Re: WELL WHICH IS IT... ?

stephen_at_nomail.com
Date: 01/21/05

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    Date: 21 Jan 2005 06:48:16 GMT

    In sci.math The Ghost In The Machine <ewill@sirius.athghost7038suus.net> wrote:
    : In sci.logic, Will Twentyman
    : <wtwentyman@read.my.sig>
    : wrote
    : on Thu, 20 Jan 2005 22:20:37 -0500
    :> So, apparently Ghost and I interpretted the above differently. I
    :> interpretted it as "Given the list of computables, and a real number X,
    :> X can have all its digits as a single entry on the list, but not be on
    :> the list."
    :>
    :> With that reading, it is equivalent to "Given the list of computables,
    :> and a real number X, X can be on the list, but not be on the list,"
    :> which is obviously false.
    :>
    :> I'm not sure how Ghost read it.
    :>

    : Hmm....well, if one takes my favorite class TX_10, then it
    : is clear that all of pi's finite prefixes are within TX_10,
    : but pi, being irrational, is not.

    : Or, one can take a constant such as Chaitin's Omega.
    : This constant is a perfectly good real number, but is
    : not computable, therefore cannot be on any list of computables.
    : However, all of its prefixes easily can be (since one way
    : of "computing" a finite sequence is simply to recite it).

    Of course, even though all of the prefixes of Chaitin's Omega
    will appear on a list of computables we cannot actually determine
    that they are prefixes of Chaitin's Omega.

    Stephen

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    Relevant Pages

    • Re: WELL WHICH IS IT... ?
      ... apparently Ghost and I interpretted the above differently. ... :> interpretted it as "Given the list of computables, and a real number X, ... is clear that all of pi's finite prefixes are within TX_10, ... even though all of the prefixes of Chaitin's Omega ...
      (sci.logic)
    • Re: WELL WHICH IS IT... ?
      ... apparently Ghost and I interpretted the above differently. ... :> interpretted it as "Given the list of computables, and a real number X, ... is clear that all of pi's finite prefixes are within TX_10, ... even though all of the prefixes of Chaitin's Omega ...
      (sci.math)
    • Re: WELL WHICH IS IT... ?
      ... even though all of the prefixes of Chaitin's Omega ... >>will appear on a list of computables we cannot actually determine ... For any n there is a TM that computes the first n digits (if that's what ...
      (comp.theory)
    • Re: WELL WHICH IS IT... ?
      ... even though all of the prefixes of Chaitin's Omega ... >>will appear on a list of computables we cannot actually determine ... For any n there is a TM that computes the first n digits (if that's what ...
      (sci.math)
    • Re: WELL WHICH IS IT... ?
      ... even though all of the prefixes of Chaitin's Omega ... >>will appear on a list of computables we cannot actually determine ... For any n there is a TM that computes the first n digits (if that's what ...
      (sci.logic)
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