ANTI-CANTOR CLAIM
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From: |-|ercules on 20 Jun 2010 15:33 The antidiagonal is too general to meaningfully define a number. It's not just based on all digits in forall n, L(n,n) The antidiagonal argument also has to work on EVERY PERMUTATION of a list. That means, you can construct an anti-diagonal using the 1st digit of *any* listed real, the second digit of *any* other listed real, and so on. As long as you choose ANY increasing unique position digit of EVERY real, and eventually the selected reals fillas out from the top, you can create a real from any 'diagonal', like so: 0. _ _ _ x _ _ 0. x _ _ _ _ _ 0. _ _ x _ _ _ 0. _ x _ _ _ _ 0. _ _ _ _ _ x 0. _ _ _ _ x _ A valid diagonal from the leftmost x to the rightmost x. You can select ANY digit {0, .. 9} except change the x value and that should be a NEW real according to Cantor. If you designed an algorithm that could alter the x positions so that any digit of the diagonal could be changed to a different digit, then that would prove the diagonal argument doesn't work! Herc -- If you ever rob someone, even to get your own stuff back, don't use the phrase "Nobody leave the room!" ~ OJ Simpson
From: Sylvia Else on 20 Jun 2010 21:16 On 21/06/2010 5:33 AM, |-|ercules wrote: > The antidiagonal is too general to meaningfully define a number. > > It's not just based on all digits in > forall n, L(n,n) > > The antidiagonal argument also has to work on EVERY PERMUTATION of a list. It does work. You often get different numbers as a result, but all the numbers so constructed are not in the list, nor any permutation of it. All permuting achieves is to provide a way of constructing multiple counter-examples to the hypothesis that every real was in the list. But you only needed one, so there was no point in permuting the list. > > That means, you can construct an anti-diagonal using the 1st digit of > *any* listed real, > the second digit of *any* other listed real, and so on. > > As long as you choose ANY increasing unique position digit of EVERY > real, and eventually the selected reals > fillas out from the top, you can create a real from any 'diagonal', like > so: > > 0. _ _ _ x _ _ > 0. x _ _ _ _ _ > 0. _ _ x _ _ _ > 0. _ x _ _ _ _ > 0. _ _ _ _ _ x > 0. _ _ _ _ x _ > > A valid diagonal from the leftmost x to the rightmost x. Diagonal or anti-diagonal? The resulting anti-diagonal is not in the list. The resulting diagonal may or may not be, and nothing turns on that. > > You can select ANY digit {0, .. 9} except change the x value and that > should be a NEW real > according to Cantor. > > If you designed an algorithm that could alter the x positions so that > any digit of the > diagonal could be changed to a different digit, then that would prove > the diagonal > argument doesn't work! That's not at all apparent. Sylvia.
From: |-|ercules on 20 Jun 2010 22:08 "Sylvia Else" <sylvia(a)not.here.invalid> wrote > On 21/06/2010 5:33 AM, |-|ercules wrote: >> The antidiagonal is too general to meaningfully define a number. >> >> It's not just based on all digits in >> forall n, L(n,n) >> >> The antidiagonal argument also has to work on EVERY PERMUTATION of a list. > > It does work. You often get different numbers as a result, but all the You just said it was a FALSE PREMISE. > numbers so constructed are not in the list, nor any permutation of it. > All permuting achieves is to provide a way of constructing multiple > counter-examples to the hypothesis that every real was in the list. But > you only needed one, so there was no point in permuting the list. > >> >> That means, you can construct an anti-diagonal using the 1st digit of >> *any* listed real, >> the second digit of *any* other listed real, and so on. >> >> As long as you choose ANY increasing unique position digit of EVERY >> real, and eventually the selected reals >> fillas out from the top, you can create a real from any 'diagonal', like >> so: >> >> 0. _ _ _ x _ _ >> 0. x _ _ _ _ _ >> 0. _ _ x _ _ _ >> 0. _ x _ _ _ _ >> 0. _ _ _ _ _ x >> 0. _ _ _ _ x _ >> >> A valid diagonal from the leftmost x to the rightmost x. > > Diagonal or anti-diagonal? The resulting anti-diagonal is not in the > list. The resulting diagonal may or may not be, and nothing turns on that. > >> >> You can select ANY digit {0, .. 9} except change the x value and that >> should be a NEW real >> according to Cantor. >> >> If you designed an algorithm that could alter the x positions so that >> any digit of the >> diagonal could be changed to a different digit, then that would prove >> the diagonal >> argument doesn't work! > > That's not at all apparent. True. But it's odd you can FLIP infinite amount of digits of the anti-diagonal (back to the diagonal) and they are "not on the list" either... Herc
From: Sylvia Else on 20 Jun 2010 22:44 On 21/06/2010 12:08 PM, |-|ercules wrote: > "Sylvia Else" <sylvia(a)not.here.invalid> wrote >> On 21/06/2010 5:33 AM, |-|ercules wrote: >>> The antidiagonal is too general to meaningfully define a number. >>> >>> It's not just based on all digits in >>> forall n, L(n,n) >>> >>> The antidiagonal argument also has to work on EVERY PERMUTATION of a >>> list. >> >> It does work. You often get different numbers as a result, but all the > > You just said it was a FALSE PREMISE. I was trying to be helpful. It is a false premise, but if you start with the premise that the reals are countable, then you can work with the notion that there is a list that can be permuted. But for the work to be valid, sooner or later you have to prove that the list exists. What actually happens, of course, is that you discover that the list doesn't exist, which means that the work you did with permutations of it was meaningless. > > >> numbers so constructed are not in the list, nor any permutation of it. >> All permuting achieves is to provide a way of constructing multiple >> counter-examples to the hypothesis that every real was in the list. >> But you only needed one, so there was no point in permuting the list. >> >>> >>> That means, you can construct an anti-diagonal using the 1st digit of >>> *any* listed real, >>> the second digit of *any* other listed real, and so on. >>> >>> As long as you choose ANY increasing unique position digit of EVERY >>> real, and eventually the selected reals >>> fillas out from the top, you can create a real from any 'diagonal', like >>> so: >>> >>> 0. _ _ _ x _ _ >>> 0. x _ _ _ _ _ >>> 0. _ _ x _ _ _ >>> 0. _ x _ _ _ _ >>> 0. _ _ _ _ _ x >>> 0. _ _ _ _ x _ >>> >>> A valid diagonal from the leftmost x to the rightmost x. >> >> Diagonal or anti-diagonal? The resulting anti-diagonal is not in the >> list. The resulting diagonal may or may not be, and nothing turns on >> that. >> >>> >>> You can select ANY digit {0, .. 9} except change the x value and that >>> should be a NEW real >>> according to Cantor. "You can select ANY digit {0, .. 9} except change the x value" I do wish you'd try to write English. What is that intended to mean? Cantor says that if you construct a number for which *every* digit differs from the corresponding digit in the diagonal, then the resulting number will not be in the list. >>> >>> If you designed an algorithm that could alter the x positions so that >>> any digit of the >>> diagonal could be changed to a different digit, then that would prove >>> the diagonal >>> argument doesn't work! >> >> That's not at all apparent. > > True. But it's odd you can FLIP infinite amount of digits of the > anti-diagonal > (back to the diagonal) and they are "not on the list" either... > Well that's pretty much what Cantor predicts. The computables are countable. The reals aren't, so for any computable, you can find uncountably many non-computable reals. You'd expect a lot of diagonals not to be in the list either. Sylvia.
From: |-|ercules on 20 Jun 2010 23:01
> Well that's pretty much what Cantor predicts. The computables are > countable. The reals aren't, so for any computable, you can find > uncountably many non-computable reals. You'd expect a lot of diagonals > not to be in the list either. > > Sylvia. Oh cripes! That's so funny how you put 1 and 3 together and get 7. So FALSE PREMISE doesn't mean not worth discussing now? BWAHAHA That's a new one, amazing twist of the discussion there, very clever. Herc |