##  [Conservativity Gap](/conservativity-gap-0) 

 Definition

A failure mode in theory extension where adding new symbols, axioms or principles yields new theorems in the language of the original theory that were not derivable before, thus violating the expected conservativity of the extension.

 

 

 

 

 

 





## Principle

Principle

A conservative extension should not prove any new sentences expressible in the original language; a conservativity gap occurs when the extension's additional expressive power or indirect consequences enable derivations of formerly unprovable base-language sentences.

 

 

 

 

 





## Demonstration

Demonstration

Adjoining a new predicate with axioms that implicitly encode combinatorial choices can lead to new arithmetic consequences not provable in the base theory; adding a truth predicate or reflection principle can similarly produce a conservativity gap over the underlying arithmetic.

 

 

 

 

## Misapplication

Misapplication

Assuming that definitional or syntactic-looking additions are automatically conservative without checking for hidden entailments or implicit coding that affects base-language sentences.

 

 

 

 

 





## Consequence

Consequence

Detecting a conservativity gap forces a re-evaluation of the extension: one may restrict axioms, provide explicit definitional reduction, or accept that the extension increases the base theory's consequences and adjust interpretations or applications accordingly.

 

 

 

 

## Reversal

Reversal

Conservative extension: an addition of symbols and axioms that preserves precisely the theorems in the original language and yields no new base-language consequences.

 

 

 

 

 





## Boundary

Boundary

Concerns relations between theories and languages; excludes changes that alter the language itself (which trivially change expressive power) unless one traces consequences back into the original vocabulary. Applicable across syntactic and semantic notions of consequence.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension lies between enriching a theory to gain expressive or explanatory power and preserving prior conclusions; the gap exposes the trade-off between conservativity and enhanced expressiveness or convenience.

 

 

 

 

 





## Synthesis

Synthesis

A conservativity gap diagnoses when an extension crosses the line from harmless enrichment to substantive increase of base-language content: it compels precise analysis of definitional status, implicit coding, and the interpretability relations between theories.