An extension of a formal theory obtained by adding new axioms (or axiom schemata) that yields at least one new theorem expressible in the original language of the theory; in other words, the extension is not conservative over the base theory for sentences of the old language.

Principle

If added principles entail sentences in the original language that were not derivable before, the extension is nonconservative; conservativity is judged relative to the original signature and sentence class.

Demonstration

Starting from Peano Arithmetic (PA), adjoin an axiom that asserts the existence of a new element satisfying a property expressible in the language of PA; if this added axiom allows proving a previously unprovable arithmetical sentence in the language of PA, the extension is nonconservative.

Misapplication

Asserting that any expansion that introduces new nonlogical symbols is nonconservative — definitional or conservative extensions can add symbols without producing new theorems in the original language.

Consequence

Nonconservative extensions change the theory's consequences in the original language, affecting what is provable and potentially altering ontological commitments or risking inconsistency if poorly chosen.

Reversal

A conservative extension adds axioms or symbols but provably yields no new sentences in the original language; it preserves all original-language consequences.

Boundary

The notion applies only relative to a specified base theory and its original language; it excludes purely definitional expansions and depends on whether one considers syntactic versus semantic conservativity.

Semantic Tension

Tension arises between syntactic conservativity (no new derivable sentences) and semantic conservativity (preservation of models), and between conservativity relative to formulas versus schemata or classes of sentences.

Synthesis

A nonconservative extension is a deliberate enlargement of a theory that produces new original-language conclusions; it signals a genuine change in the theory's deductive power and must be judged relative to the chosen signature and sentence class.