# Arithmetic as true constructions of a fictional leggo set

```Hi Bruno Marchal

It helps me if I can understand arithmetic as true
constructions of a fictional leggo set.```
```
>From what you say, the natural numbers and + and * (nn+*).
are not a priori members of Platonia (if indeed that makes
sense anyway). They can simply be invoked and used
as needed, as long as they don't produce contradictions.
That being the case, don't you need to add =, - ,  and
/ to the Leggo set ? Then we have (nn+-*/=).

I wonder if somebody could derive string theory from this set.
Then we might say that the universe is an arithmetic construction.
Probably an absurd idea.

[Roger Clough], [rclo...@verizon.net]
12/24/2012
"Forever is a long time, especially near the end." -Woody Allen

----- Receiving the following content -----
From: Bruno Marchal
Time: 2012-12-23, 09:17:09
Subject: Re: Can the physical brain possibly store our memories ? No.

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

Hi Bruno,

On Thu, Dec 20, 2012 at 1:01 PM, Roger Clough 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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