On Tue, 29 Dec 1998 00:36:21 GMT, "Bob Street" <bob@belgrave.clara.net> wrote:
> >Mike Deeth wrote in message <3686D986.745ADAB8@ashland.baysat.net>... >> >> >>Arturo Magidin wrote: >> >>> In article <3684404B.86724757@ashland.baysat.net>, >>> Mike Deeth <mad@ashland.baysat.net> wrote: >>> > >>> > >> (2/3)(1-(1/2)^(n+1)) (even) >># (1/3)(1-(1/2)^n) (odd) Diagonal Cantor's >>-------------------------------------------------------- >>0 .10 >>1 .01 .01 .10 >>2 .1010 >>3 .0101 01 10 >>4 .101010 >>5 .010101 01 10 >>6 .10101010 >>7 .01010101 01 10 >>8 .1010101010 >>9 .0101010101 01 10 >>. . >>. . >>. .10101010101010101010...101010 >>. .01010101010101010101...010101 >> > > >Arturo's *patch* solves the REAL (no pun intended) problem which the >argument has in binary - it's not easy to demonstrate that a diagonal can be >constructed which doesn't 'end' 11111111111111............ in binary, >because there aren't enough digits to play with. In any other base (10 for >example) we can construct a diagonal using only the digits 3 and 4 (for >example), thus we know that it doesn't end 999999999............. >Arturo's patch effectively gives us more 'digits' to play with by using base >4, whilst keeping a binary representation. > >
your *glitch patch* does NOT work. To see why, examine the table below. Notice that I diagonalized the odd table entries prior to the even table entries. My method of diagonalizing fixes a logical flaw in your argument. You erred, when you let the diagonal string grow twice as fast as the natural numbers (string pointers) used to index the diagonal number being constructed. I'm sure you now realize the absurdity of that. In my table, the odd entries are repressented by the infinite rational series: 1/6, 7/24, 31/96, 127/384, ... The even entries are repressented by the infinite rational series: 2/6, 14/24, 62/96, 254/384, ... Notice, for each step in the diagonal construction, Cantor's number is included, as the previous entry in the table.
Cantor's diagonal number will eventually become a Real number (completed infinity).
Should we now go back and diagonalize the even table entries? No! That would be absurd, since Cantor's diagonal number is already complete (fully determined). It is obvious that Cantor's Diagonal Method is not applicable, because only half of the table entries were/(can be) diagonalized.
