Hi Bruno,
what I undestand about the UD is that it generates all programs, a program
being simply a number from the set N.(1)
There exists an infinity of program which generates a set of growing function
(different set), all the computable growing function are generated by all
these programs(taken as a whole). Is this correct ? Is this metaset also
diagonalisable ? I said no, because if it was then there is a contradiction
with the premises that said that we generate *all* the programs that compute
growing functions, thus either we cannot generate those programs (but that
would be strange, that would means N is not enumerable ?) or the "metaset" is
not diagonalisable...
Where do I fail in my understanding ?
Thanks,
Quentin
(1) I still have another problem with that because a program to be run need
also the coding scheme (the instruction set of the turing machine that run
it), which instruction set the UD use ? or how it construct it ?
Le Mercredi 7 Juin 2006 15:11, Bruno Marchal a écrit :
> Le 06-juin-06, à 20:50, daddycaylor.domain.name.hidden a écrit :
> > Given a
> > (countably infinite) sequence of functions f1, f2, ..., you say that
> > fn(n)+1 must either be in the sequence OR not in the sequence.
>
> I am just showing constructively that if f1, f2,f3, ... is a well
> defined sequence of computable functions from N to N, then the
> "diagonal" function g (i.e. the one defined by g(n) = fn(n)+1) for each
> n) cannot belong to the sequence f1, f2, f3, ...
> The proof is constructive in the sense that if you give me some fk
> equal to g, I can generate a contradiction from that. The contradiction
> being that g(k) will be equal to g(k)+1.
>
> > But I will take some of my rare spare time (which I always have by
> > diagonalization)
>
> I hope you will explain to me how you do that :)
>
> > to think some more about this absoluteness of
> > computability and Church Thesis, etc. and try to understand this and
> > solve the puzzle of where your straw-man argument is wrong.
>
> OK, I let you think a little more then.
>
> > Speaking of straw-men, it seems you are saying that machines simply
> > running programs, without axioms and inference rules, are like zombies.
>
> Where am I saying that?
>
> > Zombies are how I would traditionally think of machines, but you seem
> > to be saying that the axioms and inference rules somehow breathe life
> > into the machine.
>
> Not really. Axioms and inference rule just makes it possible for the
> machine to develop (third person describable) beliefs. The relation
> between computation and proof are subtle. Let us be sure everyone
> understand Church thesis (and its non constructive price) before moving
> on the subject of theories and chatting machines. I could say things
> but it will adds confusions at this stage.
> Also zombie is a concept in the philosophy of mind, but we are not yet
> really talking about that.
>
> OK, i give the solution tomorrow. All right? (answer only if you prefer
> I give you more time, or else to make any other comments of course, but
> by default I give the answer tomorrow).
>
> Bruno
>
>
> http://iridia.ulb.ac.be/~marchal/
>
>
>
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Received on Wed Jun 07 2006 - 09:57:57 PDT