peteolcott

2018-11-03 20:27:48 UTC

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Permalinkwhich a certain amount of arithmetic can be carried out, there are statements of the

language of F which can neither be proved nor disproved in F. (Raatikainen Fall 2018)

Formalization of the above English:

∀F ∈ Formal_Systems

(

∃G ∈ F (G ↔ ∃Γ ⊆ F ~(Γ ⊢ G))

)

Proved(G) means the satisfaction of: (Γ ⊢ G)

thus the satisfaction of ∃Γ ⊆ F ~(Γ ⊢ G)

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

Disproved(G) means the satisfaction of: (Γ ⊢ ~G)

thus the satisfaction of ∀Γ ⊆ F (Γ ⊢ G)

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

Taking as a hypothesis that the following correctly formalize the intuitive notions of True and False:

(1) ∀x (True(x) ↔ ⊢x)

(2) ∀x (False(x) ↔ ⊢~x)

Along with this common knowledge: ∀x (Logic_Sentence(x) ↔ (True(x) ∨ False(x))

We derive a formal expression of logic that can reject semantically ill-formed WFF:

∀x (Logic_Sentence(x) ↔ (⊢x ∨ ⊢~x))

Since neither G nor ~G is satisfiable therefore neither True(G) nor False(G)

∴ Logic_Sentence(G)

Copyright 2018 Pete Olcott

Raatikainen, Panu, "Gödel's Incompleteness Theorems",

The Stanford Encyclopedia of Philosophy (Fall 2018 Edition), Edward N. Zalta (ed.),

URL = <https://plato.stanford.edu/archives/fall2018/entries/goedel-incompleteness/>.