On Jun 25, 10:24 am, "Jesse F. Hughes" <je...@phiwumbda.org> wrote: > Charlie-Boo <shymath...@gmail.com> writes: > > On Jun 24, 8:49 am, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote: > >> Charlie-Boo <shymath...@gmail.com> writes: > >> > On Jun 15, 2:15 am, "Peter Webb" > > >> >> No. You cannot form a list of all computable Reals. > > >> > Of course you can - it's just the list of Turing Machines. > > >> No, it's not. > > > I asked for a counterexample, to no avail. Don't you think you should > > substantiate your statement or retract it? > > > Each Turing Machine represents some computable real (all computable > > reals are included) and you can list those Turing Machines. The > > Turing Machine represents it as well as any other system of > > representation. > > It is not the case that every TM represents some computable real. > > Example 1: The TM that never halts and never changes the tape does not > represent a computable real.
It depends on the system of representation. An empty tape typically represents 0.
> Example 2: The TM that repeatedly changes the value in one cell, never > halting, does not represent a computable real.
The sequence of values put into a given cell defines the decimal expansion of a real number. Every machine must distinguish between scratch values and actual output. Having a certain cell represent output is also common.
Google "Turing Machines".
> Other than that, of course, your response was mighty insightful. > -- > Jesse F. Hughes > "I just define real numbers to be all those on the number line, as > they were defined before Dedekind and Cauchy." > -- Ross Finlayson's simple definition.- Hide quoted text - > > - Show quoted text -