Re: A different perspective on undecidability

On 10/16/24 8:51 PM, 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:
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The whole notion of undecidability is anchored in ignoring the fact that
some expressions of language are simply not truth bearers.

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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.
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*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.
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(2) There is a finite set of true preserving operations
(TPO) that can be applied to this finite set in theory L.
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When formula x cannot be derived by applying the TPO
of L to STBT of L then x is not a theorem of L.
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A theorem is a statement that can be demonstrated to be
true by accepted mathematical operations and arguments.
https://mathworld.wolfram.com/Theorem.html
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How can there not be a Yes or No answer to it being a statement that can be proven true?
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 I didn't say anything like that in this post.
 

You said "The whole notion of undecidabioiut is anchord in ignoring the fat that some expressions of language are simply not truth bearers"
As explain, "undeciability" of a system is based on the question of if there are some expressions in it that can not be determined if they are a provable theorem in the system (the only kind of theorems that exist) or not.
The question "Is X a Theorem of L" can not be a statement without a truth value, as X either CAN be proven or it can not (we might not KNOW if it is provable, which is what leads to undecidability, but in fact, it either is or it isn;t).
IF x is a statement without a truth value, the answer to the quesiton about x will just be false, as no consistant system can prove a non-truthbearer.
Thus, you DID says something like that, but are apparently too stupid to undertstand that you did.
My only conclusion from your remarks is that you must be assuming that all logic system are inconsistant, so the question of the provability of some statements doesn't have a truth value because the statement might be both provable and not provable at the same time.