Bruno Marchal wrote:
> 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.

I have never denied that you can count or enunumerate machines or
i.e. attach unique numerical labels to them. The problem is in
your "Fix any universal machine". Given a string
of 1's and 0s  *wihout* a universal machine, and you have no
idea of which algorithm (non-universal machine) it is. Two
things are only identical if they have *all* their properties
in common (Leibniz's law). But *none*
of the propeties of the "machine" are detectable in the number

(Yopu can also count the even numbers off against the odd
numbers , but that hardly means that even numbers
are identical to odd numbers!)

>  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".

I do prefer!

>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.

Ye-e-es. But if all this is taking place in Platonia, the
only thing it *can* be is a number. But *that* number can't
be associated with a computaiton by *another* machine,
or you get infinite regress.

>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.

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