A variant of a Gödel sentence constructed using Rosser's trick that weakens the hypotheses needed for incompleteness: it yields an undecidable sentence for any consistent, recursively enumerable theory without requiring ω-consistency, by using a modified provability predicate comparing proof lengths.

Principle

Rosser replaces the standard provability predicate with a Rosser provability relation that asserts 'there is a proof of me and there is no shorter proof of my negation', or equivalently uses a witness ordering to avoid the stronger ω-consistency assumption used in Gödel's original argument.

Demonstration

For a recursively axiomatizable T one defines a Rosser-style predicate RProv_T(x) that quantifies about shorter counterproofs and uses diagonalization to produce a sentence R satisfying T ⊢ (R ↔ ¬RProv_T(⌜R⌝)). If T is merely consistent, arguments about minimal proofs show T can neither prove R nor ¬R, so R is undecidable in T.

Misapplication

Assuming Rosser sentences remove all metamathematical subtlety: they weaken the hypothesis from ω-consistency to consistency but still require effective axiomatizability and correct formalization of proof comparisons; they do not make all incompleteness arguments trivial or erase model-theoretic distinctions.

Consequence

Shows incompleteness holds under weaker consistency hypotheses and clarifies the role of proof-comparison in undecidability; it refines Gödel's result by lowering the consistency requirement for constructing undecidable sentences.

Reversal

If a theory proves the Rosser sentence or its negation, the proof-analysis used in the Rosser construction yields a contradiction, so provability of either discloses inconsistency under the construction's assumptions.

Boundary

Applies to recursively enumerable (effectively axiomatized) theories that can formalize proof and proof-length/comparison notions; it does not apply to non-effective systems or to formalisms that cannot represent the required ordering of proofs.

Semantic Tension

Competes with the Gödel sentence by trading off hypothesis strength and simplicity of the provability predicate: Rosser sentences use more elaborate proof-comparison machinery to achieve undecidability under weaker assumptions, producing different technical behaviors and metamathematical interpretations.

Synthesis

A Rosser sentence is an arithmetized self-referential sentence crafted via a provability notion sensitive to proof comparisons; by avoiding ω-consistency it demonstrates that mere consistency plus effective axiomatizability suffices to produce undecidable sentences in formal theories.