XPost: comp.theory, sci.logic
On 8/30/23 9:07 PM, olcott wrote:
Gödel incompleteness is more aptly construed as the non-sequitur error
in the same way that the conclusion {the Moon orbits the Earth} is not a logical consequence of the premise {cats are mammals}.
Provability ONLY validates logical consequence. Whenever a conclusion is
not provable from its premises we have the non-sequitur error. This
remains true when the only premises are the axioms of the formal system.
What is "non-squitur" about it?
Do you enen understand what you are saying?
What doesn't follow?
You don't seem to understand how logic works.
The statment of G in F is TRUE, as has been proven in Meta-F, which has
been constructed in a manner that statements shown to be true in Meta-F
that don't contain any references to things that only exist in Meta-F
(and G doesn't contain such a reference) are also true in F.
The statement G has also been proven (with logic Meta-F) to not be
provable in F.
Thus G is proven to be an example of a statement True in F, and not
provable in F.
The Definition of "completness" for a logic system is that ALL True
statements in the system are provable, and a system is Incomplete if
there exists a True statement that is not provable.
Since G meets that criteria, F is not complete.
The proof applies to ALL system with sufficient axioms to provide the
needed basics of Natural Numbers, thus all such system are, by
necessity, and so proven, to be incomplete.
Your denial just shows your ignorance.
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