Re: Numbers in the Platonic Realm

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On 01 Nov 2012, at 06:19, meekerdb wrote:```
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```On 10/31/2012 6:58 PM, Stephen P. King wrote:  (actually it was Bruno)
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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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So u could be any number, depending on how you enumerated the functions and what bijection is used?
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Any number. I am not sure, the enumeration has to be given by an algorithm.
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But yes, the notion of computation, universality, etc. are intensional notion, and makes sense only relatively to the other number. That is why a often add "relative" before number.
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This should be obvious. The doctor who scan your brain will also have some flexibility in the encoding of your current local and relative state.
```I doubt it can encode it with the number 4, though.
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You might say, that 4 is for the fourth compact disk on the shell doctor, but then 4 is no more an encoding, but only a pointer to an encoding.
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Bruno

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

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