Past exam of the model theory and Gödel's incompleteness theorems course of the University of Oxford / 2010 / A / 3

Abbreviate the arithmetical provability predicate by $\square$, with the Hilbert-Bernays-Löb derivability conditions understood. Löb's theorem saysChoose a diagonal sentence $D$ with $S⊢D\leftrightarrow (\square D\to A)$. Internalizing its forward implication gives $S⊢\square D\to \square (\square D\to A)$. Internal modus ponens rule gives $\square (\square D\to A)\to (\square \square D\to \square A)$, while positive introspection gives $\square D\to \square \square D$. Combining these yields $S⊢\square D\to \square A$.

Now use the assumed theorem $\square A\to A$. It follows that $S⊢\square D\to A$, and the reverse direction of the diagonal equivalence gives $S⊢D$. External necessitation rule gives $S⊢\square D$, and the previously derived implication gives $S⊢A$. This proves the theorem from the supplied fixed-point property and the derivability conditions.

The Gödel second incompleteness theorem states that a consistent arithmetic theory in this setting cannot prove its own consistency sentence ${\mathrm{Con}}_S=\neg \square \perp$. If it could, it would prove $\square \perp \to \perp$; applying Löb's theorem with $A=\perp$ would make it inconsistent. Therefore $\boxed{S⊬{\mathrm{Con}}_S}$ when $S$ is consistent.

The language of Gödel-Löb logic has propositional variables, $\perp$, implication, and the unary operator $\square$; the other logical connectives are abbreviations. Its axioms are all classical tautologies, the distribution scheme $\square (A\to B)\to (\square A\to \square B)$, and the Löb scheme $\square (\square A\to A)\to \square A$. Its rules are modus ponens rule, necessitation rule of theorems, and uniform substitution.

For a single boxed equation $p\leftrightarrow \square C(p)$, set $F=\square C(\top )$. The identity supplied in the source gives directly $GL⊢F\leftrightarrow \square C(F)$. If $p$ does not occur, take $F=\square C$ instead.

Proceed by induction for a system of $m$ equations, allowing other propositional variables as parameters. Treat $p_1,\dots ,p_{m-1}$ as parameters in the last equation and putThe one-variable result proves $D\leftrightarrow \square C_m(p_1,\dots ,p_{m-1},D)$. Substitute this $D$ for $p_m$ in the first $m-1$ equations. They are still boxed equations, so induction supplies solutions $F_1,\dots ,F_{m-1}$. Define $F_m=D(F_1,\dots ,F_{m-1})$.

By construction, the first $m-1$ equations hold after this substitution; substituting into the theorem for $D$ proves the last. HenceThe elimination introduces no new parameter letters and removes every unknown letter. This gives a constructive simultaneous boxed fixed-point elimination in GL.

Take the outermost boxed subformulas of $X(p)$ containing the distinguished letter. Because $X$ is modalized in $p$, every occurrence of $p$ lies inside one of these boxes. The supplied decomposition therefore has the formwhere $H$ has no occurrence of $p$. Other letters and subformulas not involving $p$ may be retained as parameters in $H$ and the $C_i$.

Introduce temporary letters $q_1,\dots ,q_m$ and solve the boxed systemby part (b), obtaining $Q_i$ without the temporary letters or $p$. Put $F=H(Q_1,\dots ,Q_m)$. Each solved equation gives $GL⊢Q_i\leftrightarrow \square C_i(F)$. The supplied replacement-of-equivalents principle then givesOnly letters already present in $X$, apart from the removed $p$, occur in the constructed $F$. If $p$ is absent from $X$ to begin with, take $F=X$. This is the modalized fixed-point theorem for GL, proved by finite elimination rather than a semantic completeness argument.

One can takeFor the first formula, the Löb axiom with conclusion $\perp$ gives $\square (\square \perp \to \perp )\to \square \perp$, that is, $\square F_1\to \square \perp$. Conversely, the tautology $\perp \to F_1$, necessitation rule, and distribution give $\square \perp \to \square F_1$. Thus $GL⊢\square F_1\leftrightarrow \square \perp$. Negating both sides gives $GL⊢\neg \square F_1\leftrightarrow F_1$.

For the second formula, $\neg F_2=\top$ is a theorem, so $GL⊢\square \neg F_2$. Hence $\neg \square F_2\wedge \neg \square \neg F_2$ is false in the logic and is equivalent to $\perp =F_2$. A fixed point is a formula satisfying the equivalence; it need not itself express a nontrivial undecidable assertion.