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From: Richard Damon <richard@damon-family.org>
Newsgroups: sci.logic
Subject: Re: A different perspective on undecidability
Date: Fri, 18 Oct 2024 19:17:06 -0400
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On 10/17/24 10:53 AM, olcott wrote:
> On 10/16/2024 7:47 PM, Richard Damon wrote:
>> On 10/16/24 6:34 PM, olcott wrote:
>>> On 10/16/2024 11:37 AM, Mikko wrote:
>>>> On 2024-10-16 14:27:09 +0000, olcott said:
>>>>
>>>>> The whole notion of undecidability is anchored in ignoring the fact 
>>>>> that
>>>>> some expressions of language are simply not truth bearers.
>>>>
>>>> A formal theory is undecidable if there is no Turing machine that
>>>> determines whether a formula of that theory is a theorem of that
>>>> theory or not. Whether an expression is a truth bearer is not
>>>> relevant. Either there is a valid proof of that formula or there
>>>> is not. No third possibility.
>>>>
>>>
>>> *I still said that wrong*
>>> (1) There is a finite set of expressions of language
>>> that are stipulated to be true (STBT) in theory L.
>>>
>>> (2) There is a finite set of true preserving operations
>>> (TPO) that can be applied to this finite set in theory L.
>>>
>>> When formula x cannot be derived by applying the TPO
>>> of L to STBT of L then x is not a theorem of L.
>>>
>>> A theorem is a statement that can be demonstrated to be
>>> true by accepted mathematical operations and arguments.
>>> https://mathworld.wolfram.com/Theorem.html
>>>
>>
>> How can there not be a Yes or No answer to it being a statement that 
>> can be proven true?
>>
> 
> I didn't say anything like that in the words shown
> immediately above. Maybe the reason that you get
> so confused is that you never respond to the exact
> words that I just said right now.
> 

Then what are you referring to if other than your initial claim?

What statement are you saying simply not being a truth bearer makes the 
definition of undecidability incorrect?

I reply to your WHOLE message, as context matters.

Your statements (1) and (2) are just clearification that you understand 
the problem, but then how can the fact that we can show that there can 
be some statements we can not know if they are provable or not, not be a 
valid proof of the system being undecidable?

Note, that the fact that we haven't been able to demonstrate that a 
proof exists, is not in itself a proof that no such proof exists. If the 
Turing Machine existed, then all True Statements would be provable, all 
False statements refutable, and all non-truthbears detectable for being 
that.

The fact that it can be shown that there can exist statements in a 
language L, that are TRUE, but not provable in that language, show that 
there exist language Ls that are undecidable.