Re: Mathematical incompleteness has always been a misconception

On 2/1/2025 3:19 AM, Mikko wrote:

On 2025-01-31 13:57:02 +0000, olcott said:
 

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.

 And well undeerstood. The claim on the subject line is false.
 

a fact or piece of information that shows that something
exists or is true:
https://dictionary.cambridge.org/us/dictionary/english/proof
When you disallow assigning different idiomatic meanings to terms-of the-art such that the term "proof" retains every nuance of its base meaning and not allowed to diverge from this meaning at all then every expression of language that is only made true (by a connection through its philosophical truth maker) to its verbalized semantic meaning must necessarily remain untrue when it lacks such a connection.

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.

 Wrong. The result is the same if the omission is unintentional. For example,
Peano did not intentionally leave anything out in order to make an incomplete
theory. Only later Gödel found out that Peano's theory is incomplete.
 

--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer