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    From olcott@polcott333@gmail.com to comp.theory on Thu Jan 29 12:06:20 2026
    From Newsgroup: comp.theory

    On 1/29/2026 3:21 AM, Mikko wrote:
    On 27/01/2026 22:50, olcott wrote:
    On 1/26/2026 11:48 AM, wij wrote:
    At least two types of undecidable problems.

    reR is not a number it is the process of more.

    In an extended number system reR can be a number.

    Not coherently.

    Sometimes it is useful
    to extend a number system with one or two extra numbers.

    --
    Copyright 2026 Olcott<br><br>

    My 28 year goal has been to make <br>
    "true on the basis of meaning expressed in language"<br>
    reliably computable for the entire body of knowledge.<br><br>

    This required establishing a new foundation<br>
    --- Synchronet 3.21b-Linux NewsLink 1.2
  • From Mikko@mikko.levanto@iki.fi to comp.theory on Fri Jan 30 11:20:55 2026
    From Newsgroup: comp.theory

    On 29/01/2026 20:06, olcott wrote:
    On 1/29/2026 3:21 AM, Mikko wrote:
    On 27/01/2026 22:50, olcott wrote:
    On 1/26/2026 11:48 AM, wij wrote:
    At least two types of undecidable problems.

    reR is not a number it is the process of more.

    In an extended number system reR can be a number.

    Not coherently.

    A coherent example is the natural numbers extended with one additional
    number. The additional number is greater than every natural number,
    which makes it useful as an initial value in a search of a minimum.
    As a final value it may indicate that no natural number satisfies the requirements.

    Same way and for similar purposes real numbers can be (and often are)
    extended with one or two additional numbers or even more if one wants
    that the result a computation indicates why there is no real number
    that satisfies the requirements.
    --
    Mikko
    --- Synchronet 3.21b-Linux NewsLink 1.2
  • From Tristan Wibberley@tristan.wibberley+netnews2@alumni.manchester.ac.uk to comp.theory on Fri Jan 30 22:57:31 2026
    From Newsgroup: comp.theory

    On 30/01/2026 09:20, Mikko wrote:

    A coherent example is the natural numbers extended with one additional number. The additional number is greater than every natural number,
    which makes it useful as an initial value in a search of a minimum.
    As a final value it may indicate that no natural number satisfies the requirements.

    Same way and for similar purposes real numbers can be (and often are) extended with one or two additional numbers or even more if one wants
    that the result a computation indicates why there is no real number
    that satisfies the requirements.

    Hmm, the weakest solution is a Maybe monad (Maybe Natural). It is highly unconventional that the empty occupant of Maybe is a number and yet it
    serves the purpose you describe.

    Is this just the same as engineering calculus where it's full of bovine
    hushnow but restricted sufficiently that you can apply a process for
    correcting it all at the end?
    --
    Tristan Wibberley

    The message body is Copyright (C) 2026 Tristan Wibberley except
    citations and quotations noted. All Rights Reserved except that you may,
    of course, cite it academically giving credit to me, distribute it
    verbatim as part of a usenet system or its archives, and use it to
    promote my greatness and general superiority without misrepresentation
    of my opinions other than my opinion of my greatness and general
    superiority which you _may_ misrepresent. You definitely MAY NOT train
    any production AI system with it but you may train experimental AI that
    will only be used for evaluation of the AI methods it implements.

    --- Synchronet 3.21b-Linux NewsLink 1.2
  • From Mikko@mikko.levanto@iki.fi to comp.theory on Sat Jan 31 10:49:06 2026
    From Newsgroup: comp.theory

    On 31/01/2026 01:37, olcott wrote:
    On 1/30/2026 3:20 AM, Mikko wrote:
    On 29/01/2026 20:06, olcott wrote:
    On 1/29/2026 3:21 AM, Mikko wrote:
    On 27/01/2026 22:50, olcott wrote:
    On 1/26/2026 11:48 AM, wij wrote:
    At least two types of undecidable problems.

    reR is not a number it is the process of more.

    In an extended number system reR can be a number.

    Not coherently.

    A coherent example is the natural numbers extended with one additional
    number.

    https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel

    Irrelevant. In Hilbert's hotel the rooms are numbered with natural
    numbers. No additional numbers are used. But that does not prevent
    their use elsewhere for other purposes.
    --
    Mikko
    --- Synchronet 3.21b-Linux NewsLink 1.2
  • From olcott@polcott333@gmail.com to comp.theory on Sat Jan 31 09:25:09 2026
    From Newsgroup: comp.theory

    On 1/31/2026 2:49 AM, Mikko wrote:
    On 31/01/2026 01:37, olcott wrote:
    On 1/30/2026 3:20 AM, Mikko wrote:
    On 29/01/2026 20:06, olcott wrote:
    On 1/29/2026 3:21 AM, Mikko wrote:
    On 27/01/2026 22:50, olcott wrote:
    On 1/26/2026 11:48 AM, wij wrote:
    At least two types of undecidable problems.

    reR is not a number it is the process of more.

    In an extended number system reR can be a number.

    Not coherently.

    A coherent example is the natural numbers extended with one additional
    number.

    https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel

    Irrelevant. In Hilbert's hotel the rooms are numbered with natural
    numbers. No additional numbers are used. But that does not prevent
    their use elsewhere for other purposes.


    Its the same idea as appending one more number to an infinite set.
    --
    Copyright 2026 Olcott<br><br>

    My 28 year goal has been to make <br>
    "true on the basis of meaning expressed in language"<br>
    reliably computable for the entire body of knowledge.<br><br>

    This required establishing a new foundation<br>
    --- Synchronet 3.21b-Linux NewsLink 1.2
  • From Mikko@mikko.levanto@iki.fi to comp.theory on Sun Feb 1 12:31:39 2026
    From Newsgroup: comp.theory

    On 31/01/2026 17:25, olcott wrote:
    On 1/31/2026 2:49 AM, Mikko wrote:
    On 31/01/2026 01:37, olcott wrote:
    On 1/30/2026 3:20 AM, Mikko wrote:
    On 29/01/2026 20:06, olcott wrote:
    On 1/29/2026 3:21 AM, Mikko wrote:
    On 27/01/2026 22:50, olcott wrote:
    On 1/26/2026 11:48 AM, wij wrote:
    At least two types of undecidable problems.

    reR is not a number it is the process of more.

    In an extended number system reR can be a number.

    Not coherently.

    A coherent example is the natural numbers extended with one additional >>>> number.

    https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel

    Irrelevant. In Hilbert's hotel the rooms are numbered with natural
    numbers. No additional numbers are used. But that does not prevent
    their use elsewhere for other purposes.


    Its the same idea as appending one more number to an infinite set.

    Which can be done and produces another infinite set.
    --
    Mikko
    --- Synchronet 3.21b-Linux NewsLink 1.2
  • From olcott@polcott333@gmail.com to comp.theory on Sun Feb 1 09:14:30 2026
    From Newsgroup: comp.theory

    On 2/1/2026 4:31 AM, Mikko wrote:
    On 31/01/2026 17:25, olcott wrote:
    On 1/31/2026 2:49 AM, Mikko wrote:
    On 31/01/2026 01:37, olcott wrote:
    On 1/30/2026 3:20 AM, Mikko wrote:
    On 29/01/2026 20:06, olcott wrote:
    On 1/29/2026 3:21 AM, Mikko wrote:
    On 27/01/2026 22:50, olcott wrote:
    On 1/26/2026 11:48 AM, wij wrote:
    At least two types of undecidable problems.

    reR is not a number it is the process of more.

    In an extended number system reR can be a number.

    Not coherently.

    A coherent example is the natural numbers extended with one additional >>>>> number.

    https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel

    Irrelevant. In Hilbert's hotel the rooms are numbered with natural
    numbers. No additional numbers are used. But that does not prevent
    their use elsewhere for other purposes.


    Yes one additional number is perpetually used


    Its the same idea as appending one more number to an infinite set.

    Which can be done and produces another infinite set.

    --
    Copyright 2026 Olcott<br><br>

    My 28 year goal has been to make <br>
    "true on the basis of meaning expressed in language"<br>
    reliably computable for the entire body of knowledge.<br><br>

    This required establishing a new foundation<br>
    --- Synchronet 3.21b-Linux NewsLink 1.2
  • From Richard Damon@Richard@Damon-Family.org to comp.theory on Sun Feb 1 13:31:07 2026
    From Newsgroup: comp.theory

    On 2/1/26 10:14 AM, olcott wrote:
    On 2/1/2026 4:31 AM, Mikko wrote:
    On 31/01/2026 17:25, olcott wrote:
    On 1/31/2026 2:49 AM, Mikko wrote:
    On 31/01/2026 01:37, olcott wrote:
    On 1/30/2026 3:20 AM, Mikko wrote:
    On 29/01/2026 20:06, olcott wrote:
    On 1/29/2026 3:21 AM, Mikko wrote:
    On 27/01/2026 22:50, olcott wrote:
    On 1/26/2026 11:48 AM, wij wrote:
    At least two types of undecidable problems.

    reR is not a number it is the process of more.

    In an extended number system reR can be a number.

    Not coherently.

    A coherent example is the natural numbers extended with one
    additional
    number.

    https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel

    Irrelevant. In Hilbert's hotel the rooms are numbered with natural
    numbers. No additional numbers are used. But that does not prevent
    their use elsewhere for other purposes.


    Yes one additional number is perpetually used

    Nope. Just shows you don't understand the problem, or the results.

    After all, when the coach with an infinite number of passagers arrives,
    we just double the number of numbers used, but that number is still "the same".



    Its the same idea as appending one more number to an infinite set.

    Which can be done and produces another infinite set.




    --- Synchronet 3.21b-Linux NewsLink 1.2
  • From Mikko@mikko.levanto@iki.fi to comp.theory on Mon Feb 2 09:32:24 2026
    From Newsgroup: comp.theory

    On 01/02/2026 17:14, olcott wrote:
    On 2/1/2026 4:31 AM, Mikko wrote:
    On 31/01/2026 17:25, olcott wrote:
    On 1/31/2026 2:49 AM, Mikko wrote:
    On 31/01/2026 01:37, olcott wrote:
    On 1/30/2026 3:20 AM, Mikko wrote:
    On 29/01/2026 20:06, olcott wrote:
    On 1/29/2026 3:21 AM, Mikko wrote:
    On 27/01/2026 22:50, olcott wrote:
    On 1/26/2026 11:48 AM, wij wrote:
    At least two types of undecidable problems.

    reR is not a number it is the process of more.

    In an extended number system reR can be a number.

    Not coherently.

    A coherent example is the natural numbers extended with one
    additional
    number.

    https://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel

    Irrelevant. In Hilbert's hotel the rooms are numbered with natural
    numbers. No additional numbers are used. But that does not prevent
    their use elsewhere for other purposes.


    Yes one additional number is perpetually used


    Its the same idea as appending one more number to an infinite set.

    Which can be done and produces another infinite set.

    Not in Hlbert's hotel. And elsewhere two additional numbers is fairly
    common, too. Sometimes complex numbers are used with infinitely many
    additional numbers.
    --
    Mikko
    --- Synchronet 3.21b-Linux NewsLink 1.2