Ontologisches Beweis

Ass.1: P(φ).P(ψ) ⊃ P(φ.ψ)
Ass.2: P(φ) ∨ P(-φ)

Def.1: G(x) ≡ (φ)[P(φ) ⊃ φ(x)]
Def.2: φ Ess. x ≡ (ψ)[ψ(x) ⊃ N(y){φ(y) ⊃ ψ(y)]]

Ass.3a: P(φ) ⊃ NP(φ)
Ass.3b: ∼P(φ) ⊃ N∼P(φ)

Theo.1: G(x) ⊃ G Ess. x

Def.3: E(x) ≡ (φ)[φ Ess. x ⊃ N(∃x) φ(x)]

Ass.4: P(E)

Theo.2: G(x) ⊃ N(∃y)G(y)

therefore:
(∃x)G(x) ⊃ N(∃y)G(y)
M(∃x)G(x) ⊃ MN(∃y)G(y)
M(∃x)G(x) ⊃ N(∃y)G(y)
Ass.5: P(φ).N(φ⊃ψ):⊃P(ψ)

K. Gödel, 1970

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