Re: Mathematical incompleteness has always been a misconception

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.

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.
--
Mikko