Re: Mathematical incompleteness has always been a misconception

On 1/31/2025 3:24 AM, Mikko wrote:

On 2025-01-30 23:10:18 +0000, olcott said:
 

Within the entire body of analytical truth any expression of language that has no sequence of formalized semantic deductive inference steps from the formalized semantic foundational truths of this system are simply untrue in this system. (Isomorphic to provable from axioms).

 If there is a misconception then you have misconceived something. It is well
known that it is possible to construct a formal theory where some formulas
are neither provble nor disprovable.

This is well known. What is not so widely known is that this
is only possible because process defining what is referred to
as a math proof intentionally leaves out key required elements
that would otherwise make it complete.
Any expression of language that lacks a sequence of semantic
deductive inference steps from the basic facts stipulated truths
of this system to this expression is simply untrue in this system.
Using another more expressive system to show that the expression
is true in this other system does not make the expression true in
the original system.

Often that is done intentionally in
order to make the theory applicable to situations where some such sentence
is true as well as to situations where the same sentence is false.
 

Thus incompleteness is intentional incoherence that can always be prevented.
--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer