# Re: Numbers in the Platonic Realm

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On 30 Oct 2012, at 18:29, Stephen P. King wrote:```
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```On 10/30/2012 12:38 PM, Bruno Marchal wrote:
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No? If they do not have something equivalent to concepts, how can they dream?
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Yes, the universal numbers can have concept.
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```Dear Bruno,

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Let's start over. Please plain in detail what is a universal number and how it (and not ordinary numbers) have concepts or 1p.
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I will give more detail on FOAR, soon or later. But let me explains quickly.
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Fix your favorite Turing universal system. It can be a programming language, a universal Turing machine, or a sigma_1 complete theory, or even a computer.
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Enumerate the programs computing functions fro N to N, (or the equivalent notion according to your chosen system). let us call those functions: phi_0, phi_1, phi_2, ... (the phi_i)
```Let B be a fixed bijection from N x N to N. So B(x,y) is a number.

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The number u is universal if phi_u(B(x,y)) = phi_x(y). And the equality means really that either both phi_u(B(x,y)) and phi_x(y) are defined (number) and that they are equal, OR they are both undefined.
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In phi_u(B(x,y)) = phi_x(y), x is called the program, and y the data. u is the computer. u i said to emulate the program (machine, ...) x on the input y.
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Comp is the thesis that I can survive with a physical digital computer in place of the physical brain, as far as it emulates me close enough.
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Comp gives a special role to computer (physical incarnation of a universal number). The comp idea is that computer can supports thinking and consciousness, and makes them capable of manifestation relatively to other universal structure (physical universes if that exists, people, etc.). This should answer your question.
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The lobian machines are only universal numbers, having the knowledge that they are universal. I can prove to any patient human that he/she is Löbian (I cannot prove that he/she is sound or correct, note).
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The UDA results is that whatever you mean by physical for making comp meaningful, that physicalness has to emerge entirely and only, from a 'competition' between all universal numbers. There is no need to go out of arithmetic, and "worst", there is no possible use of going out of arithmetic, once betting on comp.
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By arithmetic I mean arithmetical truth, or the standard model of arithmetic, I don't mean a theory. I mean the whole set of true arithmetical propositions, or of their Gödel numbers.
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Bruno

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Onward!

Stephen

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http://iridia.ulb.ac.be/~marchal/

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