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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