MATH.22:4.3 - Construct models and separating cases

Give the objects and the interpretation of every operation and relation needed by the claim. Establish each retained axiom. For an infinite family, use a general argument; enumerated cases suffice only when they exhaust the admitted possibilities.

Choose a case that distinguishes the changed theories. To investigate an axiom A relative to retained assumptions T, a model of T in which A fails shows that A is not a consequence of T. A model where A holds shows that adding A is compatible with that model. These are different results, and both can matter.

Use MATH.6 to construct a countermodel. Finite structures are often a cheap first attempt because their operations and relevant failures can be displayed completely. An existing infinite structure with a short argument may be simpler.

If the changed account uses different objects or primitives, an interpretation can carry its constructions into an established theory. Check the translated axioms and the translated steps of inference needed for the result. A picture suggesting an analogy is a starting point for this work.