Working with the set of real numbers

[Marko Rauhamaa]

Chris Angelico <rosuav at gmail.com>:

> On Wed, Feb 12, 2014 at 11:48 PM, Marko Rauhamaa <marko at pacujo.net> wrote:
>> According to your definition, there's no computer in the world that can
>> work with integers or text files.
>
> Integers as far as RAM will allow, usually (which is the same caveat
> as is used when describing a programming language as "Turing complete"
> - strictly, that term is valid only if it has infinite memory
> available), but yes, technically that's a subset of integers. However,
> that subset is bounded by something other than the code, algorithms,
> or even hardware - it's theoretically possible to add two numbers
> larger than will fit in memory, by reading them in (even over the
> network), adding segments, and writing them out again.
>
> Text files. Since there's already no such thing as a "text file"
> unless you know what its encoding is, I don't see a problem with this.

Text files suffer from the same caveat as integers: there's a limit to
how much you can store on the physical computer.

A similar caveat prevents computers from dealing with real numbers. In
the case of integers, you have a finite subset of ℵ₀. In the case of
reals, you have a finite subset of ℵ₁.

Yes, integers are algorithmically much more tractable than reals.
However, in practice integer math is often computationally much harder
than real math. Take cryptography vs calculus as an example.


Marko