How Wide Do All The Possible Computed Digit Sequences Go?
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From: Joshua Cranmer on 23 Jun 2010 22:47 On 06/23/2010 10:22 PM, Graham Cooper wrote: > 3 proof by induction that the maximum digit width > of all listed permutations is infinite. I find it amusing when people claim to prove facts about infinity via induction. Proofs by inductions can prove results for some subset of natural numbers (or, sometimes, subsets of real numbers containing non-natural numbers). Infinity is not a natural number, so it is, so to speak, outside the range of induction. -- Beware of bugs in the above code; I have only proved it correct, not tried it. -- Donald E. Knuth
From: Tim Little on 23 Jun 2010 23:12 On 2010-06-23, Sylvia Else <sylvia(a)not.here.invalid> wrote: > On 23/06/2010 5:00 PM, Tim Little wrote: >> It can't be conceded, as it is simply false. The predicate "list L >> contains x" means exactly that there exists n in N such that L_n = x. >> The list does not contain pi since there is no such n. It really is >> that simple. > > Well, OK. Though I can't see how it makes any difference to Herc's > argument, since I could never see what role it played anyway. It is the very core of his argument: if pi were "contained in" the infinite list 3 3.1 3.14 3.141 ... then it would also be "contained in" any other list sharing the "all prefixes" property, such as 3.0000000 3.1000000 3.1428571 . . . 3.1414141 ... It would even be "contained in" any list having that as a sublist, e.g. 0.0000000 3.0000000 3.1000000 1.0000000 3.1428571 . . . 2.0934953 0.5829345 3.1414141 ... The list of all computable reals is exactly such a list. Most importantly, no property specific to pi is used here. Every real would be "contained in" that list if we were to use Herc's broken idea of "contained in". - Tim
From: Graham Cooper on 24 Jun 2010 00:35 On Jun 24, 12:47 pm, Joshua Cranmer <Pidgeo...(a)verizon.invalid> wrote: > On 06/23/2010 10:22 PM, Graham Cooper wrote: > > > 3 proof by induction that the maximum digit width > > of all listed permutations is infinite. > > I find it amusing when people claim to prove facts about infinity via > induction. Proofs by inductions can prove results for some subset of > natural numbers (or, sometimes, subsets of real numbers containing > non-natural numbers). Infinity is not a natural number, so it is, so to > speak, outside the range of induction. > > -- > Beware of bugs in the above code; I have only proved it correct, not > tried it. -- Donald E. Knuth That is very dubious. It would be helpful if you knew what you were talking about. Eg the proof you are disputing. Herc
From: Graham Cooper on 24 Jun 2010 00:40 On Jun 24, 1:12 pm, Tim Little <t...(a)little-possums.net> wrote: > On 2010-06-23, Sylvia Else <syl...(a)not.here.invalid> wrote: > > > On 23/06/2010 5:00 PM, Tim Little wrote: > >> It can't be conceded, as it is simply false. The predicate "list L > >> contains x" means exactly that there exists n in N such that L_n = x.. > >> The list does not contain pi since there is no such n. It really is > >> that simple. > > > Well, OK. Though I can't see how it makes any difference to Herc's > > argument, since I could never see what role it played anyway. > > It is the very core of his argument: if pi were "contained in" the > infinite list > > 3 > 3.1 > 3.14 > 3.141 > ... > > then it would also be "contained in" any other list sharing the "all > prefixes" property, such as > > 3.0000000 > 3.1000000 > 3.1428571. . . > 3.1414141 > ... > > It would even be "contained in" any list having that as a sublist, e.g. > > 0.0000000 > 3.0000000 > 3.1000000 > 1.0000000 > 3.1428571. . . > 2.0934953 > 0.5829345 > 3.1414141 > ... > > The list of all computable reals is exactly such a list. Most > importantly, no property specific to pi is used here. Every real > would be "contained in" that list if we were to use Herc's broken idea > of "contained in". > This reminds me of a classic skeptics retort. I say something like my horoscope said I was a 1 5 7 then that night my father handed me a DVD to watch "U571" then a skeptic will say I went to the beach and 3 planes flew overhead which was the same number of remaining chocolates in forrest gumps box of chocolates and last week I bought some chocolates at kmart and k is my wides initial then they explain how facts interrelate and the brain misfires and recognizes patterns that aren't there. True story about my numerology report 157! Herc > - Tim
From: George Greene on 24 Jun 2010 02:02
On Jun 21, 2:28 am, "|-|ercules" <radgray...(a)yahoo.com> wrote: > HOW WIDE ARE ALL_POSSIBLE_SEQUENCES COVERED IN THE SET OF COMPUTABLE REALS? We're going to try this again from the beginning: 1) This question IS TOTALLY STUPID because Herc canNOT DEFINE "covered". 2) The words "all" and "possible" are just irrelevant here: ANY real, BY DEFINITION, is w digits wide. No other width is relevant. In particular NO finite widths are relevant. If, however, you want to talk about ALL the infinitely many DIFFERENT finite widths, that is STILL NOT THE SAME THING AS talking about THE ONE INDIVIDUAL INfinite width. |