##  [Rosser Sentence](/rosser-sentence-0) 

 Definition

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

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

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

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

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

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

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

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

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.