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

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

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

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

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

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

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

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

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.