> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/900-index/enums/enum-axioms/axioms-per-theory/axioms-in-zfc.md).

# ZFC Axioms

* `ZFC` - the Zermelo-Fraenkel set theory with the axiom of choice
* `ZF` - the Zermelo-Fraenkel set theory without the axiom of choice

ZFC axioms - ZF axioms + The axiom of choice (AC)

* Axiom of extensionality
* Axiom of empty set
* Axiom of pairing
* Axiom of union
* Axiom of infinity
* Axiom schema of replacement
* Axiom of power set
* Axiom of regularity
* Axiom schema of specification

Links

* [Axiom of extensionality](https://en.wikipedia.org/wiki/Axiom_of_extensionality)
* [Axiom of empty set](https://en.wikipedia.org/wiki/Axiom_of_empty_set)
* [Axiom of pairing](https://en.wikipedia.org/wiki/Axiom_of_pairing)
* [Axiom of union](https://en.wikipedia.org/wiki/Axiom_of_union)
* [Axiom of infinity](https://en.wikipedia.org/wiki/Axiom_of_infinity)
* [Axiom schema of replacement](https://en.wikipedia.org/wiki/Axiom_schema_of_replacement)
* [Axiom of power set](https://en.wikipedia.org/wiki/Axiom_of_power_set)
* [Axiom of regularity](https://en.wikipedia.org/wiki/Axiom_of_regularity)
* [Axiom schema of specification](https://en.wikipedia.org/wiki/Axiom_schema_of_specification)

## Set Theory (Part 2): ZFC Axioms

<https://www.youtube.com/watch?v=op3WZRUKk_k>

Intro to the axioms of set theory using the Zermelo-Fraenkel with the axiom of choice (ZFC) formal system.

Showing how the union, the intersection, the empty set and the relative complement are derived or defined under this axiomatic system.

## The axiom of extensionality

<https://en.wikipedia.org/wiki/Axiom_of_extensionality>

Subset notation:

$$\forall a . (a\in A \to a \in B) \iff A \subseteq B$$

If set A is subset of set B and B is subset of A, then A and B are equal sets i.e. they are the same set.

$$\forall A,\forall B,(\forall X,(X\in A\Rightarrow X\in B)\to A=B)$$

## The Axiom of pairing

For all sets X and Y, there is a set C which contains them as its elements. Since X and Y are sets, it means that, besides each being a member of C, each one is also a subset of C.

$$
X={a,b} \\
Y={c,d} \\
C={X,Y} = { {a,b}, {c,d} } \\
$$

$$\forall X\ \forall Y\ \exists C\ . (X \in C \land X \subseteq C) \lor (Y\in C \land Y \subseteq C)$$