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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: sci.logic
Subject: Re: Undecidability based on epistemological antinomies V2 --Tarski Proof--
Date: Wed, 24 Apr 2024 11:35:10 +0300
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On 2024-04-23 14:31:00 +0000, olcott said:

> On 4/23/2024 3:21 AM, Mikko wrote:
>> On 2024-04-22 17:37:55 +0000, olcott said:
>> 
>>> On 4/22/2024 10:27 AM, Mikko wrote:
>>>> On 2024-04-22 14:10:54 +0000, olcott said:
>>>> 
>>>>> On 4/22/2024 4:35 AM, Mikko wrote:
>>>>>> On 2024-04-21 14:44:37 +0000, olcott said:
>>>>>> 
>>>>>>> On 4/21/2024 2:57 AM, Mikko wrote:
>>>>>>>> On 2024-04-20 15:20:05 +0000, olcott said:
>>>>>>>> 
>>>>>>>>> On 4/20/2024 2:54 AM, Mikko wrote:
>>>>>>>>>> On 2024-04-19 18:04:48 +0000, olcott said:
>>>>>>>>>> 
>>>>>>>>>>> When we create a three-valued logic system that has these
>>>>>>>>>>> three values: {True, False, Nonsense}
>>>>>>>>>>> https://en.wikipedia.org/wiki/Three-valued_logic
>>>>>>>>>> 
>>>>>>>>>> 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 some ordinary proofs of ordinary theorems are no longer valid and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.
>>>>>>>>>> 
>>>>>>>>> 
>>>>>>>>> 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.
>>>>>>>> 
>>>>>>>> You cannot teach because you lack necessary skills. Therefore you
>>>>>>>> don't need any teaching device.
>>>>>>>> 
>>>>>>> 
>>>>>>> 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.
>>>>>> 
>>>>>> No, it isn't. You introduced youtself as a topic of discussion so
>>>>>> you are a legitimate topic of discussion.
>>>>>> 
>>>>>> I didn't claim that there be any reasoning, incorrect or otherwise.
>>>>>> 
>>>>> 
>>>>> 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.
>>>> 
>>>> 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.
>>>> 
>>> 
>>> 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.
>> 
>> 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.
>> 
>>> When I refer to rejecting an invalid input math would seem to construe
>>> this as nonsense, where as computability theory would totally understand.
>> 
>> People working on computability theory do not understand "invalid input"
>> as "impossible input".
> 
> 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#
> 
> 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.

That "must" is false as it does not follow from anything.

>> 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".

> 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.

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.

> 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

In formal logic the corresponding concept is sentence.

-- 
Mikko