How Wide Do All The Possible Computed Digit Sequences Go?
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From: Sylvia Else on 23 Jun 2010 08:57 On 23/06/2010 8:00 PM, Graham Cooper wrote: > On Jun 23, 7:06 pm, Sylvia Else<syl...(a)not.here.invalid> wrote: >> On 23/06/2010 7:01 PM, Graham Cooper wrote: >> >> >> >> >> >>> On Jun 23, 6:55 pm, Sylvia Else<syl...(a)not.here.invalid> wrote: >>>> On 23/06/2010 6:43 PM, Graham Cooper wrote: >> >>>>> On Jun 23, 5:35 pm, Sylvia Else<syl...(a)not.here.invalid> wrote: >>>>>> 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 >>>>>>> ... >> >>>>> Huh? I'm not explaining trivial items to you. >> >>>> The purpose was to clarify the intent of "How many digits in order of pi >>>> are below this line if interpreted mathematically?" Don't you want it >>>> clarified? >> >>>>> Your "mutilation" of herc_can't_3 was proved erronous >>>>> and it stands. >> >>>> Where was it proved erroneous? >> >>>>> There are other terms than contains where it holds >>>>> so what is your argument? >> >>>> herc_cant_3 contained a proposition of the form >> >>>> If X has property A, then it also has property B. >> >>>> Such a proposition is demonstrable false if a possible X is demonstrated >>>> to have property A, and demonstrated not to have property B. It doesn't >>>> matter than there are some Xs with both A and B. The proposition is >>>> falsified by a single counter-example. >> >>>> Since the proposition in herc_cant_3 has a counter-example, it cannot be >>>> correct, and without a substitute proposition, herc_cant_3 cannot stand >>>> as a theorem. >> >>>>> Your disproof was wrong and you >>>>> won't admit you were wrong. Now you are shifting your erronous >>>>> claim to a termi ology issue. >> >>>> No - I'm just trying to understand what your question was intended to mean. >> >>>>> Either give a proper disproof >>>>> or ceasevyour eternal complaints. Hc3 stands. The ball >>>>> is in your court to prove otherwise. >> >>>> Sylvia. >> >>> Hc3 does not state >> >>> all finite prefixes -> all inf sequences >> >>> so your counter example is moot >> >> herc_cant_3 states >> >> "Given the increasing finite prefixes of ALL infinite expansions, >> that list contains every digit (in order) of every infinite expansion." >> >> If that's not what that statement is intended to mean, then what, >> *exactly* is it intended to mean? >> >> Sylvia. > > > > Don't you listen? > > I told you every digit in order does not mean a single > infinite sequence. > > 3 > 31 > 314 > ... > > How many digits of pi are in that list? infinity * (infinity + 1) / 2 which is nonsense, of course. The number of digits in the last element of the list tends to infinity as the list length tends to infinity. If the earlier elements in the list have some role, you'll need to explain it. Sylvia.
From: Transfer Principle on 23 Jun 2010 17:59 On Jun 22, 9:24 pm, Graham Cooper <grahamcoop...(a)gmail.com> wrote: > On Jun 23, 2:08 pm, Sylvia Else <syl...(a)not.here.invalid> wrote: > > 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. > 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 claims that that herc_cant_3 is a theorem to which he has given two proofs. I would like to know in what theory Herc has given the two proofs -- i.e., which axioms he used in the proofs. It's not because I want to belittle Cooper's theory, but just because I like to learn more about alternate theories, and also to give him a warning as to what sort of theories are likely to be belittled by others. For example, many posters criticize theories which appear to assume more structure than necessary. Apparently, this herc_cant_3 proves that if X is a set (or list) such that 3, 3.1, 3.14, etc., are elements of X, then pi must also be an element of X. Notice that in standard theory, if we claim that X is a _closed_ set (i.e., contains all of its limit points), then X does contain pi, so perhaps Herc is assuming that every set is closed. But this is the type of assumption that many posters are likely to disparage. I don't mind assuming such additional structures (the current TO thread also involves assuming some additional structure), but many other posters dislike this.
From: Rupert on 23 Jun 2010 21:22 On Jun 23, 12:17 pm, Graham Cooper <grahamcoop...(a)gmail.com> wrote: > On Jun 23, 11:31 am, Rupert <rupertmccal...(a)yahoo.com> wrote: > > > > > > > On Jun 22, 7:36 pm, Graham Cooper <grahamcoop...(a)gmail.com> wrote: > > > > On Jun 22, 4:28 pm, Graham Cooper <grahamcoop...(a)gmail.com> wrote: > > > > > On Jun 22, 3:21 pm, Rupert <rupertmccal...(a)yahoo.com> wrote: > > > > > > On Jun 22, 6:44 am, Graham Cooper <grahamcoop...(a)gmail.com> wrote: > > > > > > > On Jun 22, 12:08 am, Graham Cooper <grahamcoop...(a)gmail.com> wrote: > > > > > > > > On Jun 21, 10:40 pm, Sylvia Else <syl...(a)not.here.invalid> wrote: > > > > > > > > > On 21/06/2010 5:03 PM, Rupert wrote: > > > > > > > > > > On Jun 21, 4:28 pm, "|-|ercules"<radgray...(a)yahoo.com> wrote: > > > > > > > > >> Every possible combination X wide... > > > > > > > > > >> What is X? > > > > > > > > > >> Now watch as 100 mathematicians fail to parse a trivial question. > > > > > > > > > >> Someone MUST know what idea I'm getting at! > > > > > > > > > >> This ternary set covers all possible digits sequences 2 digits wide! > > > > > > > > > >> 0.00 > > > > > > > > >> 0.01 > > > > > > > > >> 0.02 > > > > > > > > >> 0.10 > > > > > > > > >> 0.11 > > > > > > > > >> 0.12 > > > > > > > > >> 0.20 > > > > > > > > >> 0.21 > > > > > > > > >> 0.22 > > > > > > > > > >> HOW WIDE ARE ALL_POSSIBLE_SEQUENCES COVERED IN THE SET OF COMPUTABLE REALS? > > > > > > > > > >> Herc > > > > > > > > >> -- > > > > > > > > >> If you ever rob someone, even to get your own stuff back, don't use the phrase > > > > > > > > >> "Nobody leave the room!" ~ OJ Simpson > > > > > > > > > > It would probably be a good idea for you to talk instead about the set > > > > > > > > > of all computable sequences of digits base n, where n is some integer > > > > > > > > > greater than one. Then the length of each sequence would be aleph- > > > > > > > > > null. But not every sequence of length aleph-null would be included. > > > > > > > > > That answer looks correct. > > > > > > > > > But I guarantee that Herc won't accept it. > > > > > > > > > Sylvia. > > > > > > > > It's truly hilarious. It's like using a Santa clause metaphor > > > > > > > to explain why Santa clause is not real, > > > > > > > but it will do for now. > > > > > > > > Herc > > > > > > > Actually on second reading I think Rupert threw a red herring > > > > > > > He didn't adress the question at all. How wide are all possible > > > > > > permutations of digits covered? This is different to all possible > > > > > > listed sequences he just answered that numbers are inf. long! > > > > > > > Herc- Hide quoted text - > > > > > > > - Show quoted text - > > > > > > I'm afraid I don't understand the question. > > > > > If it takes 10^x reals to have every permutation x digits wide > > > > how many digits wide would oo reals make? > > > > > Herc > > > > Where is my reference to computable reals here Rupert? > > > > This is a question with a quantity answer. > > > > If you can't answer say so. > > > > Herc- Hide quoted text - > > > > - Show quoted text - > > > There does not exist a cardinal number x, such that the set of all > > sequences of decimal digits of length x has cardinality aleph-null. > > > If you have some cardinal number x and a set of sequences of decimal > > digits of length x of cardinality aleph-null, then it must be the case > > that this set does not contain all the sequences of decimal digits of > > length x. > > > That is my answer to your question as best I understand it. But I am > > not sure I really understand what you are talking about. > > The topic of the thread is the width of permutations Well, that certainly wasn't clear to me before. It seemed to me tha we were talking about sequences of decimal digits. Permutations of which set, pray tell? > as in every permutation of a certain width > and it's relation to the size of the list of reals. > > You are refuting that this width approaches infinity > as the list of reals approaches infinity > based on > > a/. You don't know what I'm referring to > b/. Reverse engineering that there is no defined width > because it refutes transfiniteness theory > > you're avoiding the question plain and simple > I think my problem is that I don't understand the question. But that doesn't mean I'm not trying.
From: Graham Cooper on 23 Jun 2010 22:05 On Jun 24, 7:59 am, Transfer Principle <lwal...(a)lausd.net> wrote: > On Jun 22, 9:24 pm, Graham Cooper <grahamcoop...(a)gmail.com> wrote: > > > On Jun 23, 2:08 pm, Sylvia Else <syl...(a)not.here.invalid> wrote: > > > 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. > > 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 claims that that herc_cant_3 is a theorem to which he > has given two proofs. > > I would like to know in what theory Herc has given the two > proofs -- i.e., which axioms he used in the proofs. > > It's not because I want to belittle Cooper's theory, but > just because I like to learn more about alternate theories, > and also to give him a warning as to what sort of theories > are likely to be belittled by others. > > For example, many posters criticize theories which appear > to assume more structure than necessary. Apparently, this > herc_cant_3 proves that if X is a set (or list) such that > 3, 3.1, 3.14, etc., are elements of X, then pi must also be > an element of X. Notice that in standard theory, if we > claim that X is a _closed_ set (i.e., contains all of its > limit points), then X does contain pi, so perhaps Herc is > assuming that every set is closed. But this is the type of > assumption that many posters are likely to disparage. I > don't mind assuming such additional structures (the > current TO thread also involves assuming some additional > structure), but many other posters dislike this. No. I'm starting to doubt sci.math posters give more than a cursory glance at the keywords before making their retort. Indie not say pi was in the set. I said all the digits of pi were in the set. 1 1 2 1 2 3 ..... The above list "has" every natural number in order that does not mean it has an element with all N Herc
From: Graham Cooper on 23 Jun 2010 22:22
On Jun 24, 7:59 am, Transfer Principle <lwal...(a)lausd.net> wrote: > On Jun 22, 9:24 pm, Graham Cooper <grahamcoop...(a)gmail.com> wrote: > > > On Jun 23, 2:08 pm, Sylvia Else <syl...(a)not.here.invalid> wrote: > > > 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. > > 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 claims that that herc_cant_3 is a theorem to which he > has given two proofs. 3 now. 1 extrapolating the result of several finite prefix examples to all finite prefixes 2 contradicting the assertion that a read could contain a non computable finite sequence 3 proof by induction that the maximum digit width of all listed permutations is infinite. I asked everyone 50 times to clarify 1 & 2 but 3 is sufficient to put the final nail in Cantor's coffin! Herc |