# NOTE: ZFC

ZFC(Zermelo-Fraenkel set theory with the axiom of choice) is an axiomatic system used to formally define set theory. More precisely, ZFC is a collection of approximately 9 axioms, define the core of mathematics through the usage of set theory. Primitive set theory is powerful, in fact, too powerful. The power of axiom of unrestricted comprehension makes several paradoxes, which is the main reason to create ZFC.

## 1. Axioms

1. Axiom of extensionality

$\forall u (u \in X \equiv u \in Y) \implies X = Y$

1. Axiom of pairing

$\forall a \forall b \exists Z \forall x (x \in Z \equiv (x = a \lor x = b))$

1. Axiom of comprehension

$\forall X \forall p \exists Y \forall u (u \in Y \equiv (u \in X \land \phi (u,p)))$

1. Axiom of union

$\forall X \exists Y \forall u (u \in Y \equiv \exists z (z \in X \land u \in z))$

1. Axiom of power set

$\forall X \exists Y \forall u (u \in Y \equiv u \subseteq X)$

1. Axiom of infinity

$\exists S (\emptyset \in S \land (\forall x \in S (x \cup \{x\} \in S)))$

infinity set 存在

1. Axiom of replacement

If $$F$$ is any function, for any set $$X$$ there exists a set $$Y = F(X) = \{F(x), x \in X\}$$

1. Axiom of regularity

$\forall S (S \ne \emptyset \implies (\exists x \in S : S \cap x = \emptyset))$

These 8 axioms make ZF. ZFC needs one more axiom called: Axiom of choice.

Axiom of choice

$\forall x \in a \exists A(x,y) \implies \exists y \forall x \in a A(x,y(x))$

Date: 2020-06-11 Thu 00:00