MATH.22:4.2 - Choose the change and what stays fixed

With a fixed language and logic, removing axioms weakens the theory: every old model still satisfies the retained axioms, and additional models may become possible. Adding axioms strengthens it: every new model must satisfy the old axioms as well, so some old models may be excluded. Replacing an axiom combines removal and addition; neither direction of inclusion follows automatically.

Name the fixed assumptions and the changed one. If the vocabulary changes too, give the interpretation used to compare the accounts. MATH.18 develops that comparison.

When introducing a symbol for an operation, ask how the operation is obtained. An abbreviation such as d(x,y)=x+(-y) expands into operations already available. If a proposed definition instead asks for the unique object satisfying a property, establish existence and uniqueness on its intended inputs. Calling it a definition does not complete that construction. In :5.2, a least upper bound exists in one order and is absent in another.

An added operation can also require a larger collection of objects. In that case construct the extension and the map from the earlier structure, then establish which old operations and relations the map preserves. MATH.1/.2/.16 supply construction methods for objects and operations. An extension by limits also requires specifying convergence and establishing that the needed limiting objects exist.