# 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](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](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.

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

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