiom

For any sets $u$ and $v$, there exists a set having as members just $u$ and $v$:

Union Axiom

Preliminary Form

For any sets $a$ and $b$, there exists a set whose members are those sets belonging either to $a$ or to $b$ (or both):

General Form

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

Power Set Axiom

For any set $a$, there is a set whose members are exactly the subsets of $a$:

Subset Axioms

The "subset axioms" refer to the axiom schema stating:

For any formula $\varphi (x)$ not containing the letter $B$,

$$\mathrm{\forall }A,\mathrm{\exists }B,\mathrm{\forall }x,(x\in B⟺x\in A\wedge \varphi (x)).$$

Infinity Axiom

There exists an inductive set:

Replacement Axioms

The "replacement axioms" refer to an axiom schema stating:

For any formula $\varphi (x,y)$ not containing the letter $B$,

$$\begin{array}{rl}\mathrm{\forall }A,& [\mathrm{\forall }x\in A,\mathrm{\forall }{y}_1,y_2,(\varphi (x,y_1)\wedge \varphi (x,y_2)⟹y_1=y_2)]⟹\\ & [\mathrm{\exists }B,\mathrm{\forall }y,(y\in B⟺\mathrm{\exists }x\in A,\varphi (x,y)]\end{array}$$

Notice the top-level antecedent states that formula $\varphi (x,y)$ is a function-class.

Regularity Axiom

The regularity axiom states that regularity holds. That is, every nonempty set $A$ has a member $m$ such that $m\cap A=\varnothing$.

This axiom is also known as the foundation axiom.