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> >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