On 13/08/07, Bruno Marchal <[EMAIL PROTECTED]> wrote:

> Question to David, and others who could be interested:  is the notion
> of enumerable and non enumerable set clear? Can you explain why the set
> of functions from N to N is not enumerable?

Do please remind us.  "Off the top of my head", do you mean, by
non-enumerable, arbitrary extensibility by the generation of new
members via diagonalisation?

> Do you people know the difference between ordinal and cardinal (I know
> some knows 'course).

Yes

> I don't think Church thesis can be grasped
> conceptually without the understanding that the class of programmable
> functions is closed for the diagonalization procedure.

Please explain 'programmable functions' and 'closed for the
diagonalisation procedure'.

> Do everyone
> (interested) know how to prove the non enumerability of the subset of N
> by diagonalization?

Which subset do you mean?  I've encountered the
diagonalisation/enumerability argument, assuming it's the one I
referred to above.

> Let us go slow and deep so that everybody can understand, once and for
> all.  OK?

Definitely OK.

David

>
>
> Le 13-août-07, à 13:29, Kim Jones a écrit :
>
> > where he appears to serve the option of being machine or some other
> > order of being. I must confess that I still don't understand the
> > ontology of angels as opposed to machines but I'm sure his reply
> > contains the reason
>
>
> Don't worry, I will try to explain.
>
>
> Question to David, and others who could be interested:  is the notion
> of enumerable and non enumerable set clear? Can you explain why the set
> of functions from N to N is not enumerable?
>
> Just say no, and I go back to Cantor, the one who discussed with the
> pope about the question of naming infinities (!), and indeed the one
> who will discover (or invent) the varieties of infinities.
>
> Do you people know the difference between ordinal and cardinal (I know
> some knows 'course). I don't think Church thesis can be grasped
> conceptually without the understanding that the class of programmable
> functions is closed for the diagonalization procedure. Do everyone
> (interested) know how to prove the non enumerability of the subset of N
> by diagonalization?
>
> Let us go slow and deep so that everybody can understand, once and for
> all.  OK?
>
>
> Bruno
>
>
> http://iridia.ulb.ac.be/~marchal/
>
>
> >
>

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