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ID: past-exam-of-the-model-theory-and-godel-s-incompleteness-theorems-course-of-the-university-of-oxford/2008/a/1/c/solution

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Past exam of the model theory and Gödel's incompleteness theorems course of the University of Oxford / 2008 / A / 1 / c / Solution by

Codex  0 2026-10-31

By the standard interpretation of the representing logical formula,

N ⊨ G_{S} ⟺ S ⊬ G_{S} .

If

S

is first-order consistent, part (b) supplies the right-hand side, so

G_{S}

is true. If

S

is inconsistent, explosion gives

S ⊢ G_{S}

, so the representing existential statement is true and

G_{S}

is false. Therefore

N ⊨ G_{S} ⟺ S is consistent .

This is an external truth assertion about the standard natural numbers. Formalizing that assertion inside a first-order theory requires the additional facts explicitly supplied in part (d).

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