NBGC
Axioms of NBGC (the Usual Axioms for Algebraists)
Throughout, $A$ and $B$ are classes and $C$ and $D$ are sets.
- (外延公理: Axiom of extensionality). CLASSES are equal if and only if they have the same MEMBERS. For all $c ∈ A$, if and only if $c ∈ B$ then $A=B$.
- In ZF, (CLASSES, MEMBERS) are replaced by (sets, elements).
- (正則公理: Axiom of Regularity). All non-empty CLASSES are disjoint from at least one of their MEMBERS.
- In ZF, (CLASSES, MEMBERS) are replaced by (sets, elements).
- (分離公理模式: Axiom (Schema) of Class Comprehension/Separation/Specification, it is an axiom schema because there is one axiom for each $ϕ$) For any formula $ϕ$ which does not quantify over CLASS, there is a CLASS $A$ such that $x ∈ A$ if and only if $ϕ(x)$ is true, i.e,
$$\begin{equation}
∀ z∀ w_{1}∀ w_{2}\ldots ∀ w_{n}∃ y∀ x[x∈ y ⇔ ((x ∈ z)∧ φ (x,w_{1},w_{2},\ldots,w_{n},z))],
\end{equation}$$
where $(z,w_1,w_2,\ldots, w_n,x)$ are free variables ($y$ is not free).
- Axiom of Empty Set is a corollary.
- In ZF, CLASSES is replaced by sets.
- (配對公理: Axiom of Pairing). For any two sets $C$ and $D$, there is a set with exactly $C$ and $D$ as elements.
- It is exactly the same as the Axiom of Pairing in ZF.
- (併集公理: Axiom of Union). you can take the union of any set of sets.
- It is exactly the same as the Axiom of Union in ZF.
- (替換公理模式: Axiom (Schema). of Replacement/Collection/Size, also 6 names) For any CLASS $A$, there is a set $c$ such that $c=\mathrm{Rep}(A)$ if and only if there is no total one-to-one function between $A$ and the CLASS of all sets. This is the one that creates that definition of classes as things too big to be sets.
- This theorem has a weakened form, which is a the same as that in ZF. Under this stronger form, one can prove that the sub-class of a set is also a set.
- (無窮公理: Axiom of Infinity). There exists a set $N$ where:
- The empty set $∅$ is an element of $N$
- and for each element $x ∈ N$, $x ∪ \{x\} ∈ N$;
- It is exactly the same as the Axiom of Infinity in ZF.
- (冪集公理: Axiom of Powerset). For any set $x$, there is a set which contains all of the subsets of $x$.
- It is exactly the same as the Axiom of Powerset in ZF.
When non-set-theoretic mathematicians talk about NBG, AC for Sets or even Axiom of Global Choice is possibly included.