Re: A different perspective on undecidability

On 2024-10-16 22:34:51 +0000, olcott said:

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

Better. The word "theory" starts with T so instead of L the
letter T should be used as the name of a theory.
In a formal theory no set of expressions are stipuated to be
true. Instead they are defined to be the postulates of the
theory.
When discussing a formal theory the theorems are not assumed to
be true. They can be true in one interpretation and false in
another one.
Whether the inference rules of a theory are truth preserving is
a matter of separate investigation.
--
Mikko