On 18 Sep., 16:23, Bruno Marchal <[EMAIL PROTECTED]> wrote:
> So without putting any
> extra-stcruture on the set of infinite strings, you could as well have
> taken as basic in your ontology the set of subset of N, written  P(N).
> Now, such a set is not even nameable in any first order theory. In a
> first order theory of those strings you will get something equivalent
> to Tarski theory of Real: very nice but below the turing world: the
> theory is complete and decidable and cannot be used for a theory of
> everything (there is no natural numbers definable in such theories).
>  From this I can deduce that your intuition relies on second order
> arithmetic or analysis (and this is confirmed by the way you introduce
> time).

Bruno and Russell, I don't want to interfere with your discussion. But
I want to say something concerning the mathematics applied to study
the ensemble of infinite bitstrings (which is, as you, Bruno,
mentioned correctly, equivalent to the power set of the natural
numbers). For me, the Everything ensemble is something given. I'm not
forced to restrict myself to the use of mathematical structures
definable by the structure of the Everything ensemble. I can use the
whole of mathematics developed until today in order to study the
Everything ensemble.

Let's consider our universe that is studied by physics. Probably, we
won't find the set of natural numbers within this universe, the number
of identical particles (as far as we can talk about that) of any kind
is finite. Nonetheless, it is useful to define the natural numbers and
to construct rational, real and even complex numbers in order to study
the universe.

A vivid though quite ridiculous example might be: When we study the
unaffected tropics, we go there with cameras despite of the fact that
cameras don't come from the tropics.

As Everything ensemble, we use the set of infinite bitstrings. But the
Theory of Everything, which doesn't really exist so far, may use every
mathematical structure that proves to be useful. This of course
differs seriously from arithmetical realism.


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