Re: A different perspective on undecidability --- incorrect question

On 2024-10-22 02:04:14 +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.
 

 After being continually interrupted by emergencies
interrupting other emergencies...
 If the answer to the question: Is X a formula of theory Y
cannot be determined to be yes or no then the question
itself is somehow incorrect.

There are several possibilities.
A theory may be intentionally incomplete. For example, group theory
leaves several important question unanswered. There are infinitely
may different groups and group axioms must be true in every group.
Another possibility is that a theory is poorly constructed: the
author just failed to include an important postulate.
Then there is the possibility that the purpose of the theory is
incompatible with decidability, for example arithmetic.

An incorrect question is an expression of language that
is not a truth bearer translated into question form.
 When "X a formula of theory Y" is neither true nor false
then "X a formula of theory Y" is not a truth bearer.

Whether AB = BA is not answered by group theory but is alwasy
true or false about specific A and B and universally true in
some groups but not all.
--
Mikko