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Date: Fri, 14 Feb 2014 06:25:37 +1100
Subject: Re: Working with the set of real numbers
From: Chris Angelico <rosuav@gmail.com>
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On Fri, Feb 14, 2014 at 1:00 AM, Marko Rauhamaa <marko@pacujo.net> wrote:
> Well, if your idealized, infinite, digital computer had =E2=84=B5=E2=82=
=81 bytes of RAM
> and ran at =E2=84=B5=E2=82=81 hertz and Python supported transfinite iter=
ation, you
> could easily do reals:
>
>     def real_sqrt(y):
>         for x in continuum(0, max(1, y)):
>             # Note: x is not traversed in the < order but some other
>             # well-ordering, which has been proved to exist.
>             if x * x =3D=3D y:
>                 return x
>         assert False

How exactly do you iterate over a continuum, with a digital computer?
Even adding to your requirements that it have an =E2=84=B5=E2=82=81 Hz bus =
(which, by
the way, I *totally* want - the uses are endless), it would take a
finite amount of time to assign to x the "next number", ergo your
algorithm can't guarantee to finish in finite time.

ChrisA