拓扑第一课-ZhiMap思维导图

Magi

拓扑-集合论公理体系

拓扑第一课
进入思维导图模式
- Proposional logic（命题逻辑）
  a variable can take the value "T" or "F"
  - Unary operation
    - $\neg P$
    - $Id ~P$
    - $\top P( \equiv True)$
    - $\bot P( \equiv False)$
  - Binary operation
    - $p \wedge q$
    - $p \vee q$
    - $p \Leftrightarrow q$
    - $p \Rightarrow q$
- Predicate logic（谓词逻辑）
  a proposition-valued function of some variable
  - For all quantification $\forall x:P(x)$
  - Existence quantification $\exists x:P(x)$
  - Corollary : $\Big(\forall x:P(x) \Big) \Leftrightarrow \neg \Big ( \exists x:P(x) \Big)$
- Axioms of Zermelo-Fraenkel
  （Z-F集合论公理系统）
  - 1) Axiom on $\in$ relation
    - x∈y is a proposition iff x and y are both sets
    - Russell paradox（罗素悖论）
  - 2) Axiom on existence of an empty set（空集公理）
    - $\exists x:(\,\forall y:y\notin x)$
    - Therom : Empty set is unique
  - 3) Axiom on pairs（配对公理）
    - There exists a set Z that consist of the sets X and Y
    - Notation: {X,Y}
    - {X,Y}={Y,X}
      由集合之间的“相等”的定义可证
    - Def : {x]:={x,x}
  - 4) Axiom on union sets（并集公理）
    - There exists a set Y whose elements are precisely the
      elements of the elements of X
    - Notation: $Y= \cup X$
  - 5) Axiom of replacement（替换公理）
    - let R be a functional relation（函数关系）
    - m is a set $\Rightarrow$ $\{\,y~|~\exists x:R(x,y)\,\}$ is a set
    - Notation: the principle of restricted comprehension: $\{\,y \in m~|~P(y)\,\}$
  - 6) Axiom on existence of power set（幂集公理）
    - there exists a set denoted by $\rho(m)$ whose elements are
      precisely the subsets of m.
  - 7) Axiom of infinity（无限公理）
    - There exists a set thet contians the empty set ∅ and with every of its elements y, it also contians {y} as a element $\emptyse$
    - Corollary : $\mathbb N$ is a set
    - Remark : as a set $\mathbb R$ := $\rho(\mathbb N )$
  - 8) Axiom of fountation（基础公理、正则公理）
    - Every non-empty set x contains an element y that has non of
      its elements in common with x.
    - Corollary : there is no set that contains itself as an element
    - $X \in X$ is false for all set X
  - 9) Axiom of choice（选择公理）
    也叫策梅洛公理，对于任意两两不交的集合族，存在集合C，使对所给的族中的每个集合X，集合X与C的交恰好只含一个元素。
    - Every vector space has a basis
    - Zorn's lemma（佐恩引理）
      一个偏序集中，如果任何链都有上界，那么这个偏序集必然存在一个最大值
    - Lebesgue Measure
      Axiom of choice => non-measurable set
- Axiomatic systems（公理体系）
  a finite sequence of propositions
  - Proof
    a finite sequence of propsitions qi (1<=i<=m)，with qm<=>p so that for any 1<=i<=m either:
    - 1) $q_j$ is an axiom（公理）
    - 2) $q_j$ is a tautology（永真式）
    - 3） Modus ponens（分离规则）
      $\Big(\exists~ 1 \leq m,n <j:(q_m \wedge q_n \Rightarrow q_j)\Big)~is~true$
  - Consistent
    - Def : there are no contradictions
      若有冲突，则S和非S均属于公理体系，进而对于任意命题q都有（S∩非S=>q）成立，根据分离规则，任意命题q均成立，显然不合情理
    - Therom（ $G \overset{o}{¨} d e l$ ）
      - An axiomatic system,if it is powerful enough to encode the elementary arithmetic of natural numbers, is either inconsistant or contains a proposition can neither be proven nor disproven.
- The $\in$ relation（属于关系）
  - Relation ：a predicate of two variables
  - Set theroy is built on the postulation（假设）that there exists $\in$
  - Def
    - $(x\notin y)\Leftrightarrow \neg(x\in y)$
    - $(x \subseteq y)\Leftrightarrow \forall a:(a \in x \Rightarrow a \in y)$
      - 子集公理
    - $(x = y)\Leftrightarrow \big((x \subseteq y)\wedge(y \subseteq x)\big)$
      - 同一律（外延公理）

拓扑第一课

​Proposional logic（命题逻辑）

Unary operation

​ ¬P \neg P¬P

​ Id PId ~PId P

⊤P(≡True) \top P( \equiv True) ⊤P(≡True)

​ ⊥P(≡False) \bot P( \equiv False) ⊥P(≡False)

​Binary operation

​ p∧qp \wedge qp∧q

​ p∨qp \vee qp∨q

​ p⇔qp \Leftrightarrow qp⇔q

​ p⇒qp \Rightarrow qp⇒q

​Predicate logic（谓词逻辑）

​For all quantification ∀x:P(x) \forall x:P(x)∀x:P(x)

​Existence quantification ∃x:P(x) \exists x:P(x)∃x:P(x)

​Corollary : (∀x:P(x))⇔¬(∃x:P(x)) \Big(\forall x:P(x) \Big) \Leftrightarrow \neg \Big ( \exists x:P(x) \Big)(∀x:P(x))⇔¬(∃x:P(x))

​Axioms of Zermelo-Fraenkel（Z-F集合论公理系统）

​1) Axiom on ∈\in∈ relation

x∈y is a proposition iff x and y are both sets

​Russell paradox（罗素悖论）

​2) Axiom on existence of an empty set（空集公理）

​ ∃x:( ∀y:y∈/ x)\exists x:(\,\forall y:y\notin x)∃x:(∀y:y∈/​x)

​Therom : Empty set is unique

​3) Axiom on pairs（配对公理）

​There exists a set Z that consist of the sets X and Y

Notation: {X,Y}

​{X,Y}={Y,X}

​Def : {x]:={x,x}

​4) Axiom on union sets（并集公理）

​There exists a set Y whose elements are precisely the elements of the elements of X

​Notation: Y=∪XY= \cup XY=∪X

​5) Axiom of replacement（替换公理）

​let R be a functional relation（函数关系）

​m is a set ⇒\Rightarrow⇒ { y ∣ ∃x:R(x,y) }\{\,y~|~\exists x:R(x,y)\,\}{y ∣ ∃x:R(x,y)} is a set

​Notation: the principle of restricted comprehension: { y∈m ∣ P(y) }\{\,y \in m~|~P(y)\,\}{y∈m ∣ P(y)}

​6) Axiom on existence of power set（幂集公理）

​there exists a set denoted by ρ(m)\rho(m)ρ(m) whose elements areprecisely the subsets of m.

​7) Axiom of infinity（无限公理）

​There exists a set thet contians the empty set ∅ and with every of its elements y, it also contians {y} as a element∅\emptyse

​Corollary : N\mathbb NN is a set

​Remark : as a set R\mathbb RR := ρ(N)\rho(\mathbb N )ρ(N)

​8) Axiom of fountation（基础公理、正则公理）

​Every non-empty set x contains an element y that has non ofits elements in common with x.

​Corollary : there is no set that contains itself as an element

​ X∈XX \in XX∈X is false for all set X

​9) Axiom of choice（选择公理）

​Every vector space has a basis

​Zorn's lemma（佐恩引理）

Lebesgue Measure

​Axiomatic systems（公理体系）

​Proof

​1) qjq_jqj​ is an axiom（公理）

2) qjq_jqj​ is a tautology（永真式）

​3） Modus ponens（分离规则） (∃ 1≤m,n<j:(qm∧qn⇒qj)) is true\Big(\exists~ 1 \leq m,n <j:(q_m \wedge q_n \Rightarrow q_j)\Big)~is~true(∃ 1≤m,n<j:(qm​∧qn​⇒qj​)) is true

​Consistent

​Def : there are no contradictions

Therom（ Go¨del ）

​An axiomatic system,if it is powerful enough to encode the elementary arithmetic of natural numbers, is either inconsistant or contains a proposition can neither be proven nor disproven.

The ∈\in∈ ​relation（属于关系）

Relation ：​a predicate of two variables

​Set theroy is built on the postulation（假设）that there exists ∈

​Def

​ (x∈/ y)⇔¬(x∈y)(x\notin y)\Leftrightarrow \neg(x\in y)(x∈/​y)⇔¬(x∈y)

​ (x⊆y)⇔∀a:(a∈x⇒a∈y)(x \subseteq y)\Leftrightarrow \forall a:(a \in x \Rightarrow a \in y)(x⊆y)⇔∀a:(a∈x⇒a∈y)

​子集公理

​ (x=y)⇔((x⊆y)∧(y⊆x))(x = y)\Leftrightarrow \big((x \subseteq y)\wedge(y \subseteq x)\big)(x=y)⇔((x⊆y)∧(y⊆x))

​同一律（外延公理）

Proposional logic（命题逻辑）

$\neg P$

$Id ~P$

$\top P( \equiv True)$

$\bot P( \equiv False)$

Binary operation

$p \wedge q$

$p \vee q$

$p \Leftrightarrow q$

$p \Rightarrow q$

Predicate logic（谓词逻辑）

For all quantification $\forall x:P(x)$

Existence quantification $\exists x:P(x)$

Corollary : $\Big(\forall x:P(x) \Big) \Leftrightarrow \neg \Big ( \exists x:P(x) \Big)$

Axioms of Zermelo-Fraenkel
（Z-F集合论公理系统）

1) Axiom on $\in$ relation

Russell paradox（罗素悖论）

2) Axiom on existence of an empty set（空集公理）

$\exists x:(\,\forall y:y\notin x)$

Therom : Empty set is unique

3) Axiom on pairs（配对公理）

There exists a set Z that consist of the sets X and Y

{X,Y}={Y,X}

Def : {x]:={x,x}

4) Axiom on union sets（并集公理）

There exists a set Y whose elements are precisely the
elements of the elements of X

Notation: $Y= \cup X$

5) Axiom of replacement（替换公理）

let R be a functional relation（函数关系）

m is a set $\Rightarrow$ $\{\,y~|~\exists x:R(x,y)\,\}$ is a set

Notation: the principle of restricted comprehension: $\{\,y \in m~|~P(y)\,\}$

6) Axiom on existence of power set（幂集公理）

there exists a set denoted by $\rho(m)$ whose elements are
precisely the subsets of m.

7) Axiom of infinity（无限公理）

There exists a set thet contians the empty set ∅ and with every of its elements y, it also contians {y} as a element $\emptyse$

Corollary : $\mathbb N$ is a set

Remark : as a set $\mathbb R$ := $\rho(\mathbb N )$

8) Axiom of fountation（基础公理、正则公理）

Every non-empty set x contains an element y that has non of
its elements in common with x.

Corollary : there is no set that contains itself as an element

$X \in X$ is false for all set X

9) Axiom of choice（选择公理）

Every vector space has a basis

Zorn's lemma（佐恩引理）

Axiomatic systems（公理体系）

Proof

1) $q_j$ is an axiom（公理）

2) $q_j$ is a tautology（永真式）

3） Modus ponens（分离规则）
$\Big(\exists~ 1 \leq m,n <j:(q_m \wedge q_n \Rightarrow q_j)\Big)~is~true$

Consistent

Def : there are no contradictions

Therom（ $G \overset{o}{¨} d e l$ ）

An axiomatic system,if it is powerful enough to encode the elementary arithmetic of natural numbers, is either inconsistant or contains a proposition can neither be proven nor disproven.

The $\in$ relation（属于关系）

Relation ：a predicate of two variables

Set theroy is built on the postulation（假设）that there exists $\in$

Def

$(x\notin y)\Leftrightarrow \neg(x\in y)$

$(x \subseteq y)\Leftrightarrow \forall a:(a \in x \Rightarrow a \in y)$

子集公理

$(x = y)\Leftrightarrow \big((x \subseteq y)\wedge(y \subseteq x)\big)$

同一律（外延公理）