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
- Axiom of extensionality
\[ \forall u (u \in X \equiv u \in Y) \implies X = Y \]
兩集合所有元素都相同蘊含兩集合相等
- Axiom of pairing
\[ \forall a \forall b \exists Z \forall x (x \in Z \equiv (x = a \lor x = b)) \]
對任意兩元素，必然有集合正好含有這兩元素
- Axiom of comprehension
\[ \forall X \forall p \exists Y \forall u (u \in Y \equiv (u \in X \land \phi (u,p))) \]
對所有屬性 \(\phi\) 與集合 \(X\)，必然有集合包含 \(X\) 中所有滿足 \(\phi\) 之元素。換句話說子集合可通過規則由已知集合構造，例如 \(is-even\) 套上 \(Natural\) 得到 \(even\)
- Axiom of union
\[ \forall X \exists Y \forall u (u \in Y \equiv \exists z (z \in X \land u \in z)) \]
這條簡單來說就是對所有集合 \(X\) 有集合 \(Y\) 是由 \(X\) 的所有元素構成的聯集
- Axiom of power set
\[ \forall X \exists Y \forall u (u \in Y \equiv u \subseteq X) \]
簡單講就是集合 \(X\) 必有 power set(the set that all elements are subsets of \(X\))
- Axiom of infinity
\[ \exists S (\emptyset \in S \land (\forall x \in S (x \cup \{x\} \in S))) \]
infinity set 存在
- Axiom of replacement
If \(F\) is any function, for any set \(X\) there exists a set \(Y = F(X) = \{F(x), x \in X\}\)
對任何集合 \(A\)，函數 \(F: A \to B\) 生成集合 \(B\)
- Axiom of regularity
\[ \forall S (S \ne \emptyset \implies (\exists x \in S : S \cap x = \emptyset)) \]
對任何非空集合都有一個元素與自身不相交，這避免了集合指涉自己(羅素悖論)、以及對任何非空集合必然有更小的集合
These 8 axioms make ZF. ZFC needs one more axiom called: Axiom of choice.
Axiom of choice
\[ \forall x \in a \exists A(x,y) \implies \exists y \forall x \in a A(x,y(x)) \]
這表示非空集合會有一個選擇函數