My FINAL Cantor thread!
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From: George Greene on 26 Jun 2010 12:44 On Jun 26, 1:18 am, "|-|ercules" <radgray...(a)yahoo.com> wrote: > And its the width of the Complete Permutation Set THERE IS NO SUCH THING as "the complete permutation set"! WHY do you think you can keep using the word "permutation"?? No permutations are relevant here. You have not even clarified whether you are "permuting" horizontally or vertically. If you are permuting "horizontally" then it does NOT MATTER; the result of permuting the digits in ONE real IS JUST ANOTHER REAL. If the first real was computable and the permutation was computable then the resulting real WILL ALSO BE COMPUTABLE and was therefore ALREADY ON your list! You ARE NOT ADDING anything by talking about permutations! > not the reals themselves, A permutation of a real IS a real (unless you permute some of the numbers to be infinitely far from the start, which is not possible if you were doing it with transpositions), so THIS IS A MEANINGLESS distinction! People who are letting you continue to use "permutation" are just adding complexity for you to get lost in. > whoops, I mean the width of the initial permuations IN the CPS! If the original list is a list of computable reals then, yes, "the width" is appropriate BECAUSE THEY ALL HAVE THE EXACT SAME (infinite) width. There is in that case NO question about any "max". The max is only relevant if INSTEAD of using the list of computable reals, you use the list of FINITE digit-sequences. BOTH of these have the property that they match ANY real (computable OR NOT) up to ANY (and EVERY) finite width, so I don't see why it is so important to you to use the computable list, or to start talking about irrelevant permutations of it. No matter how you permute the list OF ALL AND ONLY the computable reals, it is STILL a list of all and only the computable reals. IT DOES NOT MAKE ANY DIFFERENCE whether you do or don't permute it. Different permutations will give you different anti- diagonals, but that just means you are MORE WRONG -- there are MORE numbers that CAN'T be on the list!
From: |-|ercules on 26 Jun 2010 15:46 "George Greene" <greeneg(a)email.unc.edu> wrote > On Jun 26, 1:18 am, "|-|ercules" <radgray...(a)yahoo.com> wrote: >> And its the width of the Complete Permutation Set > > THERE IS NO SUCH THING as "the complete permutation set"! > WHY do you think you can keep using the word "permutation"?? > No permutations are relevant here. > You have not even clarified whether you are "permuting" horizontally > or vertically. > If you are permuting "horizontally" then it does NOT MATTER; the > result of > permuting the digits in ONE real IS JUST ANOTHER REAL. > If the first real was computable and the permutation was computable > then > the resulting real WILL ALSO BE COMPUTABLE and was therefore ALREADY > ON > your list! You ARE NOT ADDING anything by talking about permutations! > >> not the reals themselves, > > A permutation of a real IS a real (unless you permute some of the > numbers to be > infinitely far from the start, which is not possible if you were doing > it with transpositions), > so THIS IS A MEANINGLESS distinction! > > People who are letting you continue to use "permutation" are just > adding complexity > for you to get lost in. > > >> whoops, I mean the width of the initial permuations IN the CPS! > > If the original list is a list of computable reals then, yes, "the > width" is > appropriate BECAUSE THEY ALL HAVE THE EXACT SAME (infinite) width. > There is in that case NO question about any "max". The max is only > relevant > if INSTEAD of using the list of computable reals, you use the list of > FINITE > digit-sequences. > BOTH of these have the property that they match ANY real (computable > OR NOT) > up to ANY (and EVERY) finite width, so I don't see why it is so > important to > you to use the computable list, or to start talking about irrelevant > permutations of it. > No matter how you permute the list OF ALL AND ONLY the computable > reals, > it is STILL a list of all and only the computable reals. IT DOES NOT > MAKE ANY DIFFERENCE > whether you do or don't permute it. Different permutations will give > you different anti- > diagonals, but that just means you are MORE WRONG -- there are MORE > numbers that > CAN'T be on the list! > It should be obvious to non-entrenched mathematicians that the proof holds, and the attacks don't acknowledge my comments or are refutations based on the existence of higher levels of pedantic wording! The induction shows the structure of ALL digits of computable reals. There are 2 things that are demonstrated in the proof. 1/ all possible sequences of digits 2/ the domain is the oo wide digit sequences of computable reals This trivially contradicts construction of new digit sequences. That "all finite prefixes" has a similar result and doesn't show the presence of infinite reals is not relevant. The entire list of computable reals from the first real can all be used as Complete Permutation Sets for any widths 'w'. I need a special char to use for 'complete permutation widths'. Herc
From: Mike Terry on 26 Jun 2010 17:31 "|-|ercules" <radgray123(a)yahoo.com> wrote in message news:88lga0FikkU1(a)mid.individual.net... > "Mike Terry" <news.dead.person.stones(a)darjeeling.plus.com> wrote > >> > should imply there is unlimited width of all sequences. > >> > >> Aaaargh, now you've gone back to unclear mode. What does "unlimited width > >> of all sequences" mean? > >> > >> Whatever it means, it obviously does not imply all (countably) infinite > >> permutations are in the list, because that's obviously false. (Not even 1 > >> infinite digit permuatation is in the list :-) > > > > the proof is lost on you. I will say again I am not building infinite amount of > finite sized blocks of numbers. I am sampling the pattern of digits early in > the string of reals, and using induction to show the structure of digits out to > infinity wide, i.e. the TOTAL digit strings of all the computable reals. All very well, but the question is what are you going to do now regarding your proof in the OP for this thread: a) Correct your proof so that w is well defined? b) Rewrite your proof so that w is not needed? c) Leave it as is, and say you are really trying to do something else etc.? I can only comment on your proof as presented, and it needs to be a meaningful proof. (I just mean a proof with clear mathematical meaning, not utilising vague/ambiguous/contradictory terms.) Then I can point to the exact step in the proof where a mistake is made. (Or agree with the proof if it is correct.) Mike. > > e.g. > > 006666666.. > 016666666.. > 106666666.. > 116666666.. > > The structure of the first 2 digits is a complete permutation set. I prove > the maximum width of complete permutations is infinity. > > Think of a CPS as a tall rectangle which is sampled at larger and larger sizes > with induction, revealing the digit structure of the entire list (of computable reals). > > It's a BREADTH and DEPTH wise induction so it reveals the nature of lists more reliably > than drawing a square and striking a line down the center saying NO! (that's diagonalisation > if you missed the pun) > > > > > > In any case, the list does not have any *uncomputable* reals in it, and > > these are of "width" omega (first infinite ordinal). So (4) above is still > > correct, and your w does not exist. OR... maybe you have a secret proof > > that *all* infinite digit sequences are computable? > > > > > I'm glad for the open mindedness of your final statement, because the top down attacks > on my proof are merely blind belief that ZFC is complete. > > Herc
From: |-|ercules on 26 Jun 2010 17:49 "Mike Terry" <news.dead.person.stones(a)darjeeling.plus.com> wrote > "|-|ercules" <radgray123(a)yahoo.com> wrote in message > news:88lga0FikkU1(a)mid.individual.net... >> "Mike Terry" <news.dead.person.stones(a)darjeeling.plus.com> wrote >> >> > should imply there is unlimited width of all sequences. >> >> >> >> Aaaargh, now you've gone back to unclear mode. What does "unlimited > width >> >> of all sequences" mean? >> >> >> >> Whatever it means, it obviously does not imply all (countably) infinite >> >> permutations are in the list, because that's obviously false. (Not > even 1 >> >> infinite digit permuatation is in the list :-) >> > >> >> the proof is lost on you. I will say again I am not building infinite > amount of >> finite sized blocks of numbers. I am sampling the pattern of digits early > in >> the string of reals, and using induction to show the structure of digits > out to >> infinity wide, i.e. the TOTAL digit strings of all the computable reals. > > All very well, but the question is what are you going to do now regarding > your proof in the OP for this thread: > a) Correct your proof so that w is well defined? > b) Rewrite your proof so that w is not needed? > c) Leave it as is, and say you are really trying to do something else etc.? > > I can only comment on your proof as presented, and it needs to be a > meaningful proof. (I just mean a proof with clear mathematical meaning, not > utilising vague/ambiguous/contradictory terms.) Then I can point to the > exact step in the proof where a mistake is made. (Or agree with the proof > if it is correct.) > > Mike. > I have no idea what you're on about. w is the maximum width of finite prefixes of (elements of) subsets of computable reals such that there is no missing digit sequence in the finite prefixes of any considered subset. ----------------------------------------------------------------------------- For each subset of computable reals, there exists a maximum digit length that that subset doesn't miss a possible sequence of digits. w is the maximum of those maximums. Herc
From: Mike Terry on 27 Jun 2010 07:16
"|-|ercules" <radgray123(a)yahoo.com> wrote in message news:88navnFr4iU1(a)mid.individual.net... > "Mike Terry" <news.dead.person.stones(a)darjeeling.plus.com> wrote > > "|-|ercules" <radgray123(a)yahoo.com> wrote in message > > news:88lga0FikkU1(a)mid.individual.net... > >> "Mike Terry" <news.dead.person.stones(a)darjeeling.plus.com> wrote > >> >> > should imply there is unlimited width of all sequences. > >> >> > >> >> Aaaargh, now you've gone back to unclear mode. What does "unlimited > > width > >> >> of all sequences" mean? > >> >> > >> >> Whatever it means, it obviously does not imply all (countably) infinite > >> >> permutations are in the list, because that's obviously false. (Not > > even 1 > >> >> infinite digit permuatation is in the list :-) > >> > > >> > >> the proof is lost on you. I will say again I am not building infinite > > amount of > >> finite sized blocks of numbers. I am sampling the pattern of digits early > > in > >> the string of reals, and using induction to show the structure of digits > > out to > >> infinity wide, i.e. the TOTAL digit strings of all the computable reals. > > > > All very well, but the question is what are you going to do now regarding > > your proof in the OP for this thread: > > a) Correct your proof so that w is well defined? > > b) Rewrite your proof so that w is not needed? > > c) Leave it as is, and say you are really trying to do something else etc.? > > > > I can only comment on your proof as presented, and it needs to be a > > meaningful proof. (I just mean a proof with clear mathematical meaning, not > > utilising vague/ambiguous/contradictory terms.) Then I can point to the > > exact step in the proof where a mistake is made. (Or agree with the proof > > if it is correct.) > > > > Mike. > > > > > I have no idea what you're on about. :-) > > w is the maximum width of finite prefixes of (elements of) subsets of computable reals > such that > > there is no missing digit sequence in the finite prefixes of any considered subset. > That's not the way you defined w at the start of the thread. > > -------------------------------------------------------------------------- --- > > For each subset of computable reals, there exists a maximum digit length that > that subset doesn't miss a possible sequence of digits. > > w is the maximum of those maximums. > > Herc |