Well-ordering Theorem and Zorn’s Lemma

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數學導論

學數學

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前言

學學數學學數學 . 學數學

論 . 學 , .

(Logic), (Set) 數 (Function) . , 學

論. 論學數學 .

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. , .

, . , 論

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v

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Chapter 6

Axiom of Choice,

Well-ordering Theorem and Zorn’s Lemma

Axiom of choice 數學 , 導

, Well-ordering Theorem Zorn’s Lemma (

). axiom of choice , .

數學 axiom of choice, axiom 數學

論 , 前數學 axiom of choice

Well-ordering Theorem Zorn’s Lemma. 學數學

, . ,

( 學論 ),

.

6.1. Axiom of Choice

Axiom of choice S, “ ” S

. S ﬁnite set , S

,

. inﬁnite set, . S N , N

well-ordering principle, least element .

inﬁnite set, , “ ”

. , , !

ﬁnite set , “ ” .

數學 ﬁnite set , ,

.

95

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96 6. Axiom of Choice, Well-ordering Theorem and Zorn’s Lemma

, 前 .

, . , ,

. function . S

, S power set, P(S) . P(S)

A A , 數 function f

S A f (A)∈ A. function f , S

, choice function. , axiom of choice

.

Axiom of Choice: For any nonempty set S, there exists a choice function f :P(S)\ /0 → S such that for every nonempty subset A of S ( A∈ P(S) \ /0), we have f (A) ∈ A.

, A , choice function A

, . choice function f S , S,

f (A) 前 Section 5.2 “image of A under f ”. axiom

of choice choice function , 論 choice function.

axiom of choice , constructive.

前 , axiom of choice. Proposition 5.5.9 ,

axiom of choice. S₁, . . . , S_n, . . . countable set , i∈ N, |Si| ≤ |N|, Si→ N one-to-one function fi: Si→ N, . Proposition 5.5.9 axiom of choice

. , axiom of choice .

Proposition 6.1.1. S inﬁnite set, S subset countably inﬁnite.

Proof. f :P(S) \ /0 → S choice function. g :N → S, g(1) = f (S).

S2= S\{ f (S)}, S inﬁnite set, S2̸= /0, S2∈ P(S)\ /0, f (S2) , g(2) = f (S2). 數學 k≥ 2 S_k+1= S\{ f (S), f (S2), . . . , f (S_k)}.

S inﬁnite set, S_k+1∈ P(S) \ /0, g(k + 1) = f (S_k+1). ,

g :N → S one-to-one function, g image, g(N) S

countably inﬁnite subset.

6.2. Well-ordering Theorem

partial ordered set (X,≼), total ordered X

, a, b∈ X, a≼ b b≼ a, . poset (partial ordered

set) (X,≼), well-ordered X T least element,

t₀∈ T t₀≼ t, ∀t ∈ T. min(T ) T least element.

Well-ordering Theorem, X , order ≼,

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6.2. Well-ordering Theorem 97

Well-ordering Theorem Well-ordering Principle . Well-ordering Principle, order ≤ (N,≤) well-ordered set. Well-ordering

Theorem, X , order ≼ (X ,≼)

well-ordered set. 數 Q ≤, well-ordered (

{r ∈ Q | 0 < r < 1} 數). Q countable (Corollary

5.5.8), N Q , Q , Q

well-ordered. Well-ordering Theorem, countable set, N Well-ordering Principle . uncountable set, Well-ordering Theorem

. 前 , 數 R order ≼ (R,≼)

well-ordered.

Well-ordering Theorem Zermelo’s Theorem, Zermelo Axiom of Choice

. Well-ordering Theorem Axiom of Choice.

, . Well-ordering Theorem axiom ( ).

前 , Axiom of Choice ,

Well-ordering Theorem, , Axiom of Choice

. .

Theorem 6.2.1 (Well-ordering Theorem). Let X be a nonempty set. Then there exists an order relation ≼ on X such that (X,≼) is well-ordered.

Well-ordered Theorem order ≼, (X ,≼) well-ordered.

Axiom of Choice , 前 Axiom of Choice

constructive, Well-ordered Theorem (X ,≼) well-ordered set order ≼ .

Axiom of Choice Well-ordering Theorem.

, Well-ordering Theorem Axiom of Choice. nonempty

set S, Well-ordering Theorem, order ≼ (S,≼) well-ordered.

A∈ P(S) \ /0, f (A) = min(A). f :P(S) \ /0 → S, choice function , f (A)∈ A. Axiom of Choice.

Well-ordering Theorem , N Well-ordering principle

, mathematical induction transﬁnite induction.

Corollary 2.3.6, mathematical induction P(n) n∈ N ,

(i) P(1) ; (ii) i < k, P(i) , P(k) .

(i), (ii) P(n) n∈ N . transﬁnite induction (X ,≼)

well-ordered, (1) P(min(X)) ; (ii) α ≺ β, P(α) , P(β)

. P(x), x∈ X . :

Theorem 6.2.2 (Transﬁnite Induction). (X ,≼) well-ordered set x1= min(X ).

statement , x∈ X, P(x) .

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98 6. Axiom of Choice, Well-ordering Theorem and Zorn’s Lemma

(1) P(x₁) .

(2) β ∈ X. α ∈ X α ≺ β, P(α) , P(β) .

Proof. , x∈ X P(x) . S ={x ∈ X | P(x) }.

S̸= /0. (X ,≼) well-ordered, β ∈ S β = min(S). (1) β ̸= x0.

β = min(S), α ∈ X α ≺ β, α ̸∈ S, P(α) . (2)

P(β) . β ∈ S , S = /0, x∈ X, P(x) .

Well-ordering Theorem N .

學 . 學 well-ordered,

. 1, 2,

inﬁnite set N , countable

? , well-order

, . N

order ≼: a, b∈ N , a≼ b a≤ b; a 數 b 數,

a≼ b. (N,≼) well-ordered set, ,

2 ( 數) ( 數數).

6.3. Zorn’s Lemma

Zorn’s Lemma Axiom of Choice . 前 partial ordered

set , . 數學 ,

Lemma. , Zorn’s Lemma 前 , Axiom

. 論 Zorn’s Lemma Axiom of Choice ( Well-ordering Theorem)

, Lemma.

order . partial ordered set (S,≼) , T

S subset T ≼ (T,≼) total ordered set ( t,t^′∈ T

t≼ t^′ t^′≼ T), T (S,≼) chain. S nonempty subset S^′,

u∈ S S^′ upper bound, s^′∈ S^′ s^′≼ u. µ ∈ S S

maximal element, µ ∈ S S s µ ≼ s ( S s,

s≼µ µ ). Zorn’s Lemma.

Lemma 6.3.1 (Zorn’s Lemma). (S,≼) partial ordered set. S chain S upper bound, S maximal element.

Zorn’s Lemma , partial ordered set maximal

element, Zorn’s Lemma. , poset (S,≼) maximal

element, S chain, S chain upper

bound. chain S upper bound, poset (S,≼)

maximal element. , , chain upper bound S

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6.3. Zorn’s Lemma 99

Zorn’s Lemma 論 . Axiom of Choice

Well-ordering Theorem , constructive. ,

maximal element , maximal element . Zorn’s

Lemma 數學 . 數 , ring maximal

ideal; 數 , (inﬁnite dimensional) vector space basis,

Zorn’s Lemma. Zorn’s Lemma , 前

.

Proposition 6.3.2. S sets .

⊆ partial ordered set (S ,⊆). S chain

S , S M S S M⊂ S.

Proof. S . S chain

chain , S chain S

chain upper bound. Zorn’s Lemma (S ,⊆) maximal element M,

M S , S M “ ” (

M), .

, ,

學 Proposition 6.3.2 chain S ?

前 Zorn’s Lemma chain upper bound S .

upper bound , S .

Proposition 6.3.2 “chain” , S .

Zorn’s Lemma , .