How Wide Do All The Possible Computed Digit Sequences Go?
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From: Sylvia Else on 23 Jun 2010 01:30 On 23/06/2010 2:24 PM, Graham Cooper wrote: > On Jun 23, 2:08 pm, Sylvia Else<syl...(a)not.here.invalid> wrote: >> On 23/06/2010 1:01 PM, Tim Little wrote: >> >>> On 2010-06-23, Sylvia Else<syl...(a)not.here.invalid> wrote: >>>> To recap, herc_cant_3 was based on the proposition that if a list >>>> contains all finite prefixes, then it contains all infinite >>>> sequences, and thus all reals. >> >>>> However all finite prefixes can be obtained by taking a list of all >>>> computables, permuting it and taking the diagonals. [...] >> >>> An even more straightforward counterexample was given much earlier in >>> the discussion: a list of all finite digit sequences obviously >>> contains all finite prefixes, and by definition does not contain *any* >>> infinite sequences. So it certainly does not contain all of them. >> >>> Herc didn't accept that one, >> >> Well, a list of finite digit sequences obviously contains no infinite >> sequences, as you say, by definition. >> >> But Herc talks about finite prefixes rather than finite sequences. I >> took this to mean that if one looks at the supposed list, one can find >> any finite prefix in it, said prefix being the start of some infinite >> sequence. Herc then leaps to the conclusion that this means that the >> list contains all infinite sequences. >> >> The construction I proposed generates an infinity of infinite sequences >> which contain all finite prefixes thus meeting the requirement for >> Herc's list. It just demonstrably doesn't contain all infinite sequences. >> >> So Herc's problem is that it is not inevitably true that a list that >> contains all finite prefixes also contains all infinite sequences, and >> indeed it is specifically false for a sequences generated by permuting >> the computables. >> >> Sylvia. > > > All digits in order does not mean > a single infinitely long sequence > > like this list contains all digits in order > of pi > > 3 > 31 > 314 > ... > On the face of it, line n contains the n digits of pie, sequentially, and in order. I suppose it can be conceded that the infinite list contains Pi. How that relates to herc_cant_3, or your "All digits in order..." comment is far from clear. Sylvia
From: Tim Little on 23 Jun 2010 03:00 On 2010-06-23, Sylvia Else <sylvia(a)not.here.invalid> wrote: > On the face of it, line n contains the n digits of pie, > sequentially, and in order. I suppose it can be conceded that the > infinite list contains Pi. 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. It does satisfy a much looser property: there exists a sublist S such that lim S = pi. In general you can form a set of real numbers closure(L) = { x in R | exists sublist S of L such that lim S = x } which you could call the "closure" of a list L. Then you could say that pi is in the closure of the list, but pi is certainly not in the list itself. Herc does not know the difference. - Tim
From: Graham Cooper on 23 Jun 2010 03:03 On Jun 23, 5:00 pm, Tim Little <t...(a)little-possums.net> wrote: > On 2010-06-23, Sylvia Else <syl...(a)not.here.invalid> wrote: > > > On the face of it, line n contains the n digits of pie, > > sequentially, and in order. I suppose it can be conceded that the > > infinite list contains Pi. > > 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. > > It does satisfy a much looser property: there exists a sublist S such > that lim S = pi. In general you can form a set of real numbers > closure(L) = { x in R | exists sublist S of L such that lim S = x } > which you could call the "closure" of a list L. > > Then you could say that pi is in the closure of the list, but pi is > certainly not in the list itself. > > Herc does not know the difference. > > - Tim How many digits in order of pi are below this line if interpreted mathematically? ____________ 3 31 314 ....
From: Sylvia Else on 23 Jun 2010 03:32 On 23/06/2010 5:00 PM, Tim Little wrote: > On 2010-06-23, Sylvia Else<sylvia(a)not.here.invalid> wrote: >> On the face of it, line n contains the n digits of pie, >> sequentially, and in order. I suppose it can be conceded that the >> infinite list contains Pi. > > 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. Sylvia.
From: Sylvia Else on 23 Jun 2010 03:35
On 23/06/2010 5:03 PM, Graham Cooper wrote: > On Jun 23, 5:00 pm, Tim Little<t...(a)little-possums.net> wrote: >> On 2010-06-23, Sylvia Else<syl...(a)not.here.invalid> wrote: >> >>> On the face of it, line n contains the n digits of pie, >>> sequentially, and in order. I suppose it can be conceded that the >>> infinite list contains Pi. >> >> 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. >> >> It does satisfy a much looser property: there exists a sublist S such >> that lim S = pi. In general you can form a set of real numbers >> closure(L) = { x in R | exists sublist S of L such that lim S = x } >> which you could call the "closure" of a list L. >> >> Then you could say that pi is in the closure of the list, but pi is >> certainly not in the list itself. >> >> Herc does not know the difference. >> >> - Tim > > > How many digits in order of pi are below this line > if interpreted mathematically? What does that question mean? In particular, what does "digits in order of pi" mean? Perhaps you could give some example lists, with the answer in each case. Sylvia. > > ____________ > > 3 > 31 > 314 > ... |