On Jun 25, 3:25 pm, Virgil <Vir...@home.esc> wrote: > In article > <adaa005c-c180-4ce0-8d2c-81da912c6...@w12g2000yqj.googlegroups.com>, > > > > > > Charlie-Boo <shymath...@gmail.com> wrote: > > 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. > > Aren't there Turing machines that don't represent any real at all?
No. In general terms, every TM computes some result from its input. Agree? Then if we start with an empty tape, it represents a constant. How we map its execution history determines which real number that represents. There is generally a way to indicate which actions constitute output, which is needed when we use an infinite representation such as its real number binary expansion.
If the executon history can be finite, we can map all finite strings into real numbers, as well as the infinite ones as described above.
> > > > > > C-B > > > > -- > > > Aatu Koskensilta (aatu.koskensi...@uta.fi) > > > > "Wovon man nicht sprechan kann, dar ber muss man schweigen" > > > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus- Hide quoted text - > > - Show quoted text -- Hide quoted text - > > - Show quoted text -