ZFC axioms

ZFC axioms: ZFC is the first-order theory in the language with equality and the membership relation $\in$ whose axioms are the Zermelo–Fraenkel axioms together with the Axiom of Choice . A common presentation includes:

• Extensionality: sets with the same elements are equal.
• Empty set: an empty set exists.
• Pairing: for any $a,b$ there exists a set $\{a,b\}$.
• Union: for any set $A$ there exists $\bigcup A$ (a set containing exactly the elements of the members of $A$), aligning with union constructions.
• Power set: for any set $A$ there exists its power set $\mathcal P(A)$.
• Infinity: an inductive set exists, enabling the construction of the natural numbers .
• Separation schema: definable subsets of a set are sets.
• Replacement schema: images of sets under definable functions are sets.
• Foundation (regularity): every nonempty set has an $\in$-minimal element.

Within ZFC one can develop most standard mathematics, and ZFC proves statements equivalent in strength to choice such as Zorn's lemma and the well-ordering theorem .