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
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
The one you have chosen above. But let continue to use elementary
arithmetic, as everyone learn it in school. So the answer is:
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? 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 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. ;-)
My thinking here follows the reasoning of Jaakko Hintikka. Are you
familiar with it? Game theoretic semantics for Proof theory
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