G asserts its own unprovability in F
Posted: Mon Apr 17, 2023 4:47 am
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 proved in F requires 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”
Gödel knew about this contradiction:
..."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 or meta_F 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 sums up the essence of his own proof as:
"We are therefore confronted with a proposition which asserts its own
unprovability." (Gödel 1931:39-41)
Gödel, Kurt 1931. On Formally Undecidable Propositions of Principia
Mathematica And Related Systems
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 proved in F requires 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”
Gödel knew about this contradiction:
..."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 or meta_F 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 sums up the essence of his own proof as:
"We are therefore confronted with a proposition which asserts its own
unprovability." (Gödel 1931:39-41)
Gödel, Kurt 1931. On Formally Undecidable Propositions of Principia
Mathematica And Related Systems