On 22 Dec 2012, at 17:05, Telmo Menezes wrote:
Hi Bruno,
On Thu, Dec 20, 2012 at 1:01 PM, Roger Clough <[email protected]>
wrote:
> The infinite set of natural numbers is not stored on anything,
Which causes no problem because there is not a infinite number of
anything in the observable universe, probably not even points in
space.
Perhaps, we don't know.
It causes no problem because natural numbers does not have to be
stored a priori. Only when universal machine want to use them.
Why do the natural numbers exist?
We cannot know that.
Precisely, if you assume the natural numbers, you can prove that you
cannot derived the existence of the natural number and their + and *
laws, in *any* theory which does not assume them, or does not assume
something equivalent.
That is why it is a good reason to start with them (or equivalent).
Somehow, the natural numbers, with addition and multiplication, are
necessarily "mysterious".
With the natural numbers and + and *, you can prove the existence of
all universal machines, and vice versa, if you assume any other
universal system (like the combinators K, S (K K), (K S), ...) you can
prove the existence of the natural numbers and their laws.
We have to assume at least one universal system, and I chose
arithmetic because it is the simpler one. The problem is that the
proof of its universality will be difficult, but at least it can be
found in good mathematical logic textbook, like Mendelson or Kleene,
etc.
Bruno
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