> Check out analytic TMs (Hotz, Vierke and Schieffer,
> 1995) and R-Machines (Blum, Shub, Smale, 1989):
> http://rapa.idsia.ch/~juergen/toesv2/node47.html The alphabet of such TMs
> is indeed real-valued instead of binary. This is beyond constructivism
> though. GTMs are at the extreme end of the constructive spectrum.
> They are beaten by nonconstructive devices only.
> But stuff describable by nonconstructive devices only does not even exist.
> Does it?
But stuff indescribable by us ought to be constructible by observers
with real-valued input, output and memory living in real-valued
worlds. They might communicate identities of arbitrary real numbers
as naturally as we refer to a points on a line, and execute analytic
algorithms as easily as we fiddle with finite alphabets.
Those observers may not be in reach of our analysis, but they are
within the scope of their own.