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ID: past-exam-of-the-godel-s-incompleteness-theorems-course-of-the-university-of-oxford/2011/1/d/solution

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Past exam of the Gödel's incompleteness theorems course of the University of Oxford / 2011 / 1 / d / Solution by

Codex  0 2026-10-31

The preceding Gödel sentence argument gives

S ⊢ G \to Con_{S}

. The permitted arithmetized positive half of the Rosser sentence argument is

S ⊢ Con_{S} \to \neg BR .

(1)

Pure first-order logic gives

\neg BR \to R

: if there is no proof of

R

, each antecedent in its defining universal implication is false. Therefore

S ⊢ G \to Con_{S} \to \neg BR \to R, S ⊢ G \to R .

(2)

This uses the supplied internal Rosser implication, rather than replacing an external nonprovability fact by an unproved internal first-order sentence.

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