Working with the set of real numbers

[Albert van der Horst]

In article <87fvnm7q1n.fsf at elektro.pacujo.net>,
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
>
>> 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.
>
>My assumption was you could execute ℵ₁ statements per second. That
>doesn't guarantee a finite finish time but would make it possible. That
>is because
>
>   ℵ₁ * ℵ₁ = ℵ₁ = ℵ₁ * 1
>
>This computer is definitely more powerful than a Turing machine, which
>only has ℵ₀ bytes of RAM and thus can't even store an arbitrary real
>value in memory.

You're very much off the track here. A Turing machine is an abstraction
for a computer were the limitations of size are gone.
The most obvious feature of a Turing machine is an infinite tape.
A Turing machine happily calculates Ackerman functions long after
a real machine runs out of memory to represent it, with as a result
a number of ones on that tape.
But it only happens in the mathematicians mind.

>
>
>Marko
-- 
Albert van der Horst, UTRECHT,THE NETHERLANDS
Economic growth -- being exponential -- ultimately falters.
albert at spe&ar&c.xs4all.nl &=n http://home.hccnet.nl/a.w.m.van.der.horst