"Tim Little" <tim@little-possums.net> wrote in message news:slrni1tkli.jrj.tim@soprano.little-possums.net... > On 2010-06-21, Peter Webb <webbfamily@DIESPAMDIEoptusnet.com.au> wrote: >> "Tim Little" <tim@little-possums.net> wrote in message >> I do understand that. But I don't actually ask for a "finite algorithm" >> in >> my proof, I only require that I am provided a purported list of all >> comptuable Reals. > > Later in the proof, you claim that the antidiagonal real is > computable. In other words, that there exists a finite algorithm for > computing it. > > >> Yes, in practice, this means that each item is specified by a finite >> algorithm, but so what? I ask for a purported list of all computable >> Reals >> and show there is at least one missing. > > At least one *what* missing? A real? Fine - Cantor's proof shows > that there is a missing real.
And if the nth digit of the nth term is computable, then so is the nth term of the anti-diagonal, as there is an explicit construction of the nth digit of the anti-diagonal based upon the nth digit of the nth term.
> What's that? You want the antidiagonal > to be a *computable* real? Well, then - you are required to show that > there always exists a finite algorithm to compute it, and that is not > present in Cantor's proof. >
Well, I don't actually have to write out all "infinite" decimals in its expansion to prove it is computable.
All I really have to be able to do is to compute it to arbitrary accuracy. And Cantor's proof constructs the number to arbitrary accuracy.
> So your modified "proof" is either invalid, or you must include > significant amounts of material not present in Cantor's proof. >
Well, really only one thing. I have to produce an algorithm to estimates the anti-diagonal to arbitrary accuracy. Fortunately, the digit chnage rule of Cantor uniquely specifies the nth decimal place of the anti-diagonal in a completely finite closed procedure.
> > As is well-known to competent mathematicians and computer scientists, > no amount of material will make it valid, as the modified "theorem" is > simply false. It is *not true* that any list of computable reals has > a computable antidiagonal.
You give me a list of purported computable Reals, I can compute the anti-diagonal to arbitrary precision. Just like Cantor can compute the value of the missing Real to arbitrary precision.
