4. 3
任何一個人的團體，所有人頭髮數量一定一樣多。
假設所有 n 個人的團體,都必然每個人有相同數量頭髮。
考慮一個 n + 1 人團體，編號為 1, 2, . . . , n + 1
1, 2, . . . , n 每個人頭髮數量相同。
2, . . . , n, n + 1 每個人頭髮數量相同
所以 1, 2, . . . , n, n + 1 每個人的頭髮都相同
魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 4 / 30
5. 4
證明 a = b for every a, b ∈ N
對 max (a, b) 做數學歸納法
case max (a, b) = 0: a = 0 = b, done.
設
max (a, b) = n ⇒ a = b
成立。
當 max (a, b) = n + 1 時,
max (a, b) = n + 1
⇒ max (a − 1, b − 1) = n
⇒ a−1=b−1
⇒ a=b
魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 5 / 30
6. 5
1 設 BC 中點 D
2 設 ∠A 平分線及 BC 中垂線交於O (否則 AB = AC )
3 做 OR⊥AB, OQ⊥AC
4 由 AAS, RAO ∼ QAO
=
5 由 SAS, ODB ∼ ODC
=
6 所以 ∼ QOC
ROB =
7 得 AB = AC
魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 6 / 30
7. Braess’s Paradox
Without ——————-
Total 4000 cars.
A
Route Start->A->END: 100 + 45
B
Route Start->B->END: 100 + 45
Best solution is A = B = 2000, Time = 2000 + 45 = 65.
100
魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 7 / 30
8. Braess’s Paradox
Without ——————-
Total 4000 cars.
A
Route Start->A->END: 100 + 45
B
Route Start->B->END: 100 + 45
Best solution is A = B = 2000, Time = 2000 + 45 = 65.
100
魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 7 / 30
9. Braess’s Paradox
With —————–
Total 4000 cars.
X
Route Start->A: 100 < 45
X
Route B->END: 100 < 45
Best solution is 100 + 4000 = 80.
4000
100
魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 8 / 30
10. Zeno’s paradoxes
Achilles and the tortoise
The dichotomy paradox
The arrow paradox
魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 9 / 30
11. Zeno’s paradoxes
Achilles and the tortoise
The dichotomy paradox
The arrow paradox
魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 9 / 30
12. Zeno’s paradoxes
Achilles and the tortoise
The dichotomy paradox
The arrow paradox
魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 9 / 30
13. Zeno’s paradoxes
Achilles and the tortoise
The dichotomy paradox
The arrow paradox
魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 9 / 30
14. Thomson’s Lamp
time Status
2:00 on
3:00 off
3:30 on
3:45 off
3:52.5 on
...
4:00 ?
魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 10 / 30
15. Infinite
Hilbert’s paradox of the Grand Hotel
Squares and Natural numbers
Balls and vase problem
魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 11 / 30
16. Infinite
Hilbert’s paradox of the Grand Hotel
Squares and Natural numbers
Balls and vase problem
魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 11 / 30
17. Infinite
Hilbert’s paradox of the Grand Hotel
Squares and Natural numbers
Balls and vase problem
魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 11 / 30
21. Voter’s Paradox
2 2
3 think A > B, 3 think B > C , then A > C ?
Voter 1: A > B > C
Voter 2: B > C > A
Voter 3: C > A > B
魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 15 / 30
22. Voter’s Paradox
2 2
3 think A > B, 3 think B > C , then A > C ?
Voter 1: A > B > C
Voter 2: B > C > A
Voter 3: C > A > B
魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 15 / 30
23. Arrow’s Paradox
It is impossible to satisfies
Universal Domain : Contain all possible choices.
Irrelevant Alternatives - An individual preference stated in binary
terms shouldn’t affect his ordering of other preferences
Pareto Optimality : If every one votes the same way, the society is
decisive.
Monotonicity+Non-imposition
Nondictatorship : More than one person’s vote should be used.
魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 16 / 30
24. Arrow’s Paradox
It is impossible to satisfies
Universal Domain : Contain all possible choices.
Irrelevant Alternatives - An individual preference stated in binary
terms shouldn’t affect his ordering of other preferences
Pareto Optimality : If every one votes the same way, the society is
decisive.
Monotonicity+Non-imposition
Nondictatorship : More than one person’s vote should be used.
魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 16 / 30
25. Arrow’s Paradox
It is impossible to satisfies
Universal Domain : Contain all possible choices.
Irrelevant Alternatives - An individual preference stated in binary
terms shouldn’t affect his ordering of other preferences
Pareto Optimality : If every one votes the same way, the society is
decisive.
Monotonicity+Non-imposition
Nondictatorship : More than one person’s vote should be used.
魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 16 / 30
26. Arrow’s Paradox
It is impossible to satisfies
Universal Domain : Contain all possible choices.
Irrelevant Alternatives - An individual preference stated in binary
terms shouldn’t affect his ordering of other preferences
Pareto Optimality : If every one votes the same way, the society is
decisive.
Monotonicity+Non-imposition
Nondictatorship : More than one person’s vote should be used.
魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 16 / 30
27. Arrow’s Paradox
It is impossible to satisfies
Universal Domain : Contain all possible choices.
Irrelevant Alternatives - An individual preference stated in binary
terms shouldn’t affect his ordering of other preferences
Pareto Optimality : If every one votes the same way, the society is
decisive.
Monotonicity+Non-imposition
Nondictatorship : More than one person’s vote should be used.
魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 16 / 30
30. Barber Paradox
There is a town with just one male barber.
Every man in the town keeps himself clean-shaven
some by shaving themselves
some by attending the barber.
The barber shaves all and only those men who do not shave
themselves.
seems reasonable.
Does the barber shave himself?
魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 19 / 30
31. Barber Paradox
There is a town with just one male barber.
Every man in the town keeps himself clean-shaven
some by shaving themselves
some by attending the barber.
The barber shaves all and only those men who do not shave
themselves.
seems reasonable.
Does the barber shave himself?
魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 19 / 30
32. Barber Paradox
There is a town with just one male barber.
Every man in the town keeps himself clean-shaven
some by shaving themselves
some by attending the barber.
The barber shaves all and only those men who do not shave
themselves.
seems reasonable.
Does the barber shave himself?
魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 19 / 30
33. Barber Paradox
There is a town with just one male barber.
Every man in the town keeps himself clean-shaven
some by shaving themselves
some by attending the barber.
The barber shaves all and only those men who do not shave
themselves.
seems reasonable.
Does the barber shave himself?
魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 19 / 30
34. Barber Paradox
The situation presented is in fact impossible:
If the he does not shave himself, he must shave himself.
If he does shave himself, he should not shave himself.
Barber Paradox = Russell’s Paradox {x |x ∈ x }.
Solution: modern set theory, ZFC.
ZFC is the foundation of modern mathematics.
魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 20 / 30
35. Barber Paradox
The situation presented is in fact impossible:
If the he does not shave himself, he must shave himself.
If he does shave himself, he should not shave himself.
Barber Paradox = Russell’s Paradox {x |x ∈ x }.
Solution: modern set theory, ZFC.
ZFC is the foundation of modern mathematics.
魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 20 / 30
36. Liar’s Paradox
Epimenides paradox: A Cretan says: "All Cretans are liars".
Liar paradox: "This sentence is false." or "Is the answer to this
question no?"
Godel’s incomplete theorem: “This sentence can not be proved”.
The sentence is true and can not be proved.
No formal system is complete.
Huge impact.
魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 21 / 30
37. Liar’s Paradox
Epimenides paradox: A Cretan says: "All Cretans are liars".
Liar paradox: "This sentence is false." or "Is the answer to this
question no?"
Godel’s incomplete theorem: “This sentence can not be proved”.
The sentence is true and can not be proved.
No formal system is complete.
Huge impact.
魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 21 / 30
38. Self Reference Paradoxes
Berry paradox: "the first number not nameable in under ten words".
computability and complexity implication.
Curry’s paradox: "If this sentence is true, the world will end in a
week."
logic and philosophy interests.
Grelling-Nelson paradox: "heterological" := "not applicable to itself".
Yablo’s paradox. A self reference paradoxes without self reference.
魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 22 / 30
39. Self Reference Paradoxes
Berry paradox: "the first number not nameable in under ten words".
computability and complexity implication.
Curry’s paradox: "If this sentence is true, the world will end in a
week."
logic and philosophy interests.
Grelling-Nelson paradox: "heterological" := "not applicable to itself".
Yablo’s paradox. A self reference paradoxes without self reference.
魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 22 / 30
40. Self Reference Paradoxes
Berry paradox: "the first number not nameable in under ten words".
computability and complexity implication.
Curry’s paradox: "If this sentence is true, the world will end in a
week."
logic and philosophy interests.
Grelling-Nelson paradox: "heterological" := "not applicable to itself".
Yablo’s paradox. A self reference paradoxes without self reference.
魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 22 / 30
41. Self Reference Paradoxes
Berry paradox: "the first number not nameable in under ten words".
computability and complexity implication.
Curry’s paradox: "If this sentence is true, the world will end in a
week."
logic and philosophy interests.
Grelling-Nelson paradox: "heterological" := "not applicable to itself".
Yablo’s paradox. A self reference paradoxes without self reference.
魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 22 / 30
42. More Self Reference Paradoxes
Opposite Day: "It is opposite day today."
Paradox of the Court: A law student agrees to pay his teacher after
winning his first case.
Quine’s paradox: "Yields a falsehood when appended to its own
quotation" yields a falsehood when appended to its own quotation.
Richard’s paradox:
魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 23 / 30
