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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.logic
Subject: Re: Mathematical incompleteness has always been a misconception ---
 Ultimate Foundation of Truth
Date: Tue, 25 Feb 2025 23:21:24 -0500
Organization: i2pn2 (i2pn.org)
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On 2/25/25 8:17 PM, olcott wrote:
> On 2/25/2025 5:57 PM, Richard Damon wrote:
>> On 2/25/25 9:31 AM, olcott wrote:
>>> On 2/24/2025 10:02 PM, Richard Damon wrote:
>>>> On 2/24/25 9:02 PM, olcott wrote:
>>>>> On 2/24/2025 6:12 PM, Richard Damon wrote:
>>>>>> On 2/24/25 6:11 PM, olcott wrote:
>>>>>>> On 2/24/2025 6:27 AM, Richard Damon wrote:
>>>>>>>> On 2/23/25 11:39 PM, olcott wrote:
>>>>>>>>> On 2/23/2025 8:50 PM, Richard Damon wrote:
>>>>>>>>>> On 2/23/25 1:08 PM, olcott wrote:
>>>>>>>>>>> On 2/22/2025 9:56 PM, Richard Damon wrote:
>>>>>>>>>>>> On 2/22/25 1:42 PM, olcott wrote:
>>>>>>>>>>>>> On 2/22/2025 3:25 AM, Mikko wrote:
>>>>>>>>>>>>>> On 2025-02-22 04:44:35 +0000, olcott said:
>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> On 2/21/2025 7:05 PM, Richard Damon wrote:
>>>>>>>>>>>>>>>> On 2/21/25 6:19 PM, olcott wrote:
>>>>>>>>>>>>>>>>> On 2/20/2025 2:54 AM, Mikko wrote:
>>>>>>>>>>>>>>>>>> On 2025-02-18 03:59:08 +0000, olcott said:
>>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>> Tarski anchored his whole proof in the Liar Paradox.
>>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>> By showing that given the necessary prerequisites, The 
>>>>>>>>>>>>>>>> equivalent of the Liar Paradox was a statement that the 
>>>>>>>>>>>>>>>> Truth Predicate had to be able to handle, which it can't.
>>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>>
>>>>>>>>>>>>>>> It can be easily handled as ~True(LP) & ~True(~LP), 
>>>>>>>>>>>>>>> Tarski just
>>>>>>>>>>>>>>> didn't think it through.
>>>>>>>>>>>>>>
>>>>>>>>>>>>>> No, it can't. Tarski requires that True be a predicate, 
>>>>>>>>>>>>>> i.e, a truth
>>>>>>>>>>>>>> valued function of one term. 
>>>>>>>>>>>>>
>>>>>>>>>>>>> It does not matter a whit what the Hell his misconceptions
>>>>>>>>>>>>> required. We simply toss his whole mess out the window and
>>>>>>>>>>>>> reformulate a computable Truth predicate that works correctly.
>>>>>>>>>>>>
>>>>>>>>>>>> But his logic follows from the premises.
>>>>>>>>>>>>
>>>>>>>>>>>> Maybe your logic just can't handle that level of system.
>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>> It is all ultimately anchored relations between finite
>>>>>>>>>>>>> strings even if we must toss all of logical out the window
>>>>>>>>>>>>> to do this correctly.
>>>>>>>>>>>>
>>>>>>>>>>>> And to do what you want, you have to limit your logic system 
>>>>>>>>>>>> to not be able to define the full Natural Number system, as 
>>>>>>>>>>>> that is what allows Tarski to do what he does (like Godel 
>>>>>>>>>>>> does).
>>>>>>>>>>>>
>>>>>>>>>>>>>
>>>>>>>>>>>>> We are answering the question:
>>>>>>>>>>>>> What are the relationships between arbitrary finite strings
>>>>>>>>>>>>> such that the semantic property of True(L, x)
>>>>>>>>>>>>> (where L and x are finite strings) can always be correctly
>>>>>>>>>>>>> determined for every finite string having a truth value 
>>>>>>>>>>>>> that is
>>>>>>>>>>>>> entirely verified by its relation to other finite strings.
>>>>>>>>>>>>>
>>>>>>>>>>>>
>>>>>>>>>>>> And, if the logic system can support the properties of the 
>>>>>>>>>>>> Natural Number system, and a definition of the predicate 
>>>>>>>>>>>> True, it can be shown that you can create the equivalent of
>>>>>>>>>>>>
>>>>>>>>>>>> Let P be defined as Not( True(L, P))
>>>>>>>>>>>>
>>>>>>>>>>>> in that system, and thus P is a semantically valid, 
>>>>>>>>>>>
>>>>>>>>>>> Not at all. That is the same as saying you know
>>>>>>>>>>> that it is true that all squares are always round.
>>>>>>>>>>>
>>>>>>>>>>
>>>>>>>>>> Really, then where is the error in his derivation?
>>>>>>>>>> n
>>>>>>>>>
>>>>>>>>> You clearly have no idea what "semantically sound" means.
>>>>>>>>> The only correct rebuttal to this is you proving that
>>>>>>>>> you do know this by providing the details of exactly what
>>>>>>>>> "semantically sound" means.
>>>>>>>>>
>>>>>>>>
>>>>>>>> Sure I do.
>>>>>>>>
>>>>>>>> A Systems is semantically sound if every statement that can be 
>>>>>>>> proven is actually true by the systems semantics,
>>>>>>>
>>>>>>> That is very good.
>>>>>>>
>>>>>>>> in other words, the system doesn't allow the proving of a false 
>>>>>>>> statement.
>>>>>>>>
>>>>>>>
>>>>>>> That is not too bad yet ignores that some expressions
>>>>>>> might not have any truth value.
>>>>>>
>>>>>> Which has nothing to do with "soundness".
>>>>>>
>>>>>
>>>>> When any system assumes that every expression is true
>>>>> or false and is capable of encoding expressions that
>>>>> are neither IT IS STUPIDLY WRONG.
>>>>>
>>>>
>>>> But it doesn't.
>>>>
>>>
>>> LP := ~True(LP) is semantically invalid.
>>
>> Then the predicate "True" is semantically invalid, and thus isn't a 
>> predicate.
>>
> 
> How would you propose that a correct True() predicate deal
> with random gibberish as input?

By its definition it return false.

But that can't be the return value for LP defines as ~True(LP) as then 
True(LP) would be false, and thus LP := ~false, or true.

The problem is the expression is nonsense, but also can't be if we have 
a Truth Predicate, and the abiility to express the properties of the 
Natural Numbers.

Since we want to be able to express the properties of the Natural 
Numbers, we accept that we can't have a Truth Predicate.

That you chose to try to have a Truth Predicate, even if it breaks your 
logic system just shows your ignorance.