TheoremProved

Gödel's second incompleteness theorem

Statement

Let $F$ be a consistent formal system satisfying the hypotheses of the first incompleteness theorem, and let $\mathrm{Con}(F)$ be the arithmetic sentence expressing 'no proof of a contradiction exists in $F$ '. Then $F$ cannot prove $\mathrm{Con}(F)$ .

Why is it true?

The entire argument proving the first incompleteness theorem can itself be carried out and verified inside $F$ , yielding $F\vdash \mathrm{Con}(F)\to G$ (the sentence $G$ from the first theorem). If $F$ also proved $\mathrm{Con}(F)$ , it would prove $G$ , contradicting the first theorem. So no sufficiently strong consistent system can certify its own consistency from within.

Proof sketch

Formalize the entire proof of the first incompleteness theorem as a sequence of arithmetic statements derivable inside $F$ itself (this requires care: the provability predicate $\mathrm{Prf}$ must satisfy the Hilbert–Bernays provability conditions), obtaining $F\vdash \mathrm{Con}(F)\to G$ ; since $F\nvdash G$ by the first theorem (assuming consistency), it follows $F\nvdash \mathrm{Con}(F)$ .

Proved by

proved 1931Kurt Gödel

Topics that use this theorem

Gödel's incompleteness theorems

Related theorems

Gödel's first incompleteness theorem Gödel's completeness theorem

Step-by-step proofs

No step-by-step proof yet for this theorem.

References

Kurt Gödel (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I · DOI:10.1007/s00605-006-0423-7
Martin Davis (ed.) (1965). The Undecidable