# Re: On the ontological status of elementary arithmetic

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On 01 Nov 2012, at 21:42, Stephen P. King wrote:```
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```On 11/1/2012 11:39 AM, Bruno Marchal wrote:
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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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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?
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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.
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```Dear Bruno,'

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

Bruno

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

Stephen

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

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