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Re: Formal systems that cannot possibly be incomplete except for unknowns and unknowable

From Mikko <mikko.levanto@iki.fi>
Newsgroups sci.logic
Subject Re: Formal systems that cannot possibly be incomplete except for unknowns and unknowable
Date 2025-05-05 12:50 +0300
Organization -
Message-ID <vva1ls$6uhv$1@dont-email.me> (permalink)
References <vv97ft$3fg66$1@dont-email.me>

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On 2025-05-05 02:23:56 +0000, olcott said:

> When we define formal systems as a finite list of basic facts and allow 
> semantic logical entailment as the only rule of inference we have 
> systems that can express any truth that can be expressed in language.
> 
> Also with such systems Undecidability is impossible. The only 
> incompleteness are things that are unknown or unknowable.

A formal system has a formal language. Unless the language is too
restricted for most interesting purposes the negation of every
sentence is another sentence. In a consistent system some sentence
is unprovable. If the negation of that system is also unprovable
then the system is incomplete.

None of the features of specified above prevents an expressible unprovable
sentence that has an unprovable negation. While adding one of the pair
to the basic facts there is nothing to prevent an infinite set of such
pairs.

-- 
Mikko

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Formal systems that cannot possibly be incomplete except for unknowns and unknowable olcott <polcott333@gmail.com> - 2025-05-04 21:23 -0500
  Re: Formal systems that cannot possibly be incomplete except for unknowns and unknowable Mikko <mikko.levanto@iki.fi> - 2025-05-05 12:50 +0300
    Re: Formal systems that cannot possibly be incomplete except for unknowns and unknowable olcott <polcott333@gmail.com> - 2025-05-05 10:39 -0500
      Re: Formal systems that cannot possibly be incomplete except for unknowns and unknowable Mikko <mikko.levanto@iki.fi> - 2025-05-06 13:02 +0300
  Re: Formal systems that cannot possibly be incomplete except for unknowns and unknowable Alan Mackenzie <acm@muc.de> - 2025-05-05 10:47 +0000
    Re: Formal systems that cannot possibly be incomplete except for unknowns and unknowable olcott <polcott333@gmail.com> - 2025-05-05 16:03 -0500
      Re: Formal systems that cannot possibly be incomplete except for unknowns and unknowable Alan Mackenzie <acm@muc.de> - 2025-05-06 08:30 +0000
        Re: Formal systems that cannot possibly be incomplete except for unknowns and unknowable Mikko <mikko.levanto@iki.fi> - 2025-05-06 13:04 +0300
  Re: Formal systems that cannot possibly be incomplete except for unknowns and unknowable Richard Damon <richard@damon-family.org> - 2025-05-05 07:04 -0400
    Re: Formal systems that cannot possibly be incomplete except for unknowns and unknowable Mr Flibble <flibble@red-dwarf.jmc.corp> - 2025-05-05 14:46 +0000
      Re: Formal systems that cannot possibly be incomplete except for unknowns and unknowable Richard Heathfield <rjh@cpax.org.uk> - 2025-05-05 16:51 +0100
        Re: Formal systems that cannot possibly be incomplete except for unknowns and unknowable Mr Flibble <flibble@red-dwarf.jmc.corp> - 2025-05-05 16:10 +0000
          Re: Formal systems that cannot possibly be incomplete except for unknowns and unknowable Richard Heathfield <rjh@cpax.org.uk> - 2025-05-05 17:59 +0100
      Re: Formal systems that cannot possibly be incomplete except for unknowns and unknowable Richard Damon <richard@damon-family.org> - 2025-05-05 21:08 -0400
    Re: Formal systems that cannot possibly be incomplete except for unknowns and unknowable olcott <polcott333@gmail.com> - 2025-05-05 10:31 -0500
      Re: Formal systems that cannot possibly be incomplete except for unknowns and unknowable Richard Damon <richard@damon-family.org> - 2025-05-05 21:11 -0400
        Re: Formal systems that cannot possibly be incomplete except for unknowns and unknowable olcott <polcott333@gmail.com> - 2025-05-05 21:26 -0500
          Re: Formal systems that cannot possibly be incomplete except for unknowns and unknowable Richard Damon <richard@damon-family.org> - 2025-05-06 07:16 -0400
      Re: Formal systems that cannot possibly be incomplete except for unknowns and unknowable olcott <polcott333@gmail.com> - 2025-05-05 23:27 -0500
        Re: Formal systems that cannot possibly be incomplete except for unknowns and unknowable Alan Mackenzie <acm@muc.de> - 2025-05-06 08:17 +0000
        Re: Formal systems that cannot possibly be incomplete except for unknowns and unknowable Richard Damon <richard@damon-family.org> - 2025-05-06 07:20 -0400
          Re: Formal systems that cannot possibly be incomplete except for unknowns and unknowable ---ELABORATED olcott <polcott333@gmail.com> - 2025-05-06 12:46 -0500
            Re: Formal systems that cannot possibly be incomplete except for unknowns and unknowable ---ELABORATED Richard Damon <richard@damon-family.org> - 2025-05-06 22:07 -0400
  Re: Formal systems that cannot possibly be incomplete except for unknowns and unknowable Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> - 2025-11-15 16:24 +0000
    Re: Formal systems that cannot possibly be incomplete except for unknowns and unknowable olcott <polcott333@gmail.com> - 2025-11-15 10:55 -0600
      Re: Formal systems that cannot possibly be incomplete except for unknowns and unknowable Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> - 2025-11-15 17:30 +0000
        Re: Formal systems that cannot possibly be incomplete except for unknowns and unknowable olcott <polcott333@gmail.com> - 2025-11-15 11:43 -0600
  Re: Formal systems that cannot possibly be incomplete except for unknowns and unknowable Tristan Wibberley <tristan.wibberley+netnews2@alumni.manchester.ac.uk> - 2025-11-15 16:49 +0000
    Re: Formal systems that cannot possibly be incomplete except for unknowns and unknowable olcott <polcott333@gmail.com> - 2025-11-15 11:19 -0600
  Re: Kaz ---Keith said DDD simulated by HHH is equivalent to DDD calling itself olcott <polcott333@gmail.com> - 2025-11-16 09:29 -0600
    Re: Kaz ---Keith said DDD simulated by HHH is equivalent to DDD calling itself HAL 9000 <hal@discovery.nasa> - 2025-11-16 16:46 +0000
      Re: Kaz ---Keith said DDD simulated by HHH is equivalent to DDD calling itself olcott <polcott333@gmail.com> - 2025-11-16 11:07 -0600
      DD simulated by HHH cannot possibly terminate normally olcott <polcott333@gmail.com> - 2025-11-16 11:20 -0600
      Re: Kaz ---Keith said DDD simulated by HHH is equivalent to DDD calling itself "Chris M. Thomasson" <chris.m.thomasson.1@gmail.com> - 2025-11-16 13:18 -0800

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