Re: Formal systems that cannot possibly be incomplete except for unknowns and unknowable

On 5/6/2025 10:38 AM, Alan Mackenzie wrote:

olcott <polcott333@gmail.com> wrote:

On 5/6/2025 3:17 AM, Alan Mackenzie wrote:

[ Followup-To: set ]

 

In comp.theory olcott <polcott333@gmail.com> wrote:

 [ .... ]
 

When we begin with truth and only apply truth preserving
operations then WE NECESSARILY MUST END UP WITH TRUTH.

 

You will necessarily end up with only a subset of truth, no matter how
shouty you are in writing it.  You'll also end up with undecidability,
no matter how hard you try to pretend it isn't there.

 

When we ALWAYS end up with TRUTH then we NEVER end up with
UNDECIDABILITY.

 

Shut your eyes, and you won't see it.

 

Try to provide one simple concrete example where we begin with truth
and only apply truth preserving operations and end up with
undecidability.

 There is no need to provide examples for rigorously proven mathematical
theorems, in particular for Gödel's incompleteness theorem.  I don't go
around providing examples for 2 + 2 = 4 either.
 

This is proven completely true entirely on the basis of
the meaning of its words:
When True(x) is defined to mean that x can be derived
by applying truth preserving operations to basic facts
that are stipulated to be true ONLY TRUTH CAN BE DERIVED.
Everything (including random gibberish) is divided into
True(x) or ~True(x). Since the scope of this system
includes all general knowledge that can be expressed
using language.
True(x) returns false for unknown and unknowable truths.
Thus True(x) is a membership algorithm that verifies
whether or not x is in the body of general knowledge
that can be expressed using language.
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer