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From: Mikko <mikko.levanto@iki.fi>
Newsgroups: sci.logic
Subject: Re: A different perspective on undecidability
Date: Mon, 21 Oct 2024 12:36:16 +0300
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On 2024-10-18 23:43:15 +0000, olcott said:

> On 10/18/2024 6:17 PM, Richard Damon wrote:
>> 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.
> 
> When one thinks of proofs as finite string transformation
> rules then one finite string can be transformed into another
> according to the transformation rules or not.

Typical logic systems have transformation rules that transform two
strings to one. For example, you cannot infer A ∧ B from A nor from
B but if you have both A and B then you can infer A ∧ B.

-- 
Mikko