Re: A different perspective on undecidability

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