Hi,
I have come back from Bergen (it was very nice) and I have read the
last posts and I will make some comments in order.
Peter D. Jones said some time ago, after I said that I will identify
"(digital) machines" with number; he said:
"You can't".
Of course I can. This is a key point, and it is not obvious. But I can,
and the main reason is Church Thesis (CT). Fix any universal machine,
then, by CT, all partial computable function can be arranged in a
recursively enumerable list F1, F2, F3, F4, F5, etc. It is the list of
the Fi, which has this fundamental and amazing property that it is
close for the diagonalization operation. I have explain this at length
in some posts to George and Tom. The identification between number and
machine is similar to the geometric identification of real numbers and
points once a coordinate system has been fixed. If you prefer I should
have said "associate" instead of "identifying". In computer science, a
fixed universal machine plays the role of a coordinate system in
geometry. That's all. With Church Thesis, we don't even have to name
the particular universal machine, it could be a universal cellular
automaton (like the game of life), or Python, Robinson Aritmetic,
Matiyasevich Diophantine universal polynomial, Java, ... rational
complex unitary matrices, universal recursive group or ring, billiard
ball, whatever. Then just list the programs describable in the language
of that machine to get the Fi. The domain of the Fi gives the Wi which
can be shown to be the mechanically generable sets (of numbers, or
entities nameable, associable or identifiable with numbers in some
context like the partial recursive (computable) functions).
David Nyman wrote:
> [Scene: Night-time. Fathers Ted and Dougal are in bed.
>
> Ted: "Dougal, that's a great idea! Can you tell me more?"
> Dougal: "Whoa, Ted - I want out! I can't take the pressure."]
I love them :)
Bruno
PS For a while I will let Colin and David continue their discussion
before interfering. I have other comments but will regroup them for
making minimal the number of posts. Just try not to confuse
computability and provability (in a formal system or by a machine).
Computability is an absolute notion (with CT), but provability is a
relative (with respect to a machine) notion. Put in another way:
computations admits a universal dovetailer which generates and run all
computations, but there is no universal dovetailer for proofs. By
Godel's theorem for any proof system (with checkable proofs) you can
build a richer proof system. Without this all 'hypostases' would
collapse, for example, and the interview with a universal machine would
be ... infinitely boring, and probably unrelated to both quanta and
qualia.
Not also that the relative aspect of provability does not prevent the
finding of universal feature of provability (like obeying G and G* for
example). Note also that most provability systems (like RA and PA or
ZF) are universal computers, but still only relative (and different)
theorem provers. Seen as a universal machine, RA and PA and ZF can
simulate each others. As provability systems, ZF extends properly PA
which extends properly RA.
ZF and PA are lobian machines. RA isn't.
http://iridia.ulb.ac.be/~marchal/
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Received on Wed Oct 18 2006 - 09:13:47 PDT