Discussion:
Refuting Incompleteness and Undefinability Version(14)
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peteolcott
2018-11-05 17:33:46 UTC
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A world class expert provided some coaching. They have published very much in the field of Incompleteness and many related fields.

They changed my formulation of a correct simplification of Gödel's 1931 Incompleteness Theorem:
∀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)

As long as any simplification of the original theorem sufficiently captures the essence of the original theorem, any refutation of this simplification applies equally to the original theorem by analogy.

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

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

If the above expression evaluates to False it refutes every Gödel sentence in every F ⊇ Q.

Copyright 2018 Pete Olcott
Arnaud Fournet
2018-11-05 21:20:57 UTC
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Post by peteolcott
A world class expert provided some coaching. They have published very much in the field of Incompleteness and many related fields.
∀F ∈ Formal_Systems (∃G ∈ F (G ↔ ∃Γ ⊆ F ~(Γ ⊢ G)))
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)
As long as any simplification of the original theorem sufficiently captures the essence of the original theorem, any refutation of this simplification applies equally to the original theorem by analogy.
Since the following G is neither Provable nor Refutable in F it forms a Gödel sentence in F.
∃F ∈ Formal_Systems ⊇ Q (∃G ∈ L(F) (G ↔ ~(F ⊢ G)))
If the above expression evaluates to False it refutes every Gödel sentence in every F ⊇ Q.
Copyright 2018 Pete Olcott
could you avoid posting twice the same junk??
Franz Gnaedinger
2018-11-06 08:59:55 UTC
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Post by peteolcott
Copyright 2018 Pete Olcott
Peter Olcott knows the absolute and complete and total truth, he is the author
of life and creator of life, he has hundred reasons to believe that he is God,
he creates our future minds in order that we can go on existing, he is both
a human being and God, he is the one Creator of the Universe (claims he made
in sci.lang). His message in nuce, nella luce della verità: Godel was a doedel,
Olcott is Allgod.

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