Imo talk2

Paradox

魏澤人

國立東華大學應用數學學系

2013-04-13

魏澤人 (國立東華大學應用數學學系) Paradox 2013-04-13 1 / 30

Proof.

1−1 =

−11√

1−1 =

√−11

√1√−1

√−1√1√

1√1 =

√−1√−1

1 = −1

Proof.

−1 = (−1)3 = (−1)62 = ((−1)2)

32 = 1

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 + 11, 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

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

1 設 BC 中點 D2 設 ∠A 平分線及 BC 中垂線交於O (否則 AB = AC)3 做 OR⊥AB, OQ⊥AC4 由 AAS, 4RAO ∼= 4QAO5 由 SAS, 4ODB ∼= 4ODC6 所以 4ROB ∼= 4QOC7 得 AB = AC

Braess’s Paradox

Without ——————-Total 4000 cars.Route Start->A->END: A

100 + 45Route Start->B->END: B

100 + 45Best solution is A = B = 2000, Time =2000

100 + 45 = 65.

Braess’s Paradox

Without ——————-Total 4000 cars.Route Start->A->END: A

100 + 45Route Start->B->END: B

100 + 45Best solution is A = B = 2000, Time =2000

100 + 45 = 65.

Braess’s Paradox

With —————–Total 4000 cars.Route Start->A: X

100 < 45Route B->END: X

100 < 45Best solution is 4000

100 + 4000100 = 80.

Zeno’s paradoxes

Achilles and the tortoiseThe dichotomy paradoxThe arrow paradox

Zeno’s paradoxes

Thomson’s Lamp

time Status2:00 on3:00 off3:30 on3:45 off3:52.5 on...4:00 ?

Infinite

Hilbert’s paradox of the Grand HotelSquares and Natural numbersBalls and vase problem

Infinite

玫瑰小木

中午前 1 分鐘，放 10 個球到花瓶中，然後拿出一個。中午前1/2分鐘，放 10 個球到花瓶中，然後拿出一個。中午前1/4分鐘，放 10 個球到花瓶中，然後拿出一個。. . .中午時，花瓶有幾顆球？

玫瑰小木

球編號 1, 2, 3, ...中午前 1 分鐘，放 1-10號球到花瓶中，然後拿出 1 號。中午前1/2分鐘，放 11-20 個球到花瓶中，然後拿出 2 號。中午前1/4分鐘，放 21-30 個球到花瓶中，然後拿出 3 號。. . .中午時，花瓶有幾顆球？

Banach–Tarski paradox

Voter’s Paradox

23 think A > B, 2

3 think B > C , then A > C?

Voter 1: A > B > CVoter 2: B > C > AVoter 3: C > A > B

Voter’s Paradox

23 think A > B, 2

3 think B > C , then A > C?

Voter 1: A > B > CVoter 2: B > C > AVoter 3: C > A > B

Arrow’s Paradox

It is impossible to satisfiesUniversal Domain : Contain all possible choices.Irrelevant Alternatives - An individual preference stated in binaryterms shouldn’t affect his ordering of other preferencesPareto Optimality : If every one votes the same way, the society isdecisive.

Monotonicity+Non-imposition

Nondictatorship : More than one person’s vote should be used.

Arrow’s Paradox

錢包

A和B兩人進行一場賭博。賭法是：由第三者計算A、B二君錢包裡面的錢，錢少者可以贏走錢多者的錢。A對於這場賭博的想法為：若B君的錢比我少，我可能輸掉我現有的錢。但若B君的錢比我多，我贏了，就會得到多於我現有的錢。我能夠贏的錢比輸的錢多，所以這場賭博對我有利。而B的想法也是如此。二人想法的邏輯都正確，但若認為二人的想法都正確，又將做出這場賭博對A、B二人都有利的錯誤結論。這顯然是一個悖論。

信封

兩個外觀一樣的信封，都放了錢，一個裡面的金額是另外一個的兩倍。你隨機的拿起了一個，然後給你一個機會來交換，你是否要交換？

Barber Paradox

There is a town with just one male barber.Every man in the town keeps himself clean-shaven

some by shaving themselvessome by attending the barber.

The barber shaves all and only those men who do not shavethemselves.

seems reasonable.

Does the barber shave himself?

Barber Paradox

seems reasonable.

Barber Paradox

seems reasonable.

Barber Paradox

seems reasonable.

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.

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.

Liar’s Paradox

Epimenides paradox: A Cretan says: "All Cretans are liars".Liar paradox: "This sentence is false." or "Is the answer to thisquestion 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.

Liar’s Paradox

Epimenides paradox: A Cretan says: "All Cretans are liars".Liar paradox: "This sentence is false." or "Is the answer to thisquestion 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.

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 aweek."

logic and philosophy interests.

Grelling-Nelson paradox: "heterological" := "not applicable to itself".Yablo’s paradox. A self reference paradoxes without self reference.

More Self Reference Paradoxes

Opposite Day: "It is opposite day today."Paradox of the Court: A law student agrees to pay his teacher afterwinning his first case.Quine’s paradox: "Yields a falsehood when appended to its ownquotation" yields a falsehood when appended to its own quotation.Richard’s paradox:

聖彼得堡

擲硬幣，若第一次擲出正面，你就賺1元。若第一次擲出反面，那就要再擲一次，若第二次擲的是正面，你便賺2元。若第二次擲出反面，那就要擲第三次，若第三次擲的是正面，你便賺2*2元. . . . . .如此類推，即可能擲一次遊戲便結束，也可能反覆擲沒完沒了。問題是，你最多肯付多少錢參加這個遊戲？

Penney Ante

HHH TTTHHT TTHHTH THTHTT THH

Simpson’s Paradox

Men WomenDept A 250/500, 50% accepted 9/10, 90% acceptedDept B 1/10, 10% accepted 100/500, 20% acceptedTotals 251/510 accepted 109/510 accepted

Simpson’s Paradox

Treatment A Treatment BBoth 78% (273/350) 83% (289/350)

Small Stones 93% (81/87) 87% (234/270)Large Stones 73% (192/263) 69% (55/80)

UC Berkeley 1973

Applicants % admittedMen 8442 44%

Women 4321 35%

UC Berkeley 1973

Major Men % Women %A 825 62% 108 82%B 560 63% 25 68%C 325 37% 593 34%D 417 33% 375 35%E 191 28% 393 24%F 272 6% 341 7%

Related Paradox

Low birth weight paradoxBatting average

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