Le 03-sept.-06, à 12:17, Stathis Papaioannou a écrit :
> Sure, the computation is the same (although I find it much harder to
> imagine the computation as a pure Platonic
> object than I do numbers), but its expression and implementation are
> infinitely variable.
With CT you can see "all the computations" as the collection of the
computational states get by the Universal Dovetailer.
In term of the Fi, you can related "all the computations" with the set
of the trace of length z of the computation of Fx on the input y, for
all z, x, y positive integers.
This is not obvious at all, and can be seen as a consequence of Church
thesis. As I have shown or try to show this already entails
incompleteness of theories.
Note that a computation of Fx(y) = t can be seen as a proof that it
exists a z such that z codes a proof of Fx(y) = t . It has the shape of
"ExP(x,y)", that is a sigma1 sentence. So, with CT, "All the
computational states" is captured by the set of all true sigma1
sentences, and their proof (finite or infinite) gives the finite
computations: the DU-accessible states. Note the DU generates also the
infinite proof (thanks to God) of the *wrong* sigma1 sentence which
will determine in part the first person measure on all computations. I
guess I am too technical ...
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