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1-11 Set Theory (IV): Infinity
魏恒峰
2019 年 12 月 17 日
Hengfeng Wei ([email protected]) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 1 / 49
SetTheory
A Branchof Math-ematics
N,R
ℵ0
ω
Foundationof Math-ematics(+ Logic)
{}(a, b)
A × BR ⊆A × B
f :
A → B
Hengfeng Wei ([email protected]) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 2 / 49
Georg Cantor (1845 – 1918) David Hilbert (1862 – 1943)
Leopold Kronecker(1823 – 1891)
Henri Poincaré(1854 – 1912)
Ludwig Wittgenstein(1889 – 1951)
Hengfeng Wei ([email protected]) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 3 / 49
From his paradise that Cantor with us unfolded, we hold ourbreath in awe; knowing, we shall not be expelled.
— David Hilbert
“没有人能把我们从 Cantor 创造的乐园中驱逐出去”
Hengfeng Wei ([email protected]) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 4 / 49
“das wesen der mathematik liegt in ihrer freiheit”
“The essence of mathematics lies in its freedom”
Hengfeng Wei ([email protected]) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 5 / 49
Before Cantor
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公理: “整体大于部分”
Hengfeng Wei ([email protected]) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 7 / 49
Galileo Galilei (1564 – 1642) “关于两门新科学的对话” (1638)
“用我们有限的心智来讨论无限 · · · ”
Hengfeng Wei ([email protected]) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 8 / 49
S1 = {1, 2, 3, · · · , n, · · · }
S2 = {1, 4, 9, · · · , n2, · · · }
|S1| = |S2| S2 ⊂ S1
“部分等于全体”
说到底,“等于”、“大于” 和 “小于” 诸性质不能用于无限,而只能用于有限的数量。 — Galileo Galilei
无穷数是不可能的。 — Gottfried Wilhelm Leibniz
Hengfeng Wei ([email protected]) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 9 / 49
这些证明一开始就期望那些数要具有有穷数的一切性质,或者甚至于把有穷数的性质强加于无穷。
相反,这些无穷数,如果它们能够以任何形式被理解的话,倒是由于它们与有穷数的对应,它们必须具有完全新的数量特征。
这些性质完全依赖于事物的本性,· · · 而并非来自我们的主观任意性或我们的偏见。
— Georg Cantor (1885)
Hengfeng Wei ([email protected]) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 10 / 49
Definition (Dedekind-infinite & Dedekind-finite (Dedekind, 1888))A set A is Dedekind-infinite if there is a bijective function from A ontosome proper subset B of A.
A set is Dedekind-finite if it is not Dedekind-infinite.
This is a theorem in our theory of infinity.
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We have not defined “finite” and “infinite”!
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Comparing Sets
Function
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Hengfeng Wei ([email protected]) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 14 / 49
Definition (|A| = |B| (A ≈ B) (1878))A and B are equipotent if there exists a bijection from A to B.
A (two abstractions)
Abstract from elements: {1, 2, 3} vs. {a, b, c}
Abstract from order: {1, 2, 3, · · · } vs. {1, 3, 5, · · · , 2, 4, 6, · · · }
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Definition (|A| = |B| (A ≈ B) (1878))A and B are equipotent if there exists a bijection from A to B.
Q : Is “≈” an equivalence relation?
Theorem (The “Equivalence Concept” of Equipotent)For any sets A, B, C:(a) A ≈ B
(b) A ≈ B =⇒ B ≈ A
(c) A ≈ B ∧B ≈ C =⇒ A ≈ C
Hengfeng Wei ([email protected]) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 16 / 49
Definition (Finite)X is finite if
∃n ∈ N : |X| = n.
|X| = |{0, 1, · · · , n− 1}|
Theorem (UD Theorem 22.6)Let A be a finite set. There is a unique n ∈ N such thatA ≈ {0, 1, · · · , n− 1}.
Hengfeng Wei ([email protected]) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 17 / 49
Definition (Infinite)X is infinite if it is not finite:
∀n ∈ N : |X| ̸= n.
Theorem (UD Theorem 22.3)N is infinite. (So are Z, Q, R.)
By Contradiction.
∃n ∈ N : |N| = n.
∃f : N 1−1←−→onto
{0, 1, · · · , n− 1}
g ≜ f |{0,1,...,n} : {0, 1, · · · , n} → {0, 1, · · · , n− 1}
By the Pigeonhole Principle : g is not 1-1 =⇒ f is not 1-1
Hengfeng Wei ([email protected]) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 18 / 49
Definition (Infinite)For any set X,Countably Infinite
|X| = |N| ≜ ℵ0
Countable(finite ∨ countably infinite)
Uncountable(¬ countable)
(infinite) ∧(¬ (countably infinite)
)
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Hengfeng Wei ([email protected]) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 20 / 49
Theorem (Z is Countable.)
|Z| = |N|
0 1 − 1 2 − 2 · · ·
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Theorem (Q is Countable. (Cantor 1873-11; Published in 1874))
|Q| = |N|
|Q| = |N| (UD Problem 23.12)
q ∈ Q+ : a/b (a, b ∈ N+)
Hengfeng Wei ([email protected]) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 22 / 49
Theorem (N× N is Countable.)
|N× N| = |N|
π : N× N→ N
π(k1, k2) = 12
(k1+k2)(k1+k2+1)+k2
Cantor Pairing Function
Hengfeng Wei ([email protected]) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 23 / 49
Theorem (Nn is Countable.)
|Nn| = |N|
TheoremThe Cartesian product of finitely many countable sets is countable.
Nn vs. NN
π(n) : Nn → N
π(n)(k1, . . . , kn−1, kn) = π(π(n−1)(k1, . . . , kn−1), kn)
Hengfeng Wei ([email protected]) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 24 / 49
TheoremAny finite union of countable sets is countable.
A = {an | n ∈ N} B = {bn | n ∈ N} C = {cn | n ∈ N}
a0 b0 c0 a1 b1 c1 · · ·
Hengfeng Wei ([email protected]) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 25 / 49
TheoremThe union of countably many countable sets is countable.
Counting by Diagonals.
We need Axiom of (Countable) Choice!
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Beyond
Hengfeng Wei ([email protected]) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 27 / 49
Theorem (R is Uncountable. (Cantor 1873-12; Published in 1874))
|R| ̸= |N|
Different “Sizes” of Infinity
Cantor’s Diagonal Argument (1890)
Hengfeng Wei ([email protected]) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 28 / 49
Theorem (R is Uncountable. (Cantor 1873-12; Published in 1874))
|R| ̸= |N|
By Contradiction.
f : R 1−1←−→onto
N
By Diagonal Argument.Hengfeng Wei ([email protected]) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 29 / 49
c ≜ |R|
Hengfeng Wei ([email protected]) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 30 / 49
Theorem (|R| (Cantor 1877))
|(0, 1)| = |R| = |R× R| = |Rn∈N|
Proof.
f(x) = tan (2x− 1)π2
|(0, 1)| = |(−π
2,π
2)| = |R|
(x = 0.a1a2a3 · · ·, y = 0.b1b2b3 · · ·) 7→ 0.a1b1a2b2a3b3 · · ·
Hengfeng Wei ([email protected]) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 31 / 49
Theorem (|R| (Cantor 1877))
|(0, 1)| = |R| = |R× R| = |Rn|
“Je le vois, mais je ne le crois pas !”“I see it, but I don’t believe it !”
— Cantor’s letter to Dedekind (1877).
Q : Then, what is “dimension”?
Theorem (Brouwer (Topological Invariance of Dimension))There is no continuous bijections between Rm and Rn for m ̸= n.
Hengfeng Wei ([email protected]) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 32 / 49
Beyond
cHengfeng Wei ([email protected]) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 33 / 49
Theorem (Cantor’s Theorem (1891))
|A| ̸= |P(A)|
Theorem (Cantor Theorem (ES Theorem 24.4))If f : A→ P(A), then f is not onto.
Hengfeng Wei ([email protected]) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 34 / 49
Theorem (Cantor Theorem)If f : A→ P(A), then f is not onto.
Hengfeng Wei ([email protected]) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 35 / 49
Theorem (Cantor Theorem)If f : A→ P(A), then f is not onto.
Understanding this problem:
A = {1, 2, 3}
P(A) ={∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}
}Onto
∀B ∈ P(A) :(∃a ∈ A : f(a) = B
)
Not Onto∃B ∈ P(A) :
(∀a ∈ A : f(a) ̸= B
)Hengfeng Wei ([email protected]) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 36 / 49
Theorem (Cantor Theorem)If f : A→ P(A), then f is not onto.
∃B ∈ P(A) :(∀a ∈ A : f(a) ̸= B
)▶ Constructive proof (∃):
B = {a ∈ A | a /∈ f(a)}
▶ By contradiction (∀):
∃a ∈ A : f(a) = B.
Q : a ∈ B?
a ∈ B ⇐⇒ a /∈ B
Hengfeng Wei ([email protected]) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 37 / 49
Theorem (Cantor Theorem)If f : A→ P(A), then f is not onto.
Diagonal Argument (以下仅适用于可数集合 A).
a f(a)1 2 3 4 5 · · ·
1 1 1 0 0 1 · · ·2 0 0 0 0 0 · · ·3 1 0 0 1 0 · · ·4 1 1 1 1 1 · · ·5 0 1 0 1 0 · · ·...
......
......
... · · ·
B = {0, 1, 1, 0, 1}
Hengfeng Wei ([email protected]) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 38 / 49
Theorem (Cantor Theorem)
|A| < |P(A)|
A P(A) P(P(A)) . . .
There is no largest infinity.
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Hengfeng Wei ([email protected]) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 40 / 49
Definition (|A| ≤ |B|)|A| ≤ |B| if there exists an one-to-one function f from A into B.
Q : What about onto function f : A→ B?
|B| ≤ |A| (Axiom of Choice)
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Definition (|A| < |B|)|A| < |B| ⇐⇒ |A| ≤ |B| ∧ |A| ̸= |B|
|N| < |R|
|X| < |2X |
|N| < |2N|
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Definition (Countable Revisited)X is countable:
(∃n ∈ N : |X| = n) ∨ |X| = |N|
Theorem (Proof for Countable)X is countable iff
|X| ≤ |N|.
X is countable iff there exists a one-to-one function
f : X → N.
Subsets of Countable Set (UD Corollary 23.4)Every subset B of a countable set A is countable.
f : A1−1−−→ N g = f |B
Hengfeng Wei ([email protected]) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 43 / 49
Slope (UD Problem 23.3 (a))(a) The set of all lines with rational slopes
(Q, R)
|R| ≤ |Q× R| ≤ |R× R| = |R|
Hengfeng Wei ([email protected]) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 44 / 49
Q : Is “≤” a partial order?
Theorem (Cantor-Schröder–Bernstein (1887))
|X| ≤ |Y | ∧ |Y | ≤ |X| =⇒ |X| = |Y |
∃ one-to-one f : X → Y ∧ g : Y → X =⇒ ∃ bijection h : X → Y
Schröder–Bernsteintheorem @ wiki
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Q : Is “≤” a total order?
Theorem (PCC)Principle of Cardinal Comparability (PCC) ⇐⇒ Axiom of Choice
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Theorem (UD Theorem 24.11)
|R| = |P(N)|
|R| ≤ |P(N)| |P(N)| ≤ |R|
c ≜ |R| = |P(N)| = |2N| ≜ 2ℵ0
c = 2ℵ0
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Continuum Hypothesis (CH)
∄A : ℵ0 < |A| < c
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R Dangerous Knowledge (22:20; BBC 2007)
Independence from ZFC:
Kurt Gödel (1940) CH cannot be disproved from ZF.Paul Cohen (1964) CH cannot be proven from the ZFC axioms.
Hengfeng Wei ([email protected]) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 49 / 49
Hengfeng Wei ([email protected]) 1-11 Set Theory (IV): Infinity 2019 年 12 月 17 日 49 / 49
Office 302Mailbox: H016
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