From:  Mikko <mikko.levanto@iki.fi>
Date:  21 Oct 2024 17:30:59 Hong Kong Time
Newsgroup:  news.alt119.net/sci.logic
Subject:  

Re: A different perspective on undecidability

NNTP-Posting-Host:  null

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