# Re: on formally describable universes and measures

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
> >computable infinite reals in the sense that it outputs their
> >finite descriptions or programs.
>
> I am not sure I understand you here.
> Are you aware that the set of descriptions of computable reals
> is not closed for the diagonalisation procedure.
> That is: you cannot generate all (and only) descriptions of
> computable reals. The algorithm you are mentionning does not exist.
> You can only generate a superset of the set of all computable reals.
> That set (of description of all computable reals) is even
> constructively not *recursively enumerable* in the sense that,
> if you give me an algorithm generating the (description of)
> computable real, I can transform it for building a computable
> real not being generated by your algorithm. I guess you know that.
>
> That is why most formal constructivists consider their "set of
> constructive reals" as subset
> of the Turing computable reals. For exemple you can choose the
> set of reals which are provably
> computable in some formal system (like the system F by Girard,
> in which you can formalize ..., well Hilbert space and probably
> the whole of the *constructive* part of Tegmark mathematical ontology!
> That is very nice and big but not enough big for my purpose which
> has some necessarily non constructive feature.
> About natural numbers and machines I am a classical
> 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
> for the sake of the reasoning?

I am not even sure what exactly a Platonist is.

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

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

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