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From: <juergen.domain.name.hidden>

Date: Thu, 25 Jan 2001 14:17:46 +0100

On Sun Jan 21 15:29:39 2001 Bruno Marchal wrote

*> There is nothing magical hidden in my reasoning, and I have no doubt
*

*> you will understand it soon or late :-)
*

*> (I surely agree that in some steps I am a little quick ; my
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*> pedagogical problem is that nobody agrees on which steps are
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*> easy or difficult).
*

I think you tend to write too much, not too little. None of your

"pedagogical" examples answers my question though. Let me cut and paste

once more:

You are repeatedly talking about universes generated by a dovetailer. All

these universes obviously must be computable, otherwise the dovetailer

could not compute them. You claim that there is no computable universe

to which we belong, but the very tool you are using generates lots of

universes to which we belong. Can you explain this contradiction?

On Sat Jan 20 15:32:06 2001 Bruno Marchal wrote

*> Juergen wrote
*

*> >The infinite computational histories are countable. The continuum is not.
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*> >The concepts of dovetailing and continuum are incompatible.
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*>
*

*> But you can write a program which dovetails on the reals !
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*> I have already explain in the list why there is no contradiction
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*> with cantor diagonal proof of the non enumerability of the real.
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*> It is no more astonishing than the existence of a short program
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*> which generate all the finite string, including those which
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*> are very long and chaitin incompressible. The trick is to generate
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*> them *all*.
*

*> ... but the dovetailer generates, for each real, all its
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*> bigger and bigger prefixes, and that is called traditionnaly,
*

*> generating the real. And the dovetailer do that for each real,
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*> and so generates all the uncountably many reals.
*

I am afraid this is nonsense. Obviously I can count the outputs of

your dovetailer. I can count the time steps it consumes. Hence

the dovetailer cannot possibly generate uncountably many things.

You are just generating countably many prefixes of reals, each prefix

being the start of uncountably many reals (most of which do not exist

from any constructive point of view). What you are really generating

is not the reals but countably many representatives of subsets of the

reals, each subset containing uncountably many elements. This has

nothing to do with generating or representing the elements themselves.

Here is the best you can achieve with a dovetailer. You can

extract representations of all the countably many computable reals.

How? Systematically enumerate all possible input strings, viewing them as

starts of programs to be executed in dovetail fashion. This excludes

many strings that cannot be input programs because one of their prefixes

already is a self-delimiting program that will never request additional

bits. Some input strings, however, are valid, finite, possibly

nonhalting programs (such as the algorithm computing the decimal expansion

of 1/3). They generate possibly infinite objects (such as 0.33333...).

Make a list containing all current programs or program prefixes.

Update the list whenever one of its elements requests a new bit. This

will eventually give you finite representations of each computable real.

Furthermore, each infinite string computable in countable time will

be computed in countable time by an optimal dovetailer like

the one in http://rapa.idsia.ch/~juergen/toesv2/node28.html

But your own so-called "dovetailer" essentially is

the extremely inefficient algorithm called ALPHABET in

http://rapa.idsia.ch/~juergen/toesv2/node27.html . ALPHABET simply

lists all bitstrings ordered by size and separated by blanks. Note that

countably many steps are insufficient for ALPHABET to print any infinite

string! So ALPHABET clearly is the wrong approach when it comes to

generating all computable universes.

Once more: The concepts of dovetailing and continuum are incompatible!

Juergen

Received on Thu Jan 25 2001 - 05:22:54 PST

Date: Thu, 25 Jan 2001 14:17:46 +0100

On Sun Jan 21 15:29:39 2001 Bruno Marchal wrote

I think you tend to write too much, not too little. None of your

"pedagogical" examples answers my question though. Let me cut and paste

once more:

You are repeatedly talking about universes generated by a dovetailer. All

these universes obviously must be computable, otherwise the dovetailer

could not compute them. You claim that there is no computable universe

to which we belong, but the very tool you are using generates lots of

universes to which we belong. Can you explain this contradiction?

On Sat Jan 20 15:32:06 2001 Bruno Marchal wrote

I am afraid this is nonsense. Obviously I can count the outputs of

your dovetailer. I can count the time steps it consumes. Hence

the dovetailer cannot possibly generate uncountably many things.

You are just generating countably many prefixes of reals, each prefix

being the start of uncountably many reals (most of which do not exist

from any constructive point of view). What you are really generating

is not the reals but countably many representatives of subsets of the

reals, each subset containing uncountably many elements. This has

nothing to do with generating or representing the elements themselves.

Here is the best you can achieve with a dovetailer. You can

extract representations of all the countably many computable reals.

How? Systematically enumerate all possible input strings, viewing them as

starts of programs to be executed in dovetail fashion. This excludes

many strings that cannot be input programs because one of their prefixes

already is a self-delimiting program that will never request additional

bits. Some input strings, however, are valid, finite, possibly

nonhalting programs (such as the algorithm computing the decimal expansion

of 1/3). They generate possibly infinite objects (such as 0.33333...).

Make a list containing all current programs or program prefixes.

Update the list whenever one of its elements requests a new bit. This

will eventually give you finite representations of each computable real.

Furthermore, each infinite string computable in countable time will

be computed in countable time by an optimal dovetailer like

the one in http://rapa.idsia.ch/~juergen/toesv2/node28.html

But your own so-called "dovetailer" essentially is

the extremely inefficient algorithm called ALPHABET in

http://rapa.idsia.ch/~juergen/toesv2/node27.html . ALPHABET simply

lists all bitstrings ordered by size and separated by blanks. Note that

countably many steps are insufficient for ALPHABET to print any infinite

string! So ALPHABET clearly is the wrong approach when it comes to

generating all computable universes.

Once more: The concepts of dovetailing and continuum are incompatible!

Juergen

Received on Thu Jan 25 2001 - 05:22:54 PST

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