How Wide Do All The Possible Computed Digit Sequences Go?
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From: Tim Little on 22 Jun 2010 23:01 On 2010-06-23, Sylvia Else <sylvia(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, so I doubt he'll accept (or even understand) yours. - Tim
From: Graham Cooper on 22 Jun 2010 23:07 On Jun 23, 1:01 pm, Tim Little <t...(a)little-possums.net> 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, so I doubt he'll accept (or even > understand) yours. > > - Tim I don't remeber herc_cant_3 saying that! Herc
From: Sylvia Else on 22 Jun 2010 23:53 On 23/06/2010 1:07 PM, Graham Cooper wrote: > On Jun 23, 1:01 pm, Tim Little<t...(a)little-possums.net> 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, so I doubt he'll accept (or even >> understand) yours. >> >> - Tim > > > > > I don't remeber herc_cant_3 saying that! Let me remind you then: http://groups.google.com/group/aus.tv/msg/d618a6632bbd27ce "Given the increasing finite prefixes of ALL infinite expansions, that list contains every digit (in order) of every infinite expansion." Sylvia.
From: Sylvia Else on 23 Jun 2010 00:08 On 23/06/2010 1:01 PM, Tim Little wrote: > On 2010-06-23, Sylvia Else<sylvia(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.
From: Graham Cooper on 23 Jun 2010 00:24
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 .... So herc_cant_3 stands so it should I gave 2 proofs Herc |