ZFC Axioms

• 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

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.

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

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