On 22 Dec 2012, at 17:05, Telmo Menezes wrote:

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Hi Bruno,On Thu, Dec 20, 2012 at 1:01 PM, Roger Clough <rclo...@verizon.net>wrote:> The infinite set of natural numbers is not stored on anything,Which causes no problem because there is not a infinite number ofanything in the observable universe, probably not even points inspace.Perhaps, we don't know.It causes no problem because natural numbers does not have to bestored 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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