Re: A different perspective on undecidability

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?
Either X IS or it IS NOT a theory of L, as either a proof of its truth exists or it doesn't.
If X is non-sense, then it isn't a theory of L, as you can't prove non-sense to be true in a non-contradictory L.
So, how can THOSE questions not be a truth bearers?
You don't seem to understad what Truth actually is.
I guess your logic is that there is no such thing as a non-contradictory field of study.
But that is just because you don't seem to understand how logic actually works.