Zermelo-Fraenkel体系(1)

1. 前言

本文的预备文章为一般拓扑/数学基础——数理逻辑。

现在我们可以正式开始公理化集合论的介绍。前文提到，在演绎理论（deductive theory）$X$中，不能产生歧义（Ambiguity）和无限回归（Regress argument），也就是说

• 理论$X$建立在逻辑的基础上。我们将从一般拓扑/数学基础——数理逻辑中的primitive notions中挑选一部分作为理论$X$的primitive notions，使得$X$的primitive notions没有歧义。显然，为了做到这一点，我们需要把很多primitive notions去掉，例如“太阳”、“春天”、“我”等。另外，我们还需要添加一些primitive notions到$X$中。例如，在Euclidean几何中，“点”是primitive notions，“点”在Euclidean几何中的解释是没有长度、大小和体积的对象，点的intuitive meaning和axioms一起，构成了Euclidean几何中的“点”。
• 理论$X$尽量是自洽（consistent）的。称一个理论是自洽的，如果该理论中没有既是真又是假的陈述（矛盾），我们希望建立自洽的理论，但这是一件非常困难的事，我们即将介绍的ZF体系目前还没有被证明是自洽的。事实上，我们有Godel第二不完备定理：自洽理论$X$的自洽性不能在$X$的框架内被证明。
• 理论$X$尽量是完备（complete）的。称一个理论是完备的，如果该理论中没有既不能证明是真也不能证明是假的陈述。这同样非常困难，ZF体系不是完备的，例如，连续统假设在ZF体系（或者ZFC体系）中既不能证明是真也不能证明是假。事实上，我们有Godel第一不完备定理：自洽理论一定是不完备的。

我们看到，尽管ZF体系（ZFC体系）既不是完备的，也未被证明是自洽的，但对绝大多数数学家而言，它已经够用了，我们只能在ZF体系（ZFC体系）出现矛盾时，修改公理以排除矛盾，这件事在二百年前已经被做过一次了，Russell悖论产生后，原先的Naive集合论变成了公理化集合论（包括很多不同的体系，如ZF体系，ZFC体系，ZF+A体系等等，其中最常用的就是ZFC体系），从而排除了Russel悖论。

Hilbert: “We must know, we will know.” Godel: “We won’t know, we can’t know.”

为了避免许多名词的翻译错误，以下将使用英文书写。

2. Zermelo-Fraenkel set theory

The modern study of set theory was initiated by Georg Cantor and Richard Dedekind in the 1870s. In that times, mathematician thought set as a collection that satisfy some properties, i.e., a set is
{ x : ϕ ( x ) } \{x:\phi(x)\} { x:ϕ(x)}
where $\varphi (x)$ is a property whose variable is $x$. However, Russell found that if the property $\varphi (x)$ is $x\notin x$. Then the set { x : x ∉ x } \{x:x\notin x\} { x:x∈/​x} cannot be asked if it is a member of itself. These kind of paradoxed lead to a more rigorous form of set theory that was free of these paradoxes.
In 1908, Ernst Zermelo proposed the first axiomatic set theory, Zermelo set theory. In 1922, Abraham Fraenkel and Thoralf Skolem independently proposed operationalizing a “definite” property as one that could be formulated as a well-formed formula in a first-order logic whose atomic formulas were limited to set membership and identity. They also independently proposed replacing the axiom schema of specification with the axiom schema of replacement. Appending this schema, as well as the axiom of regularity (first proposed by John von Neumann), to Zermelo set theory yields the theory denoted by ZF. Adding to ZF either the axiom of choice (AC) or a statement that is equivalent to it yields ZFC.
The structure of ZFC in the frame of mathematical logic displays as following

Comparison	Naive set theory	ZFC
Primitive notions	sets restricted by any property, membership $\in$ and identity $=$ by the meaning of common language.	sets described by ten axioms, membership $\in$ and identity $=$ by the meaning of common language, universe of discourse is the universe of all sets, and other primitive notions in mathematical logic.
Propositions/Formulas	—	Basic proposiitions $x=y,x\in z$ and propositional form constituted by basic propositions, connectives and quantifiers, ofen denoted by $\varphi ,\psi ,\eta ,$ etc. .

Next, let us study the ten axioms in ZFC.

3. Extensionality axiom

Extensionality axiom. $\forall x,y[\forall z(z\in x\iff z\in y)\Rightarrow x=y].$

Interpretation: For any two sets $x$ and $y$, the proposition
$$\forall z(z\in x\iff z\in y)\Rightarrow x=y$$
is always true, so that there are three cases:

• The proposition ∀ z ( z ∈ x ⇔ z ∈ y ) \forall z(z\in x\Leftrightarrow z\in y) ∀z(z∈x⇔<