Working with the set of real numbers

[Oscar Benjamin]

On 12 February 2014 10:07, Ben Finney <ben+python at benfinney.id.au> wrote:
> Chris Angelico <rosuav at gmail.com> writes:
>
>> On Wed, Feb 12, 2014 at 7:56 PM, Ben Finney <ben+python at benfinney.id.au> wrote:
>> > So, if I understand you right, you want to say that you've not found
>> > a computer that works with the *complete* set of real numbers. Yes?
>>
>> Correct. [...] My point is that computers *do not* work with real
>> numbers, but only ever with some subset thereof [...]
>
> You've done it again: by saying that "computers *do not* work with real
> numbers", that if I find a real number - e.g. the number 4 - your
> position is that, since it's a real number, computers don't work with
> that number.
>
> That's why I think you need to be clear that your point isn't "computers
> don't work with real numbers", but rather "computers work only with a
> limited subset of real numbers".

I think Chris' statement above is pretty clear. Also I didn't find the
original statement confusing and it is a reasonable point to make.

While computers can (with some limitations) do a pretty good job of
integers and rational numbers they cannot truly represent real
computation. Other people have mentioned that there are computer
algebra systems that can handle surds and other algebraic numbers or
some transcendental numbers but none of these comes close to the set
of reals.

This isn't even a question of resource constraints: a digital computer
with infinite memory and computing power would still be limited to
working with countable sets, and the real numbers are just not
countable. The fundamentally discrete nature of digital computers
prevents them from being able to truly handle real numbers and real
computation.

A hypothetical idealised analogue computer would be able to truly do
real arithmetic (but I think in practice the errors would be worse
than single precision floating point).


Oscar