Gödel's Second Incompleteness Theorem

The second incompleteness theorem strengthens the first by showing that a consistent, sufficiently strong theory cannot prove its own consistency. This has profound implications for the foundations of mathematics and Hilbert's program.

Statement

Definition6.7Consistency Statement

For a theory $T$ , the consistency statement $\mathrm{Con}(T)$ is the arithmetic sentence $\neg \mathrm{Prov}_T(\ulcorner 0 = 1 \urcorner)$ , which formalizes " $T$ does not prove $0 = 1$ ." This is equivalent to " $T$ is consistent" within reasonable formalizations.

Definition6.8Second Incompleteness Theorem

If $T$ is a consistent theory extending Peano arithmetic (or any sufficiently strong, recursively axiomatized theory), then $T \not\vdash \mathrm{Con}(T)$ . In words: $T$ cannot prove its own consistency.

ExampleZFC Cannot Prove Con(ZFC)

Assuming ZFC is consistent, ZFC $\not\vdash \mathrm{Con}(\mathrm{ZFC})$ . If we add $\mathrm{Con}(\mathrm{ZFC})$ as a new axiom, the resulting theory ZFC + Con(ZFC) is stronger, but by the second incompleteness theorem applied to itself, it cannot prove its own consistency either. This creates an infinite ascending chain of consistency strengths.

Proof Idea

RemarkProof Sketch

The key insight is that the proof of the first incompleteness theorem can be formalized within $T$ itself. Specifically, one can prove in $T$ that: $T \vdash \mathrm{Con}(T) \to G,$ where $G$ is the Gödel sentence. Since $G$ is unprovable in $T$ (by the first theorem), $\mathrm{Con}(T)$ must also be unprovable. The formalization requires the Hilbert-Bernays derivability conditions for $\mathrm{Prov}_T$ .

Hilbert-Bernays Conditions

Definition6.9Derivability Conditions

The Hilbert-Bernays derivability conditions for $\mathrm{Prov}_T$ are:

If $T \vdash \sigma$ , then $T \vdash \mathrm{Prov}_T(\ulcorner \sigma \urcorner)$ .
$T \vdash \mathrm{Prov}_T(\ulcorner \sigma \to \tau \urcorner) \to (\mathrm{Prov}_T(\ulcorner \sigma \urcorner) \to \mathrm{Prov}_T(\ulcorner \tau \urcorner))$ .
$T \vdash \mathrm{Prov}_T(\ulcorner \sigma \urcorner) \to \mathrm{Prov}_T(\ulcorner \mathrm{Prov}_T(\ulcorner \sigma \urcorner) \urcorner)$ .

These conditions suffice for the proof of the second incompleteness theorem.

RemarkLöb's Theorem

A beautiful extension: Löb's theorem states that if $T \vdash \mathrm{Prov}_T(\ulcorner \sigma \urcorner) \to \sigma$ , then $T \vdash \sigma$ . Setting $\sigma = \bot$ (falsehood) recovers the second incompleteness theorem as a special case.