Working with the set of real numbers

[Devin Jeanpierre]

On Thu, Feb 13, 2014 at 11:47 AM, Marko Rauhamaa <marko at pacujo.net> wrote:
> Chris Angelico <rosuav at gmail.com>:
>
>> On Fri, Feb 14, 2014 at 1:00 AM, Marko Rauhamaa <marko at pacujo.net> wrote:
>>> Well, if your idealized, infinite, digital computer had ℵ₁ bytes of RAM
>>> and ran at ℵ₁ hertz and Python supported transfinite iteration, you
>>> could easily do reals:
>>>
>>>         for x in continuum(0, max(1, y)):
>>
>> How exactly do you iterate over a continuum, with a digital computer?
>
> How "digital" our idealized computers are is a matter for a debate.
> However, iterating over the continuum is provably "possible:"
>
>   http://en.wikipedia.org/wiki/Transfinite_induction

You missed the most important point on that page, which is the "limit case".

There is no way to iterate over all the reals one at a time, no matter
how fast you execute instructions. If you could, it would be trivial
to show that the reals have the same cardinality as the positive
integers: correspond n with the whatever is returned by the nth call
to it.next.

It doesn't matter if you call your magical iterator "transfinite",
that doesn't make it so.

-- Devin