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morrowasted
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The Tristram Shandy Paradox
#14535695 - 05/30/11 12:32 PM (12 years, 7 months ago) |
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Tristram Shandy is an author writing an auto-biography. Unfortunately, he writes very slowly; each day of his life takes him a year to write about.
The Tristram Shandy paradox asks: If Shandy continues at this rate for eternity then will his book ever be finished?
Bertrand Russell, who invented this paradox, suggested that the book would be finished. Given an infinite amount of time, for every day in Shandy's life there is a year to spend writing about it; there are, after all, an infinite number of years in which to write the autobiography. The autobiography therefore can be completed.
This doesn't seem right though. With each passing year, Shandy completes his writing about one day, but leaves another three hundred and sixty-four days undocumented. Every year, then, there are three hundred and sixty-four days more for Shandy to write about; the more time passes, the further behind he falls.
How can it be that Shandy progressively further and further behind as time progresses, and at the same time that given an eternity he will complete his work?
Common approaches:
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If you deny that Tristam completes his autobiography, then I challenge you to name the first day that he does not write about.
If you claim he did not write about day N, you are wrong; because he writes about day N in year N.
In other words, you can easily see that for any N, Tristram writes about day N in year N. Therefore for all N, day N gets written about. Hence the autobiography is complete.
Day N gets documented in year N. There is no day of Tristram's life that does not get documented.
Quote:
In principle, there is a bijection between completed pages and years lived. The problem is that arithmetically we don't get that product.
There are days that he has completed writing, and there are days that he has lived.
His productivity is 365 days : 1 written day.
Our intuitions should tell us that he could never write about every day because he would be ((365 * written days) + (autobiography start date - date of birth)) days behind the completion of his full autobiography. Unless there is an intersection for x and y, where x is the number of lived days and y is the number of written days in the biography, he will not ever finish it.
The only natural-number solution is to assume that he has lived 0 days, and by vacuous truth, had written his complete autobiography before he was ever born:
| x = 365y + (a - b), Assp. | x = y, Assp. | y = 0 ⇒ a - b = 0 Assp. | y = 365y + (a - b), 1,2 Sub || y = 0, Ass. || a - b = 0, 3,5 MP || y = 365y + 0, 4,6 Sub || 0 = 365*0 + 0, 5,7 Sub || 0 = 0 + 0 8,Mult. || 0 = 0, 9,Add. | y = 0 ⇒ 0 = 0 ((x = 365y + (a - b)) • (x = y) • (y = 0 ⇒ a - b = 0)) ⇒ (y = 0 ⇒ 0 = 0)
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husmmoor
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Re: The Tristram Shandy Paradox [Re: morrowasted]
#14536655 - 05/30/11 04:16 PM (12 years, 7 months ago) |
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Given an infinite amount of time, I will solve this paradox. 
Interesting.
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xFrockx


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Re: The Tristram Shandy Paradox [Re: husmmoor]
#14537574 - 05/30/11 08:04 PM (12 years, 7 months ago) |
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"If you deny that Tristam completes his autobiography, then I challenge you to name the first day that he does not write about."
This is a bad refutation. He will fail to complete his autobiography not because he will stop writing, but because he will never stop writing, which means he will also be succeeding at it forever. Completed, though, it will never be.
Edited by xFrockx (05/30/11 08:58 PM)
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OrgoneConclusion
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Re: The Tristram Shandy Paradox [Re: morrowasted]
#14537725 - 05/30/11 08:33 PM (12 years, 7 months ago) |
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Given an infinite amount of time, he will eventually grow bored and write about something else, thus finishing his autobiography.
OC solves another seeming paradox.
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DieCommie

Registered: 12/11/03
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Re: The Tristram Shandy Paradox [Re: morrowasted]
#14537843 - 05/30/11 08:56 PM (12 years, 7 months ago) |
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I think the heart of this paradox lies at the impossible premise. You cannot derive anything useful from an impossible premise, and any paradox derived as such is merely a linguistic inconsistency.
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teknix
𓂀⟁𓅢𓍝𓅃𓊰𓉡 𓁼𓆗⨻


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Re: The Tristram Shandy Paradox [Re: morrowasted]
#14538512 - 05/30/11 11:15 PM (12 years, 7 months ago) |
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Quote:
morrowasted said: Tristram Shandy is an author writing an auto-biography. Unfortunately, he writes very slowly; each day of his life takes him a year to write about.
The Tristram Shandy paradox asks: If Shandy continues at this rate for eternity then will his book ever be finished?
Depends, does he intend to write about writing? Or about his life prior to undertaking the endeavour to write?
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OrgoneConclusion
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Re: The Tristram Shandy Paradox [Re: teknix]
#14538635 - 05/30/11 11:43 PM (12 years, 7 months ago) |
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Of course the whole premise is flawed as you point out that no author starting an autobiography intends to keep writing about writing about writing about their autobiography.
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Rhizoid
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Re: The Tristram Shandy Paradox [Re: xFrockx]
#14538734 - 05/31/11 12:13 AM (12 years, 7 months ago) |
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I agree with xFrockx. I personally would avoid using the word "completed" for anything that takes forever. Compare with Turing's halting problem: saying that a computer program halts after an infinite amount of time is a sort of informal way of stating that it is a non-halting program.
But if the question is will Tristram document all days of his life, then the answer is yes due to the bijection between the infinite set of years and the infinite set of days. This is similar to Hilbert's Hotel, which contains an infinite number of rooms that are occupied. A new guest can still be accommodated by moving the guest in room 1 to room 2, the guest in room 2 to room 3, and so on. This is possible because there is a bijection between {1,2,3,...} and {2,3,4,...}.
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Rhizoid
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Quote:
OrgoneConclusion said: Of course the whole premise is flawed as you point out that no author starting an autobiography intends to keep writing about writing about writing about their autobiography.
Maybe he could write about what he did during his extremely long and interesting coffee breaks?
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husmmoor
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Re: The Tristram Shandy Paradox [Re: xFrockx]
#14540508 - 05/31/11 12:45 PM (12 years, 7 months ago) |
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Quote:
xFrockx said:He will fail to complete his autobiography not because he will stop writing, but because he will never stop writing, which means he will also be succeeding at it forever. Completed, though, it will never be.
That's certainly a very logical response, but it doesn't resolve the paradox, it just kills the babydox in its cradle before becoming a paradox.
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