Re: Undecidability based on epistemological antinomies V2 --H(D,D)--

On 4/24/24 11:33 AM, olcott wrote:

On 4/24/2024 3:35 AM, Mikko wrote:

On 2024-04-23 14:31:00 +0000, olcott said:
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On 4/23/2024 3:21 AM, Mikko wrote:

On 2024-04-22 17:37:55 +0000, olcott said:
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On 4/22/2024 10:27 AM, Mikko wrote:

On 2024-04-22 14:10:54 +0000, olcott said:
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On 4/22/2024 4:35 AM, Mikko wrote:

On 2024-04-21 14:44:37 +0000, olcott said:
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On 4/21/2024 2:57 AM, Mikko wrote:

On 2024-04-20 15:20:05 +0000, olcott said:
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On 4/20/2024 2:54 AM, Mikko wrote:

On 2024-04-19 18:04:48 +0000, olcott said:
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When we create a three-valued logic system that has these
three values: {True, False, Nonsense}
https://en.wikipedia.org/wiki/Three-valued_logic

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Such three valued logic has the problem that a tautology of the
ordinary propositional logic cannot be trusted to be true. For
example, in ordinary logic A ∨ ¬A is always true. This means that
some ordinary proofs of ordinary theorems are no longer valid and
you need to accept the possibility that a theory that is complete
in ordinary logic is incomplete in your logic.
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I only used three-valued logic as a teaching device. Whenever an
expression of language has the value of {Nonsense} then it is
rejected and not allowed to be used in any logical operations. It
is basically invalid input.

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You cannot teach because you lack necessary skills. Therefore you
don't need any teaching device.
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That is too close to ad homimen.
If you think my reasoning is incorrect then point to the error
in my reasoning. Saying that in your opinion I am a bad teacher
is too close to ad hominem because it refers to your opinion of
me and utterly bypasses any of my reasoning.

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No, it isn't. You introduced youtself as a topic of discussion so
you are a legitimate topic of discussion.
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I didn't claim that there be any reasoning, incorrect or otherwise.
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If you claim I am a bad teacher you must point out what is wrong with
the lesson otherwise your claim that I am a bad teacher is essentially
an as hominem attack.

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You are not a teacher, bad or otherwise. That you lack skills that
happen to be necessary for teaching is obvious from you postings
here. A teacher needs to understand human psychology but you don't.
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You may be correct that I am a terrible teacher.
None-the-less Mathematicians might not have very much understanding
of the link between proof theory and computability.

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Sume mathematicians do have very much understanding of that. But that
link is not needed for understanding and solving problems separately
in the two areas.
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When I refer to rejecting an invalid input math would seem to construe
this as nonsense, where as computability theory would totally understand.

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People working on computability theory do not understand "invalid input"
as "impossible input".

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The proof then shows, for any program f that might determine whether
programs halt, that a "pathological" program g, called with some input,
can pass its own source and its input to f and then specifically do the
opposite of what f predicts g will do. No f can exist that handles this
case, thus showing undecidability.
https://en.wikipedia.org/wiki/Halting_problem#
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So then they must believe that there exists an H that does correctly
determine the halt status of every input, some inputs are simply
more difficult than others, no inputs are impossible.

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That "must" is false as it does not follow from anything.
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 Sure it does. If there are no "impossible" inputs that entails
that all inputs are possible. When all inputs are possible then
the halting problem proof is wrong.
 *Termination Analyzer H is Not Fooled by Pathological Input D*
https://www.researchgate.net/publication/369971402_Termination_Analyzer_H_is_Not_Fooled_by_Pathological_Input_D
 Everyone that objects to the statement that H(D,D) correctly determines the halt status of its inputs say that believe that H(D,D) must report on the behavior of the D(D) that invokes H(D,D).

Right, because that IS the definition of a Halt Decider.
To say otherwise just proves you don't actually know the meanings of the words you are using.

 They say this knowing full well that computable functions only operate on their inputs. This also violates the definition of a decider that only computes the mapping from its inputs. Thus expecting H(D,D) to report on the behavior of the D(D) that invokes H(D,D) violates two core principles of of computer science.

Nope, and the fact you think so shows you don't understand those core principles.

 Finally the behavior of the simulated D(D) before H aborts its simulation is different than the behavior of the executed D(D) after H has aborted its simulation. H(D,D) must report on the behavior that it actually sees.

Nope, BECAUSE the definition of "Correct Simulation" of a Program Description is replicating the behavior of the direct execution of the program.

 

They understand it as an input that must be
handled differently from ordinary input. Likewise, mathematicians do
understand that some inputs must be considered separately and differently.
But mathematicians don't call those inputs "invalid".

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It is so dead obvious that the whole world must be wired with a short
circuit in their brains. Formal bivalent mathematical systems of logic
must reject every expression that cannot possibly have a value of true
or false as a type mismatch error.

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Gödel's completeness theorem proves that every consistent first order
theory has a model, i.e., there is an interpretation that assigns a
truth value to every formula of the theory. No such proof is known for
second or higher order theories.
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 By switching from model theory to proof theory we need no
interpretations. Every system of logic is simply relations
between finite strings.
 To get rid of undecidability and incompleteness we simply encode all of
the facts of the general knowledge of the actual world as axioms of a
formal system of logic.

Which doesn't help at all if the system isn't supposed to be a "model" of the actual world. (and most Formal Logic systems are not).
And that also, doesn't handle the problem, as Godel proves. A Formal System needs a finite set of axioms, which your above will have, as our knowledge WILL be finite. And your system will meet the logical power requirements. (I suppose the one question is if it meets the consistancy requirement, it is actually quite possible that the some total of everything we THINK is true might have an error in it that makes the system inconsistant, or your method of accumulating "All Knowledge" might not be able to keep the different contradictory sub-fields distinct enough to keep them apart, like Euclidean Geometry and non-Euclidean Geometry)
This means we CAN form the meta-system that assigns ALL those axioms to a prime number, and thus we can encode all statements to a composite number, and build the Primiative Recursive Relationship that represents the proof checker for his statement G in your logic system, and thus we can show that G MUST be true (or your system is inconsistent as we can prove a false statement with the value of g that satisfies that relationship), and thus there can not be a proof for G in the system, or a true statement is false.
We fundamentally know that a number g that satisfies the PRR either will exist or will not.
And, your proposal also shows that you just don't understand the basis of formal logic.

 True(L, x)  ≡ ∃x ∈ L (L ⊢ x)
False(L, x) ≡ ∃x ∈ L (L ⊢ x)
Truth_Bearer(L, x) ≡ ∃x ∈ L (True(L, x) ∨ False(L, x))

And since you definitions do not match the definitions of classical logic (where Truth isn't defined by Proof, but by establishment, which allows for infinite chains of steps which proof doesn't) this mean you have to start from ground zero to show what your new system can do.
GOOD LUCK, you will need it.
I don't think you have the knowledge base to do it, or the time.
Without doing that, all you are doing is making UNPROVEN claims, and doing exactly the same thing as the election deniers are doing.

 

A proposition is a central concept in the philosophy of language,
semantics, logic, and related fields, often characterized as the primary
bearer of truth or falsity. https://en.wikipedia.org/wiki/Proposition

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In formal logic the corresponding concept is sentence.
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