集合论基础：数学的基石与应用探索

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"Notes on Set Theory, 2nd (Undergraduate Texts in Mathematics)" 是一本介绍集合论的大学数学教材，作者旨在平衡集合论的基础概念、重要成果与深度开放问题的探讨，以及它作为数学基础的角色。集合论是现代数学的核心理论之一，其基本思想和优雅的公理体系为其他数学理论提供了基础。虽然将所有数学对象简单地视为集合并不完全准确，但在标准数学实践中，用集合论来精确定义概念是非常普遍的做法。因此，集合论可以被视为数学的官方语言，而数学又扮演着科学的语言角色。本书作者在编写过程中，考虑到了集合论的两个主要方面：一是作为独立的数学理论，拥有自身的结构和挑战；二是作为数学基础，为其他理论提供逻辑框架。作者试图在入门级的教材中，既展示集合论的内在美和深度，也解释它如何应用于更广泛的数学领域。书中的内容可能涵盖了以下知识点： 1. **集合的基本概念**：如元素、集合的定义、集合的并、交、差等基本运算，以及幂集的概念。 2. **集合的性质**：讨论集合的性质，如有限集、无限集、空集、基数（势）的概念，以及良序原理等。 3. **Zermelo-Fraenkel集合论**：包括集合论的基本公理，如扩展原则、选择公理等，以及这些公理如何支撑数学的其他分支。 4. **类和集合的区别**：在标准数学实践中，有些对象太大不能是集合，比如包含所有集合的类，这展示了集合论的局限性。 5. **集合的构造**：例如通过迭代、映射或幂集构造新的集合，以及康托尔的对角线法和无穷级别的构造。 6. **集合论与逻辑的关系**：集合论如何为形式逻辑提供基础，以及如何通过逻辑推理来证明集合论的定理。 7. **应用示例**：集合论在数论、拓扑学、实分析、抽象代数等领域的应用，例如实数的构造和实数轴的定义。 8. **基础数学对象的集合论定义**：例如自然数、整数、有理数和实数的构造，以及函数、关系等概念的集合论表述。 9. **开放问题**：书中可能提及了一些集合论中尚未解决的重要问题，如连续统假设或者选择公理的独立性问题。 10. **数学基础的探讨**：讨论集合论作为数学基础的哲学意义，以及它与其他基础系统（如类型理论）的比较。这本书适合数学专业的本科生阅读，它将帮助学生理解集合论的基本概念，并引导他们进入更深层次的数学探索。通过阅读此书，学生不仅会学习到集合论本身，还会了解到数学思维和表达方式的深度和广度。

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Chapter 1. Introduction 5

x ∈ U =⇒x ∈ V and x ∈ V =⇒x ∈ U. For example, if f : X → Y and

A, B ⊆ X , then

f[A ∪ B]=f[A] ∪ f[B].

To prove this, we show ﬁrst that

x ∈ f[A ∪ B] =⇒x ∈ f[A] ∪ f[B];

this holds because if x ∈ f[A ∪ B], then there is some y ∈ A ∪ B such that

x = f(y); and if y ∈ A,thenx = f(y) ∈ f

[A] ⊆ f[A] ∪ f[B], while if

y ∈ B,thenx = f(y) ∈ f[B] ⊆ f[A] ∪ f[B]. Next we show the converse

implication, that

x ∈ f[A] ∪ f[B] =⇒x ∈ f[A ∪ B];

this holds because if x ∈ f[A], then x = f(y) for some y ∈ A ⊆ A ∪B,and

so x ∈ f[A ∪B], while if x ∈ f[B], then x = f(y

) for some y ∈ B ⊆ A ∪B,

and so, again, x ∈ f[A ∪ B].

Problems for Chapter 1

x1.1. For any three sets A, B, C ,

A ∪ (B ∩C )=(A ∪ B) ∩ (A ∪ C ),

A ∩ (B ∪C )=(A ∩ B) ∪ (A ∩ C ),

A \ (A ∩ B)=A \ B.

x1.2. (De Morgan’s laws) For any three sets A, B, C ,

C \(A ∪ B)=(C \ A) ∩ (C \ B),

C \(A ∩ B)=(C \ A) ∪ (C \ B).

x1.3. (De Morgan’s laws for sequences) For any set C and any sequence of

sets {A

}

= A

,...,

C \ (





(C \A

C \ (





(C \A

x1.4. For every injection f : X  Y , and all A, B ⊆ X ,

f[A ∩ B]=f[A] ∩ f[B],

f[A \ B]=f[A] \ f[B].

Show also that these identities do not always hold if f is not an injection.

x1.5. For every f : X → Y , and all A, B ⊆ Y ,

−1

[A ∪B]=f

−1

[A] ∪ f

−1

[B],

−1

[A ∩B]=f

−1

[A] ∩ f

−1

[B],

−1

[A \ B]=f

−1

[A] \ f

−1

[B].

Chapter 2. Equinumerosity 9

The deﬁnition works (because f is an injection, and so f

−1

(i) is uniquely

determined in the second case), and it deﬁnes a surjection  : N →→ A, because

∈ A and for every x ∈ A, x = (f(x)).

(3) =⇒ (1). If A is ﬁnite then (1) is automatically true, so assume that

A is inﬁnite but it has an enumeration  : N →→ A. We must ﬁnd another

enumeration f : N →→ A which is without repetitions,sothatitisinfacta

bijection of N with A, and hence A =

N. The proof is suggested by Figure

2.1: we simply delete the repetitions from the given enumeration  of A.To

get a precise deﬁnition of f by recursion, notice that because A is not ﬁnite,

for every ﬁnite sequence a

,... ,a

of members of A there exists some m such

that (m) /∈{a

,... ,a

}.Set

f(0) = (0),

= the least m such that (m) /∈{f(0),... ,f(n)},

f(n +1)=(m

It is obvious that f is an injection, so it is enough to verify that every x ∈ A

is a value of f, i.e., that for every n ∈ N, (n) ∈ f[N]. This is immediate for

0, since (0) = f(0). If x = (n + 1) for some n and x ∈{f(0),... ,f(n)},

then x = f(i) for some i ≤ n;andifx/∈{f(0),... ,f(n)}, then m

= n +1

and f(n +1)=(m

)=x by the deﬁnition. 

2.8. Exercise. If A is countable and there exists an injection f : B  A, then

B is also countable; in particular, every subset of a countable set is countable.

2.9. Exercise. If A is countable and there exists a surjection f : A →→ B, then

B is also countable.

The next, simple theorem is one of the most basic results of set theory.

2.10. Theorem (Cantor). For each sequence A

,... of countable sets, the

union

A =



∞

n=0

= A

∪ A

∪ ...

is also a countable set.

In particular, the union A ∪ B of two countable sets is countable.

Proof. The second claim follows by applying the ﬁrst to the sequence

A, B, B, ···

For the ﬁrst, it is enough (why?) to consider the special case where none

of the A

is empty, in which case we can ﬁnd for each A

an enumeration



: N →→ A

.Ifwelet

= 

(i)

to simplify the notation, then for each n

= {a

,...},

and we can construct from these enumerations a table of elements which lists

all the members of the union A. This is pictured in Figure 2.2,andthearrows

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