On 01 Nov 2012, at 21:42, Stephen P. King wrote:

On 11/1/2012 11:39 AM, Bruno Marchal wrote:

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.

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.

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.



OK, but this does not answer my question. What is the ontological level mechanism that distinguishes the u and the x and the y from each other?

The one you have chosen above. But let continue to use elementary arithmetic, as everyone learn it in school. So the answer is: elementary arithmetic.

Dear Bruno,'

If there is no entity to chose the elementary arithmetic, how is it chosen or even defined such that there exist arithmetic statements that can possibly be true or false?


Nobody needs to do the choice, as the choice is irrelevant for the truth. If someone choose the combinators, the proof of "1+1= 2" will be very long, and a bit awkward, but the proof of KKK = K, will be very short. If someone chose elementary arithmetic, the proof of 1+1=2 will be very short (Liz found it on FOAR), but the proof that KKK = K, will be long and a bit awkward. The fact is that 1+1=2, and KKK=K, are true, independently of the choice of the theory, and indeed independently of the existence of the theories.





We can assume some special Realm or entity does the work of choosing the consistent set of arithmetical statements or, as I suggest, we can consider the totality of all possible physical worlds

As long as you make your theory clearer, I can't understand what you mean by "physical world", "possible", "totality", etc.




as the implementers of arithmetic statements and thus their "provers". Possible physical worlds, taken as a single aggregate, is just as timeless and non-located as the Platonic Realm and yet we don't need any special pleading for us to believe in them. ;-)

?

Bruno



My thinking here follows the reasoning of Jaakko Hintikka. Are you familiar with it? Game theoretic semantics for Proof theory
 --
Onward!

Stephen

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



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