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From: <juergen.domain.name.hidden>

Date: Wed, 28 Feb 2001 10:33:38 +0100

*> >The best you can achieve is an algorithm that outputs at least the
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*> >computable infinite reals in the sense that it outputs their
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*> >finite descriptions or programs.
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*>
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*> I am not sure I understand you here.
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*> Are you aware that the set of descriptions of computable reals
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*> is not closed for the diagonalisation procedure.
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*> That is: you cannot generate all (and only) descriptions of
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*> computable reals. The algorithm you are mentionning does not exist.
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*> You can only generate a superset of the set of all computable reals.
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*> That set (of description of all computable reals) is even
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*> constructively not *recursively enumerable* in the sense that,
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*> if you give me an algorithm generating the (description of)
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*> computable real, I can transform it for building a computable
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*> real not being generated by your algorithm. I guess you know that.
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*>
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*> That is why most formal constructivists consider their "set of
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*> constructive reals" as subset
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*> of the Turing computable reals. For exemple you can choose the
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*> set of reals which are provably
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*> computable in some formal system (like the system F by Girard,
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*> in which you can formalize ..., well Hilbert space and probably
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*> the whole of the *constructive* part of Tegmark mathematical ontology!
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*> That is very nice and big but not enough big for my purpose which
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*> has some necessarily non constructive feature.
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*> About natural numbers and machines I am a classical
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*> platonist. About real numbers I have no definite opinion.
*

The describable reals are those computable in the limit by finite GTM

programs. There is a program that eventually outputs all finite programs

for a given GTM, by listing all possible program prefixes. Sure, in

general one cannot decide whether a given prefix in the current list

indeed is a complete program, or whether a given prefix will still grow longer

by requesting additional input bits, or whether it will even grow forever,

or whether it will really compute a real in the limit, or whether it won't

because some of its output bits will flip back and forth forever:

http://rapa.idsia.ch/~juergen/toesv2/node9.html

But what do such undecidability results really mean? Are they relevant in

any way? They do not imply that I cannot write down finite descriptions

of the describable reals - they just mean that in general I cannot know

at a given time which of the current list elements are indeed complete

descriptions of some real, and which are not. Still, after finite time

my list of symbol sequences will contain a complete description of any

given computable real.

Thus undecidable properties do not necessarily make things nonconstructive.

*> Can you imagine yourself as a Platonist for a while, if only
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*> for the sake of the reasoning?
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I am not even sure what exactly a Platonist is.

*> > Do I need any additional
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*> >preliminaries to realize why I "genuinely fail to understand your
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*> >invariance lemma"?
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*>
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*> Sure. The "delays" question for exemple. Let us follow Jesse Mazer
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*> idea of torture. Suppose I duplicate you and reconstitute you, not
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*> in Washington and Moscow but in some Paradise and some Hell.
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*> Would you feel more comfortable if I tell you
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*> I will reconstitute you in paradise tomorow and in hell only in
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*> 3001 after C. ? Is that what you would choose?
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Choose? Do I have a choice? Which is my choice?

*> Computationalism is more a human right than a doctrinal truth.
*

I skipped this statement and related ones...

Juergen

Received on Wed Feb 28 2001 - 01:36:39 PST

Date: Wed, 28 Feb 2001 10:33:38 +0100

The describable reals are those computable in the limit by finite GTM

programs. There is a program that eventually outputs all finite programs

for a given GTM, by listing all possible program prefixes. Sure, in

general one cannot decide whether a given prefix in the current list

indeed is a complete program, or whether a given prefix will still grow longer

by requesting additional input bits, or whether it will even grow forever,

or whether it will really compute a real in the limit, or whether it won't

because some of its output bits will flip back and forth forever:

http://rapa.idsia.ch/~juergen/toesv2/node9.html

But what do such undecidability results really mean? Are they relevant in

any way? They do not imply that I cannot write down finite descriptions

of the describable reals - they just mean that in general I cannot know

at a given time which of the current list elements are indeed complete

descriptions of some real, and which are not. Still, after finite time

my list of symbol sequences will contain a complete description of any

given computable real.

Thus undecidable properties do not necessarily make things nonconstructive.

I am not even sure what exactly a Platonist is.

Choose? Do I have a choice? Which is my choice?

I skipped this statement and related ones...

Juergen

Received on Wed Feb 28 2001 - 01:36:39 PST

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