Axioms of ZFC

The eight axioms of Zermelo-Fraenkel Set Theory, ZF for short, (and the Axiom of Choice, AC)¹ are essential elements of the foundations of modern mathematics. They will be frequently referred to on this post, so I have compiled all of them on this page. They will not be numbered² for consistency. Sometimes, slightly less formal notation will be used. Note that all objects are sets in ZFC Set Theory.

NOTE: This is meant to be a point of reference for anything else that I write on this website. Refer elsewhere if you wish to learn Axiomatic Set Theory.^[3][4]

Russel’s Paradox, which showed that Frege’s Set Theory (now commonly referred to as Naive Set Theory) was flawed, motivated the development of ZFC Set Theory. The Axiom of Comprehension, which defines sets as collections of sets based on a certain property in the form of a logical formula, leads to contradictions. Sets in that form, like \(\set{x: \phi(x)}\) for some formula \(\phi\), are considered classes in ZFC (defined objects that are not necessarily sets). Those that cannot be proven to be sets by ZFC axioms are considered proper classes. The Axiom Schema of Specification and Replacement allows us to show that any subset of a set is a well-defined set, while many of the other axioms allow us to construct bigger sets from existing sets.

The Axiom Schema of Specification and Replacement are not singular axoims, but rather infinite collections of axioms for which each formula corresponds to an axiom within the schema.

Axiom of Extensionality

The Axiom of Extensionality defines what it means for two sets to be equal, it defines the \(=\) operator. It states

\[\forall x \forall y [\forall z (z \in x \iff z \in y) \implies x = y].\]

Two sets are equal if they have exactly the same elements.

Axiom of Pairing

The Axiom of Pairing allows us to construct a set from any pair of sets that contains only those two sets. It states

\[\forall x \forall y \exists z \forall w (w \in z \iff w = x \vee w = y).\]

in other words

Given any two sets, there is a set containing precisely these two sets

Under the Axiom Schema of Specification, we do not need to guarantee that \(x\) and \(y\) are the only sets in \(z\); therefore, in ZFC, it could also be written as

\[\forall x \forall y \exists z (x \in z \wedge y \in z).\]

The Pairing Axiom also allows us to compound sets by pairing them with themselves. It implies that for any set \(a\) there is a set \(\set{a}\).

Axiom Schema of Specification

Also known as The Subset Axiom, Axiom Schema of Separation, and Axiom Schema of Restricted Comprehension.

Omitted for now.

Axiom of Union

The Axiom of Union allows us to “glue together” the contents of sets within a collection⁵. It states

\[\forall \mcal{F} \, \exists A \, \forall Y \, \forall x \left[(x \in Y \wedge Y \in \mcal{F}) \implies x \in A \right].\]

And then we can recover the union of \(\mcal{F}\), which is a subset of \(A\).

\[\bigcup\mcal{F} = \set{ x \in A : \exists Y (x \in Y \wedge Y \in \mcal{F})} \]

There is a set \(\bigcup\mcal{F}\) that contains the members of each set in \(\mcal{F}\).

By the Axiom of Pairing, we can construct a set \(\set{A, B}\) for any sets \(A\) and \(B\), and then define \[A\cup B = \bigcup \set{A, B}.\] This binary operation version of union is associative and commutative (which follows from the associativity and commutativity of the logical disjunction).

Lastly, note that \(\bigcup\varnothing\) is nothing but the empty set itself.

Axiom of Infinity

First, let’s define the successor function

\[S(x) = x \cup \set{x}.\]

Then, the Axiom of Infinity could be stated as follows.

\[\exists X [\varnothing \in X \wedge \forall u (u \in X \implies S(u) \in X)].\]

Sets satisfying the conditions of \(X\) are called inductive sets. Thus, the Infinity Axiom guarantees the existence of a set in the following form.

\[\omega = \set{\varnothing, \set{\varnothing}, \set{\varnothing, \set{\varnothing}}, \ldots}.\]

Axiom of Powerset

Let’s define the subset relation

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

Then the Powerset Axiom could be stated as⁶

\[\forall X \, \exists Y \, \forall u (u \subseteq X \implies u \in Y).\]

And then we can recover the powerset of \(X\), which is a subset of \(Y\). \[\mcal{P}(X) = \set{u \in Y : u \subseteq X}\]

Given a set \(X\), there is a set whose members are all subsets of \(X\).

Axiom Schema of Replacement

Omitted for now.

Axiom of Regularity

Also known as The Axiom of Foundation

\[\forall x (x \neq \varnothing \implies (\exists y \in x)(y \cap x = \varnothing))\]

Every nonempty set in ZFC set theory contains at least one element who is disjoint from itself.

It is critical to note that every ZFC object is a set in order to properly understand this axiom. One elementary consequence of this axiom is that a set \(A\) cannot be a member of itself, i.e. \(A\notin A\), since \(A \cap \set{A} = \varnothing\).

1. Zermelo Fraenkel Set Theory (ZF) together with the Axiom of Choice (AC) is sometimes referred to as Zermelo Fraenkel Choice or ZFC. ↩

2. I sometimes number the axioms due to their frequent use and refer to them as such. However, the axioms will be referred to by name here for consistency sake. These axioms are listed in no particular order. Furthermore, the Axiom of Regularity is introduced last, and a similar order to Jech’s Set Theory (see citation below). ↩

3. Jech, Thomas J. Set Theory: The Third Millennium Edition, Revised and Expanded. 3rd ed. Springer Monographs in Mathematics. Berlin, Germany: Springer, 2003. ↩

4. Cunningham, Daniel W. Set Theory: A First Course. New York City, NY: Cambridge University Press, 2016. ↩

5. The term “collection” is synonymous with “set”. ↩

6. The same applies as the Pairing Axiom, where a weaker statement–a conditional–is stated instead of a biconditional; n.b. that some sources would state this axiom using a biconditional. ↩