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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