From olcott@21:1/5 to All on Sat Apr 15 15:33:26 2023
XPost: sci.logic, sci.math, alt.philosophy
"We are therefore confronted with a proposition which asserts its own unprovability." (Gödel 1931:39-41)
If we take the simplest possible essence of Gödel's logic sentence we
have: G asserts its own unprovability in F.
This means that G is asserting that there is no sequence of inference
steps in F that derives G.
For G to be satisfied in F there would have to be a sequence of
inference steps in F that proves there is no such sequence of inference
steps in F.
This is like René Descartes saying:
“I think therefore thoughts do not exist”
..."there is also a close relationship with the “liar” antinomy,14"
(Gödel 1931:39-41)
"14 Every epistemological antinomy can likewise be used for a similar undecidability proof."(Gödel 1931:39-41)
So we can see from the above that it is true that G is unprovable in F,
yet without arithmetization and diagonalization hiding the reason why G
is unprovable in F we can see that G is unprovable in F because G is self-contradictory in F, not because F is in anyway incomplete.
Gödel, Kurt 1931. On Formally Undecidable Propositions of Principia Mathematica And Related Systems
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Copyright 2023 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer