Past exam of the model theory and Gödel's incompleteness theorems course of the University of Oxford / 2007 / A / 1 / c / Solution

Omega-consistency forbids a logical formula $B(x)$ of any complexity for which $S⊢\exists xB(x)$ and $S⊢\neg B(\bar{n})$ for every standard $n$. The difference from one-consistency is that the matrix is unrestricted. It implies first-order consistency.

For a complete first-order consistent classical first-order theory, theorem membership obeys the ordinary Boolean truth tables. For example, it proves a conjunction exactly when it proves both conjuncts; completeness and first-order consistency also make theoremhood of a negation the complement of theoremhood of the first-order sentence. Omega-consistency additionally gives the witness propertyThe reverse implication is existential introduction. For the forward implication, if no standard instance were provable, completeness would give a formal proof of each negated instance, violating omega-consistency. This works also after replacing additional free variables by numerals.

Now induct on logical formulas with their remaining free variables replaced by standard numerals. At the atomic level, the equivalence between theoremhood and standard truth follows from bounded completeness of an arithmetic theory and bounded soundness of an arithmetic theory from (b). The Boolean steps use the truth-table facts just proved. The existential step uses the witness property and the induction hypothesis on its matrix; universal quantification follows by negation. Hencefor every first-order sentence $X$. Completeness ensures it decides each first-order sentence, and this equivalence ensures every decision is correct.