MATH.22:2 - Problem

A familiar axiom set can make a useful construction impossible or unnecessarily restrictive. Dropping a premise admits more structures, but a familiar proof may then fail. Adding a convenient operation can conceal an existence assumption, and adding a desired law can conflict with laws already retained.

There are different outcomes to establish. A proof might survive unchanged, admit a new proof, need an extra condition, or have a counterexample. A stopped proof search leaves these possibilities unresolved. The revised theory must expose the difference because later reasoning and computation consume its consequences.