ZFC｜Zermelo–Fraenkel set theory with the Axiom of Choice

Zermelo–Fraenkel set theory with the Axiom of Choice (ZFC) is the standard foundational system for most of modern mathematics. It is a formal axiomatic system in first-order logic that defines the behavior of sets, which are the basic objects of mathematics.

🧱 Definition of ZFC

ZFC consists of nine axioms (technically eight Zermelo–Fraenkel axioms plus the Axiom of Choice):

Zermelo–Fraenkel Axioms (ZF)

1. Axiom of Extensionality
   Sets with the same elements are equal.
2. Axiom of Empty Set
   There exists a set with no elements (∅ exists).
3. Axiom of Pairing
   For any two sets, there exists a set that contains exactly those two sets.
4. Axiom of Union
   For any set of sets, there exists a set that contains all elements of those sets.
5. Axiom of Power Set
   For any set, there exists a set of all its subsets.
6. Axiom of Infinity
   There exists an infinite set (usually used to define natural numbers).
7. Axiom of Replacement
   The image of a set under a definable function is also a set.
8. Axiom of Regularity (Foundation)
   Every non-empty set contains a member that is disjoint from itself (no infinite descending membership chains).
9. Axiom Schema of Separation (Subset Axiom)
   From any set, one can form a subset by selecting elements that satisfy a property (restricted comprehension).

+ Axiom of Choice (AC)

For any set of non-empty sets, there exists a choice function that selects one element from each set.

📜 Historical Background

1908 – Ernst Zermelo

• Zermelo first formulated an axiomatic set theory to resolve Russell’s paradox and to provide a foundation for mathematics.
• He included a form of the Axiom of Choice in his system, which was controversial at the time.

1922–1930 – Abraham Fraenkel & Thoralf Skolem

• Fraenkel and Skolem extended Zermelo’s axioms to make them more robust, especially to enable the construction of the natural numbers and to allow for transfinite recursion.
• The addition of the Axiom of Replacement was critical in enabling the development of larger set hierarchies.

Mid-20th Century – Standardization

• ZFC emerged as the consensus foundation for mathematics, especially after Gödel showed (1938) that AC is consistent with ZF, and Paul Cohen (1963) showed it is independent from ZF using forcing.

🔁 Existence and Consistency

• ZFC is widely accepted but, due to Gödel’s Incompleteness Theorems, we know that:
  • If ZFC is consistent, it cannot prove its own consistency.
  • There are statements independent of ZFC, such as the Continuum Hypothesis (CH).