A BLATENT FLAW in Cantor's diag proof
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From: William Hughes on 8 Jun 2010 07:40 On Jun 8, 12:40 am, "|-|ercules" <radgray...(a)yahoo.com> wrote: > "Marshall" <marshall.spi...(a)gmail.com> wrote in messagenews:753b820e-d8b1-4ecd-b448-283171e2ee02(a)a39g2000prb.googlegroups.com... > > On Jun 7, 7:51 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: > > >> I can compute the list of all computable reals. There's just some numbers that show > >> up blank. > > >> It's trivial to compute a list that covers every digit sequence to all (infinite) finite lengths. > > > How are you going to do that? > > > Write a program that first prints out an infinite sequence of zeroes, > > and then... > > > Oops! Already a problem! There is no "then" that comes after writing > > the > > zeroes, because the process of writing the zeroes will never finish. > > > Please show us this trivial program that computes every infinite digit > > sequence. > > > Marshall > > Here you go: > > 1 000000 > 2 31415 > 3 2818 > 4 141 > 5 22 > 6 7 > > It's not finished yet! Next digit is the 7th 0 on the first number. > Where do the digit sequences that do not have a last digit go? - William Hughes
From: jbriggs444 on 8 Jun 2010 09:06 On Jun 8, 7:40 am, William Hughes <wpihug...(a)hotmail.com> wrote: > On Jun 8, 12:40 am, "|-|ercules" <radgray...(a)yahoo.com> wrote: > > > > > > > "Marshall" <marshall.spi...(a)gmail.com> wrote in messagenews:753b820e-d8b1-4ecd-b448-283171e2ee02(a)a39g2000prb.googlegroups.com... > > > On Jun 7, 7:51 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: > > > >> I can compute the list of all computable reals. There's just some numbers that show > > >> up blank. > > > >> It's trivial to compute a list that covers every digit sequence to all (infinite) finite lengths. > > > > How are you going to do that? > > > > Write a program that first prints out an infinite sequence of zeroes, > > > and then... > > > > Oops! Already a problem! There is no "then" that comes after writing > > > the > > > zeroes, because the process of writing the zeroes will never finish. > > > > Please show us this trivial program that computes every infinite digit > > > sequence. > > > > Marshall > > > Here you go: > > > 1 000000 > > 2 31415 > > 3 2818 > > 4 141 > > 5 22 > > 6 7 > > > It's not finished yet! Next digit is the 7th 0 on the first number. > > Where do the digit sequences that do not have a last digit > go? It appears that his algorithm generates its output by a serpentine traversal of all the digits in all the lines. First it fills in the corner (first digit on first line). Then the diagonal below and to the right (first digit on second line and second digit on first line). Then the diagonal below that and to the right. (3,1), (2,2), (1,3). [I con't claim to know exactly what traversal scheme is used, but it's almost certainly serpentine and will hit each location on the grid in finitely many steps] If you let the algorithm run forever it will indeed amount to a computation of the digit value for every position on every line. Even the lines with non-terminating decimals. Non-terminating decimals are not a problem for this output convention. At a guess, lines 2 and 3 at least are just such digit sequences (pi and e). Of course, the completed array still has no line that matches the anti-diagonal. Despite the fact that the anti-diagonal is clearly computable.
From: William Hughes on 8 Jun 2010 09:14 On Jun 8, 10:06 am, jbriggs444 <jbriggs...(a)gmail.com> wrote: > Non-terminating decimals are not a problem for this output > convention. At a guess, lines 2 and 3 at least are just > such digit sequences (pi and e). You live on a very strange planet. On my planet line 2 has five digits and line 3 has four digits. - William Hughes
From: jbriggs444 on 8 Jun 2010 09:57 On Jun 8, 9:14 am, William Hughes <wpihug...(a)hotmail.com> wrote: > On Jun 8, 10:06 am, jbriggs444 <jbriggs...(a)gmail.com> wrote: > > > Non-terminating decimals are not a problem for this output > > convention. At a guess, lines 2 and 3 at least are just > > such digit sequences (pi and e). > > You live on a very strange planet. On my planet line > 2 has five digits and line 3 has four digits. I flatter myself that I understand roughly where Herc is heading. His algorithm has not yet generated the remaining digits on lines 2 or 3. As he wrote, it's just getting ready to output the next digit on line 1. This overlooks the fact that he promised us an algorithm and in its place gave us some partial output.
From: Marshall on 8 Jun 2010 10:03
On Jun 7, 8:40 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: > "Marshall" <marshall.spi...(a)gmail.com> wrote: > > On Jun 7, 7:51 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: > > >> I can compute the list of all computable reals. There's just some numbers that show > >> up blank. > > >> It's trivial to compute a list that covers every digit sequence to all (infinite) finite lengths. > > > How are you going to do that? > > > Write a program that first prints out an infinite sequence of zeroes, > > and then... > > > Oops! Already a problem! There is no "then" that comes after > > writing the zeroes, because the process of writing the > > zeroes will never finish. > > > Please show us this trivial program that computes every infinite digit > > sequence. > > Here you go: > > 1 000000 > 2 31415 > 3 2818 > 4 141 > 5 22 > 6 7 > > It's not finished yet! Next digit is the 7th 0 on the first number. At no point will this process ever produce even a single infinite string of digits. Also I see no reason to think it's going to be anything vaguely comprehensive in the vertical direction either. Marshall |