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  • Re: Proving the consistency of the body of knowledge expressed in langu

    From Richard Damon@21:1/5 to olcott on Tue Apr 1 21:03:44 2025
    XPost: sci.math

    On 4/1/25 7:22 PM, olcott wrote:
    On 4/1/2025 5:30 PM, Richard Damon wrote:
    On 4/1/25 1:56 PM, olcott wrote:
    On 4/1/2025 1:33 AM, Mikko wrote:
    On 2025-03-31 18:33:26 +0000, olcott said:


    Anything the contradicts basic facts or expressions
    semantically entailed from these basic facts is proven
    false.

    Anything that follows from true sentences by a truth preserving
    transformations is true. If you can prove that a true sentence
    is false your system is unsound.


    Ah so we finally agree on something.
    What about the "proof" that detecting inconsistent
    axioms is impossible? (I thought that I remebered this).


    No, the proof is that it is impossible to prove that a system is
    consistant. (sort of the opposite of what you are thinking of).

    Proving inconsistancy is easy, you just need one example.

    Proving the non-existance isn't as easy, and for a complicated enough
    system, can't be done, as you need to search an infinite space for the
    problem, which we can't be sure we have finished,


    I have always only been referring to the consistency
    of a finite set of axioms. Just test each one against
    all the others. When we use a type hierarchy we only
    have to do this for axioms with compatible types.

    And, if they can support the needed level of logic, Godel has shown that
    they can not prove their own consistancy.


    If we are only allowed to apply the single truth
    preserving operation of semantic logical entailment
    then we know the whole system must be consistent. https://en.wikipedia.org/wiki/Logical_consequence#Semantic_consequence

    You mean your logic has no "And" or "Or" or "Not" operations?


    We bypass any need for model theory by having the full
    semantics embedded directly in the formal language.

    Not sure you can do that. You haven't been very good at being right in
    the past.

    So, which step of Tarski's proof doesn't follow that requirement?

    (Not that you disagree with his conclusion, but is logical operation
    violated this rule).


    Sort of like we can easily prove that a machine halts, but simulating
    it to that point (like a real emulator can do for DDD), but showing
    that a machine is non-halting can be more of a problem. Sometimes we
    can find an induction property to let us prove it, but not always.



    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Richard Damon@21:1/5 to olcott on Tue Apr 1 23:00:23 2025
    XPost: sci.math

    On 4/1/25 10:13 PM, olcott wrote:
    On 4/1/2025 8:03 PM, Richard Damon wrote:
    On 4/1/25 7:22 PM, olcott wrote:
    On 4/1/2025 5:30 PM, Richard Damon wrote:
    On 4/1/25 1:56 PM, olcott wrote:
    On 4/1/2025 1:33 AM, Mikko wrote:
    On 2025-03-31 18:33:26 +0000, olcott said:


    Anything the contradicts basic facts or expressions
    semantically entailed from these basic facts is proven
    false.

    Anything that follows from true sentences by a truth preserving
    transformations is true. If you can prove that a true sentence
    is false your system is unsound.


    Ah so we finally agree on something.
    What about the "proof" that detecting inconsistent
    axioms is impossible? (I thought that I remebered this).


    No, the proof is that it is impossible to prove that a system is
    consistant. (sort of the opposite of what you are thinking of).

    Proving inconsistancy is easy, you just need one example.

    Proving the non-existance isn't as easy, and for a complicated
    enough system, can't be done, as you need to search an infinite
    space for the problem, which we can't be sure we have finished,


    I have always only been referring to the consistency
    of a finite set of axioms. Just test each one against
    all the others. When we use a type hierarchy we only
    have to do this for axioms with compatible types.

    And, if they can support the needed level of logic, Godel has shown
    that they can not prove their own consistancy.


    How is it that each element of a finite set of axioms
    can not simply be tested against all of the others?

    Because you can combine the axioms in various combinations.

    The possible combinations can become like a finite string composed of
    all the axioms as symbols, which can be infinite.


    Note, part of the requirements to reach that point of not being able to
    prove consistancy, is the ability to create an infinite number of items
    in the system (like the natural numbers).



    If we are only allowed to apply the single truth
    preserving operation of semantic logical entailment
    then we know the whole system must be consistent.
    https://en.wikipedia.org/wiki/Logical_consequence#Semantic_consequence

    You mean your logic has no "And" or "Or" or "Not" operations?


    Those are part of its semantics.

    So, what keeps us from using Godel's or Tarski's proof in the system?

    Or are you admitting you can't have the full properties of the Natural
    Numbers?



    We bypass any need for model theory by having the full
    semantics embedded directly in the formal language.

    Not sure you can do that. You haven't been very good at being right in
    the past.


    Spend few years carefully studying Montague grammar
    and you might get it.

    Which is just a way to encode the meaning of a sentence.

    Doesn't do anything about the logic system.

    He also doesn't handle all the inherent ambiguity of Natural Language
    words, which is sometimes used INTENTIONALLY as a special from of meaning.


    So, which step of Tarski's proof doesn't follow that requirement?


    He started with falsehoods as his basic facts.

    Which one?

    Remember, the point you keep on pointing out is late in the proof, and
    follows from the previous steps.


    (Not that you disagree with his conclusion, but is logical operation
    violated this rule).


    True(X) inherently exists for the entire body of
    knowledge that can be expressed in language.

    Which isn't a logic system, and is just a knowledge predicate.


    For Pete's sake it is like you don't understand
    that 3 > 2.

    Sure I do, you just don't undetstand how logic works, or what truth
    actually is.



    Sort of like we can easily prove that a machine halts, but
    simulating it to that point (like a real emulator can do for DDD),
    but showing that a machine is non-halting can be more of a problem.
    Sometimes we can find an induction property to let us prove it, but
    not always.






    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Mikko@21:1/5 to olcott on Wed Apr 2 12:37:02 2025
    On 2025-04-02 02:13:36 +0000, olcott said:

    On 4/1/2025 8:03 PM, Richard Damon wrote:
    On 4/1/25 7:22 PM, olcott wrote:
    On 4/1/2025 5:30 PM, Richard Damon wrote:
    On 4/1/25 1:56 PM, olcott wrote:
    On 4/1/2025 1:33 AM, Mikko wrote:
    On 2025-03-31 18:33:26 +0000, olcott said:


    Anything the contradicts basic facts or expressions
    semantically entailed from these basic facts is proven
    false.

    Anything that follows from true sentences by a truth preserving
    transformations is true. If you can prove that a true sentence
    is false your system is unsound.


    Ah so we finally agree on something.
    What about the "proof" that detecting inconsistent
    axioms is impossible? (I thought that I remebered this).


    No, the proof is that it is impossible to prove that a system is
    consistant. (sort of the opposite of what you are thinking of).

    Proving inconsistancy is easy, you just need one example.

    Proving the non-existance isn't as easy, and for a complicated enough
    system, can't be done, as you need to search an infinite space for the >>>> problem, which we can't be sure we have finished,


    I have always only been referring to the consistency
    of a finite set of axioms. Just test each one against
    all the others. When we use a type hierarchy we only
    have to do this for axioms with compatible types.

    And, if they can support the needed level of logic, Godel has shown
    that they can not prove their own consistancy.


    How is it that each element of a finite set of axioms
    can not simply be tested against all of the others?

    You can't do so if there is no test method.

    --
    Mikko

    --- SoupGate-Win32 v1.05
    * Origin: fsxNet Usenet Gateway (21:1/5)
  • From Mikko@21:1/5 to olcott on Sat Apr 5 10:33:32 2025
    On 2025-04-02 16:04:33 +0000, olcott said:

    On 4/2/2025 4:37 AM, Mikko wrote:
    On 2025-04-02 02:13:36 +0000, olcott said:

    On 4/1/2025 8:03 PM, Richard Damon wrote:
    On 4/1/25 7:22 PM, olcott wrote:
    On 4/1/2025 5:30 PM, Richard Damon wrote:
    On 4/1/25 1:56 PM, olcott wrote:
    On 4/1/2025 1:33 AM, Mikko wrote:
    On 2025-03-31 18:33:26 +0000, olcott said:


    Anything the contradicts basic facts or expressions
    semantically entailed from these basic facts is proven
    false.

    Anything that follows from true sentences by a truth preserving >>>>>>>> transformations is true. If you can prove that a true sentence >>>>>>>> is false your system is unsound.


    Ah so we finally agree on something.
    What about the "proof" that detecting inconsistent
    axioms is impossible? (I thought that I remebered this).


    No, the proof is that it is impossible to prove that a system is
    consistant. (sort of the opposite of what you are thinking of).

    Proving inconsistancy is easy, you just need one example.

    Proving the non-existance isn't as easy, and for a complicated enough >>>>>> system, can't be done, as you need to search an infinite space for the >>>>>> problem, which we can't be sure we have finished,


    I have always only been referring to the consistency
    of a finite set of axioms. Just test each one against
    all the others. When we use a type hierarchy we only
    have to do this for axioms with compatible types.

    And, if they can support the needed level of logic, Godel has shown
    that they can not prove their own consistancy.

    How is it that each element of a finite set of axioms
    can not simply be tested against all of the others?

    You can't do so if there is no test method.

    I need a concrete example.

    A concerete example of what?

    --
    Mikko

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