Past exam of the logic and set theory course of the University of Oxford / 2009 / A / 3 / c

A graph must assign exactly one output to every input, while the distinguished unary predicate must select exactly one element. Suitable sentences areThe existential clauses provide totality and nonemptiness of the selected output, respectively; the universal clauses provide uniqueness. Thus a structure satisfying $\varphi ,\psi$ uniquely determines a total unary function $f$ and a constant $c$ on its unchanged domain. Conversely any interpretations of $f,c$ determine such relations by $P(x,y)⟺f(x)=y$ and $Q(x)⟺x=c$.

For each term $t$ define an $L_2$-formula $E_t(z)$ expressing its value. Use fresh bound variables at every step:Freshness includes avoiding all relevant free variables of the formula being translated. Define the atomic translation byand extend through the logical operations: $\sigma (\neg \theta )=\neg \sigma (\theta )$, $\sigma (\theta \to \eta )=\sigma (\theta )\to \sigma (\eta )$, and $\sigma (\forall x\theta )=\forall x\sigma (\theta )$. Fresh witnesses prevent variable capture and ensure the translation preserves the original free-variable set.

For an $L_1$-structure $A$, let $A^r$ have the graph relation of $f^A$ and singleton relation $\{c^A\}$. Induction on terms proves that $E_t(z)$ holds precisely when $z$ is the value of $t$. Induction on formulas then provesConversely every $L_2$-structure satisfying $\varphi ,\psi$ is uniquely the relational version of an $L_1$-structure on that same domain. Hence, for arbitrary possibly infinite $\Gamma$, semantic entailment on the left is equivalent to semantic entailment from $\sigma (\Gamma )\cup \{\varphi ,\psi \}$ on the right, with assignments retained in the free-formula case. Applying soundness theorem for first-order logic and Gödel's completeness theorem givesThe prohibition on completeness in the preceding direct equality question is local to that proof; here it supplies the clean semantic-to-syntactic transfer after the actual translation and model correspondence have been established.