• a Proof of G in F is contradictory

    From olcott@21:1/5 to All on Thu Apr 20 11:04:50 2023
    XPost: sci.logic, sci.math, alt.philosophy

    It turns out that the only reason that Gödel’s G is not provable in F is that G is contradictory in F.

    When Gödel’s G asserts that it is unprovable in F it is asserting that
    there is no sequence of inference steps in F that derives G.

    *A proof of G in F requires a sequence of inference steps in F that*
    *proves there is no such sequence of inference steps in F, a*
    *contradiction*

    *This is like René Descartes saying*
    “I think therefore thoughts do not exist”

    The reason why G cannot be proved in F is that the proof of G in F is contradictory in F, thus Gödel was wrong when he said the reason is that
    F is incomplete. No formal system is ever supposed to be able to prove a contradiction.

    Now we can see both THAT G cannot be proved in F and perhaps for the
    first time see WHY G cannot be proved in F. The “incompleteness”
    conclusion has been refuted.

    To be honest we would have to rename Gödel’s “incompleteness” theorem to Olcott’s “can’t prove a contradiction” theorem.

    --
    Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
    hits a target no one else can see." Arthur Schopenhauer

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Richard Damon@21:1/5 to olcott on Fri Apr 21 08:00:03 2023
    XPost: sci.logic, sci.math, alt.philosophy

    On 4/20/23 12:04 PM, olcott wrote:
    It turns out that the only reason that Gödel’s G is not provable in F is that G is contradictory in F.

    When Gödel’s G asserts that it is unprovable in F it is asserting that there is no sequence of inference steps in F that derives G.

    *A proof of G in F requires a sequence of inference steps in F that*
    *proves there is no such sequence of inference steps in F, a*
    *contradiction*

    *This is like René Descartes saying*
    “I think therefore thoughts do not exist”

    The reason why G cannot be proved in F is that the proof of G in F is contradictory in F, thus Gödel was wrong when he said the reason is that
    F is incomplete. No formal system is ever supposed to be able to prove a contradiction.

    Now we can see both THAT G cannot be proved in F and perhaps for the
    first time see WHY G cannot be proved in F. The “incompleteness” conclusion has been refuted.

    To be honest we would have to rename Gödel’s “incompleteness” theorem to
    Olcott’s “can’t prove a contradiction” theorem.


    But you keep using the wrong statement for G, ikely because you just
    don't understand the proof.

    G does NOT assert that it is unprovable in F, that is just a conclusion
    derived from G in Meta-F.

    G is actually a statement that there does not exist a whole number that satisfies a property expressed as a primitive recursive relationship.

    Such a statement CAN'T be "contradictory" as either such a number exists
    or it doesn't.

    Of course, since you are too stupid to understand that statement, you
    mix up which system you are talking in and what statement you are
    working on.

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From olcott@21:1/5 to Richard Damon on Fri Apr 21 10:27:12 2023
    XPost: sci.logic, sci.math, alt.philosophy

    On 4/21/2023 7:00 AM, Richard Damon wrote:
    On 4/20/23 12:04 PM, olcott wrote:
    It turns out that the only reason that Gödel’s G is not provable in F is >> that G is contradictory in F.

    When Gödel’s G asserts that it is unprovable in F it is asserting that
    there is no sequence of inference steps in F that derives G.

    *A proof of G in F requires a sequence of inference steps in F that*
    *proves there is no such sequence of inference steps in F, a*
    *contradiction*

    *This is like René Descartes saying*
    “I think therefore thoughts do not exist”

    The reason why G cannot be proved in F is that the proof of G in F is
    contradictory in F, thus Gödel was wrong when he said the reason is that
    F is incomplete. No formal system is ever supposed to be able to prove a
    contradiction.

    Now we can see both THAT G cannot be proved in F and perhaps for the
    first time see WHY G cannot be proved in F. The “incompleteness”
    conclusion has been refuted.

    To be honest we would have to rename Gödel’s “incompleteness” theorem to
    Olcott’s “can’t prove a contradiction” theorem.


    But you keep using the wrong statement for G, ikely because you just
    don't understand the proof.

    G does NOT assert that it is unprovable in F, that is just a conclusion derived from G in Meta-F.


    Gödel sums up his own G as simply:

    "...a proposition which asserts its own unprovability." 15 (Gödel
    1931:39-41)

    Thus this summary is accurate.
    *G asserts its own unprovability in F*

    When you simply hypothesize that it is an accurate representation
    of the essence of Gödel's G then it is easy to see that

    *G asserts its own unprovability in F*
    The reason that G cannot be proved in F is that this requires a
    sequence of inference steps in F that proves no such sequence
    of inference steps exists in F.

    Since we already have the reason why G cannot be proved in F
    (the proof of G is F is contradictory)
    and Gödel said it was another different reason then Gödel was incorrect.

    Gödel, Kurt 1931.
    On Formally Undecidable Propositions of Principia Mathematica And
    Related Systems



    G is actually a statement that there does not exist a whole  number that satisfies a property expressed as a primitive recursive relationship.

    Such a statement CAN'T be "contradictory" as either such a number exists
    or it doesn't.

    Of course, since you are too stupid to understand that statement, you
    mix up which system you are talking in and what statement you are
    working on.

    --
    Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
    hits a target no one else can see." Arthur Schopenhauer

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Richard Damon@21:1/5 to olcott on Fri Apr 21 20:53:25 2023
    XPost: sci.logic, sci.math, alt.philosophy

    On 4/21/23 11:27 AM, olcott wrote:
    On 4/21/2023 7:00 AM, Richard Damon wrote:
    On 4/20/23 12:04 PM, olcott wrote:
    It turns out that the only reason that Gödel’s G is not provable in F is >>> that G is contradictory in F.

    When Gödel’s G asserts that it is unprovable in F it is asserting that >>> there is no sequence of inference steps in F that derives G.

    *A proof of G in F requires a sequence of inference steps in F that*
    *proves there is no such sequence of inference steps in F, a*
    *contradiction*

    *This is like René Descartes saying*
    “I think therefore thoughts do not exist”

    The reason why G cannot be proved in F is that the proof of G in F is
    contradictory in F, thus Gödel was wrong when he said the reason is that >>> F is incomplete. No formal system is ever supposed to be able to prove a >>> contradiction.

    Now we can see both THAT G cannot be proved in F and perhaps for the
    first time see WHY G cannot be proved in F. The “incompleteness”
    conclusion has been refuted.

    To be honest we would have to rename Gödel’s “incompleteness” theorem to
    Olcott’s “can’t prove a contradiction” theorem.


    But you keep using the wrong statement for G, ikely because you just
    don't understand the proof.

    G does NOT assert that it is unprovable in F, that is just a
    conclusion derived from G in Meta-F.


    Gödel sums up his own G as simply:

    "...a proposition which asserts its own unprovability." 15 (Gödel 1931:39-41)

    Right, which was shown in META-F, not F.


    Thus this summary is accurate.
    *G asserts its own unprovability in F*

    Nope, the statement you are referencing is a statement that was DERIVED
    from G, using knowledge available in Meta-F that isn't available in F.


    When you simply hypothesize that it is an accurate representation
    of the essence of Gödel's G then it is easy to see that

    But you are mixing up the domains of the statements.


    *G asserts its own unprovability in F*
    The reason that G cannot be proved in F is that this requires a
    sequence of inference steps in F that proves no such sequence
    of inference steps exists in F.

    Which isn't what G says, either in F or Meta-F. The statement is
    something DERIVABLE from G in Meta-F.


    Since we already have the reason why G cannot be proved in F
    (the proof of G is F is contradictory)
    and Gödel said it was another different reason then Gödel was incorrect.

    Nope. You are using incorrect logic, because you don't understand what
    you are reading.


    Gödel, Kurt 1931.
    On Formally Undecidable Propositions of Principia Mathematica And
    Related Systems



    Which you don't understand but are quoting statements out of context.


    G is actually a statement that there does not exist a whole  number
    that satisfies a property expressed as a primitive recursive
    relationship.

    Such a statement CAN'T be "contradictory" as either such a number
    exists or it doesn't.

    Of course, since you are too stupid to understand that statement, you
    mix up which system you are talking in and what statement you are
    working on.


    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From olcott@21:1/5 to Richard Damon on Fri Apr 21 20:44:22 2023
    XPost: sci.logic, sci.math, alt.philosophy

    On 4/21/2023 7:53 PM, Richard Damon wrote:
    On 4/21/23 11:27 AM, olcott wrote:
    On 4/21/2023 7:00 AM, Richard Damon wrote:
    On 4/20/23 12:04 PM, olcott wrote:
    It turns out that the only reason that Gödel’s G is not provable in >>>> F is
    that G is contradictory in F.

    When Gödel’s G asserts that it is unprovable in F it is asserting that >>>> there is no sequence of inference steps in F that derives G.

    *A proof of G in F requires a sequence of inference steps in F that*
    *proves there is no such sequence of inference steps in F, a*
    *contradiction*

    *This is like René Descartes saying*
    “I think therefore thoughts do not exist”

    The reason why G cannot be proved in F is that the proof of G in F is
    contradictory in F, thus Gödel was wrong when he said the reason is
    that
    F is incomplete. No formal system is ever supposed to be able to
    prove a
    contradiction.

    Now we can see both THAT G cannot be proved in F and perhaps for the
    first time see WHY G cannot be proved in F. The “incompleteness”
    conclusion has been refuted.

    To be honest we would have to rename Gödel’s “incompleteness”
    theorem to
    Olcott’s “can’t prove a contradiction” theorem.


    But you keep using the wrong statement for G, ikely because you just
    don't understand the proof.

    G does NOT assert that it is unprovable in F, that is just a
    conclusion derived from G in Meta-F.


    Gödel sums up his own G as simply:

    "...a proposition which asserts its own unprovability." 15 (Gödel
    1931:39-41)

    Right, which was shown in META-F, not F.

    In exactly the same way that Tarski "proves" that the Liar Paradox is
    true in his meta-theory.

    This sentence is not true: "This sentence is not true" is true.


    --
    Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
    hits a target no one else can see." Arthur Schopenhauer

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Richard Damon@21:1/5 to olcott on Fri Apr 21 22:49:16 2023
    XPost: sci.logic, sci.math, alt.philosophy

    On 4/21/23 9:44 PM, olcott wrote:
    On 4/21/2023 7:53 PM, Richard Damon wrote:
    On 4/21/23 11:27 AM, olcott wrote:
    On 4/21/2023 7:00 AM, Richard Damon wrote:
    On 4/20/23 12:04 PM, olcott wrote:
    It turns out that the only reason that Gödel’s G is not provable in >>>>> F is
    that G is contradictory in F.

    When Gödel’s G asserts that it is unprovable in F it is asserting that >>>>> there is no sequence of inference steps in F that derives G.

    *A proof of G in F requires a sequence of inference steps in F that* >>>>> *proves there is no such sequence of inference steps in F, a*
    *contradiction*

    *This is like René Descartes saying*
    “I think therefore thoughts do not exist”

    The reason why G cannot be proved in F is that the proof of G in F is >>>>> contradictory in F, thus Gödel was wrong when he said the reason is >>>>> that
    F is incomplete. No formal system is ever supposed to be able to
    prove a
    contradiction.

    Now we can see both THAT G cannot be proved in F and perhaps for the >>>>> first time see WHY G cannot be proved in F. The “incompleteness” >>>>> conclusion has been refuted.

    To be honest we would have to rename Gödel’s “incompleteness” >>>>> theorem to
    Olcott’s “can’t prove a contradiction” theorem.


    But you keep using the wrong statement for G, ikely because you just
    don't understand the proof.

    G does NOT assert that it is unprovable in F, that is just a
    conclusion derived from G in Meta-F.


    Gödel sums up his own G as simply:

    "...a proposition which asserts its own unprovability." 15 (Gödel
    1931:39-41)

    Right, which was shown in META-F, not F.

    In exactly the same way that Tarski "proves" that the Liar Paradox is
    true in his meta-theory.

    This sentence is not true: "This sentence is not true" is true.


    You aren't reading what he is saying correctly if you are referencing
    what I think you are refeencing.

    I think the issue is you don't understand how a proof by contradiction
    works.

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)