The theorem stating that no consistent, effectively axiomatized formal theory that is capable of representing enough arithmetic can prove its own consistency, when consistency is formalized within the theory itself; informally, a theory cannot certify its own freedom from contradiction by means available inside it.

Principle

By formalizing a provability predicate and using the same diagonalization techniques as in the first theorem, one shows that if a theory T proved Con(T) (a sentence that formalizes 'there is no proof of a contradiction in T'), then T would be able to prove a sentence that it cannot consistently prove, yielding inconsistency.

Demonstration

In Peano Arithmetic one represents the predicate 'x is a proof in PA' and formulates a sentence Con(PA). Gödel-style reasoning produces a sentence G such that if PA proved Con(PA), then PA would prove G and hence would contradict the conclusion of the first theorem; therefore, if PA is consistent it cannot prove Con(PA).

Misapplication

Claiming the theorem forbids any external or stronger-system consistency proofs — the theorem only blocks a theory from proving its own consistency by resources formalizable inside it; it does not preclude metamathematical or stronger-system proofs of consistency.

Consequence

Consistency proofs for a theory must either come from stronger theories, non-formal methods, or assumptions not formalizable in the theory itself; this shapes the hierarchy of foundational justifications and indicates unavoidable relativization of consistency claims.

Reversal

The contrapositive consequence is often stated: if a theory T proves Con(T) then T is inconsistent; thus proving one's own consistency inside the same formal system is tantamount to inconsistency under the theorem's formal hypotheses.

Boundary

Requires effective axiomatizability and sufficient arithmetic to represent provability and proof predicates; it does not apply to systems lacking the ability to represent those syntactic notions or to frameworks that change the formalization of consistency.

Semantic Tension

Creates tension between internal and external perspectives on proof: internal provability of Con(T) is ruled out, yet externally (in a stronger meta-theory) one may prove Con(T); this distinguishes syntactic self-certification from metatheoretic validation.

Synthesis

Gödel's second theorem formalizes the intuition that a system cannot, by its own formal resources, establish that those resources never produce contradiction: by coding proofs and provability inside the theory and applying diagonal arguments one shows any such internal proof collapses into inconsistency.