The Axiom of Empty Set explicitly states the existence of a unique set with no elements, denoted by the symbol ∅ or {}.

The Axiom of Regularity prevents the formation of circular or self-referential sets, ensuring that every non-empty set has a member that is not a subset of the set itself.

The Axiom of Pairing states that for any two sets A and B, there exists a set {A, B} containing exactly A and B as its members.

The Axiom of Choice asserts that for any collection of non-empty sets, there exists a function that selects a unique element from each set in the collection.

The Axiom of Power Set states that for any set A, there exists a set P(A) consisting of all subsets of A.

The Axiom of Infinity asserts the existence of an infinite set, typically denoted by ℕ or ω, which serves as the foundation for constructing the natural numbers.

The Axiom of Union states that for any set A and any collection of sets B ∈ A, there exists a set C whose elements are exactly the elements of all the sets in B.

The Axiom of Replacement asserts that if a function f is defined on a set A and takes values in a set B, then the image of A under f is a set.

The Axiom of Function states that for any two sets A and B, there exists a set F such that for every element x ∈ A, there exists a unique element y ∈ B such that (x, y) ∈ F.

The Axiom of Collection states that if a property P(x) is satisfied by all elements of a set A, then there exists a set B whose elements are exactly the elements x ∈ A that satisfy P(x).

The Axiom of Intersection states that for any two sets A and B, there exists a set C whose elements are exactly the elements that are common to both A and B.

The Axiom of Separation states that for any set A and any property P(x), there exists a set B whose elements are exactly the elements x ∈ A that satisfy P(x).

The Axiom of Cartesian Product states that for any two sets A and B, there exists a set C whose elements are exactly the ordered pairs (a, b) where a ∈ A and b ∈ B.

The Axiom of Replacement asserts that if a function f is defined on a set A and takes values in a set B, then the image of A under f is a set.

The Axiom of Well-Ordering states that every non-empty set A has a well-ordering, which is a linear ordering of A such that every non-empty subset of A has a least element.