##  [Conservative Extension](/conservative-extension-0) 

 Definition

An extension T' of a theory T is conservative if T' proves no new sentences in the language of T beyond those provable in T itself; equivalently, every sentence in the original language provable in the extension was already provable in the base theory.

 

 

 

 

 

 





## Principle

Principle

Conservativity requires that added axioms or symbols serve only to abbreviate or organize content without yielding novel consequences expressible in the original vocabulary; the extension cannot settle additional original-language questions that T left open.

 

 

 

 

 





## Demonstration

Demonstration

If T is a first-order theory and T' is obtained by adding a fresh function symbol f together with an axiom that defines f(x) as a term t(x) already expressible in T, then T' is a conservative extension of T because any sentence in T's language provable in T' can be rewritten to a proof in T by eliminating f via its definition.

 

 

 

 

## Misapplication

Misapplication

Assuming that any syntactic enlargement that preserves consistency is conservative. An extension can be consistent yet nonconservative: it may prove new theorems in the old language (for example, adding an axiom scheme that implies a formerly independent arithmetic statement), so consistency alone does not imply conservativity.

 

 

 

 

 





## Consequence

Consequence

A conservative extension is safe for theoretical practice: one can introduce definitions, abbreviations, or convenient machinery without changing the theory's commitments about the original subject matter; results about the original vocabulary remain unchanged.

 

 

 

 

## Reversal

Reversal

A nonconservative extension is one that does add new theorems in the original language; such an extension changes the theory's content on the original topics and can be used to settle previously independent sentences.

 

 

 

 

 





## Boundary

Boundary

Conservativity is a claim about sentences in the base language; it says nothing about new theorems that mention added symbols. It presupposes a fixed base language and does not automatically apply when the extension changes signature in a way that alters what counts as 'original' sentences.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Tension arises between conservative extension and definitional extension: all definitional extensions are conservative, but not every conservative extension is definitional. There is also tension with interpretability: an interpretable extension may or may not be conservative depending on how the interpretation affects original-language sentences.

 

 

 

 

 





## Synthesis

Synthesis

A conservative extension augments a theory with new apparatus while guaranteeing that no additional claims about the original vocabulary become provable; it formalizes the intuition of safe notational or methodological enrichment without altering core consequences.