peteolcott

2018-11-05 02:11:10 UTC

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Permalinkin the field of Incompleteness and many related fields.

They changed this:

∀F ∈ Formal_Systems (∃G ∈ F (G ↔ ∃Γ ⊆ F ~(Γ ⊢ G)))

into this:

L(F) means the language of formal system F.

∀F (F ∈ Formal_Systems & Q ⊆ F) → ∃G ∈ L(F) (G ↔ ~(F ⊢ G))

Q here is Robinson Arithmetic (the theorem fails for some weaker formal systems)

I realized that Q is not needed if the following expression evaluates to False:

∃F ∈ Formal_Systems ∃G ∈ L(F) (G ↔ ~(F ⊢ G))

The following analysis seems to refute Gödel 1931 Incompleteness as long as

the term "satisfiable" is interpreted using the conventional meanings of the

symbols within symbolic logic.

If G was Provable in F this contradicts its assertion: G is not Provable in F

If ~G was Provable in F this contradicts its assertion: G is Provable in F.

Since G is neither Provable nor Refutable in F it forms a Gödel sentence in F.

Because G is not satisfiable in any Formal System F, the Gödel sentence does not exist.

Copyright 2018 Pete Olcott