Marshall
Posts:
1,928
Registered:
8/9/06
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Re: A BLATENT FLAW in Cantor's diag proof
Posted:
Jun 8, 2010 2:42 PM
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On Jun 8, 9:48 am, jbriggs444 <jbriggs...@gmail.com> wrote:
> On Jun 8, 10:03 am, Marshall <marshall.spi...@gmail.com> wrote:
> > On Jun 7, 8:40 pm, "|-|ercules" <radgray...@yahoo.com> wrote:
> > > "Marshall" <marshall.spi...@gmail.com> wrote:
> > > > On Jun 7, 7:51 pm, "|-|ercules" <radgray...@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.
>
> With a serpentine traversal, for every position on the grid
> there will be (for an algorithm that doesn't end up looping
> quietly or halting) a time by which the algorithm will have
> provided a digit for that position.
>
> In this sense, the algorithm specifies every digit on every
> line.
Sure. However, at no point will this process ever produce
even a single infinite line of digits.
When we talk about enumerating the natural numbers,
for example:
f(x) = { emit x; f(x+1); }
f(0);
then at no point will we be finished with all of them.
However, for every natural number n, there is
a point at which n will be emitted.
At no point, ever, in the serpentine traversal, will
we have even a single infinitely long line of digits.
We never ever produce even one infinite digit
sequence this way; we certainly cannot say that
we are producing all of them.
Marshall
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