MATH.6:1 - Problem frame

Use this pattern when a proposed mathematical implication might fail and you need a construction that settles the failure. Examples include a claimed property of every relation of a certain kind, a proposed inverse, or an interchange of quantifiers that would make a result more useful.

A countermodel gives mathematical objects, operations or relations in which the assumptions hold and the proposed conclusion fails. A counterexample to a statement about fixed objects supplies the particular values that make it fail. Both let you stop trying to prove the original claim and identify a useful change of question or assumptions.

Start by stating what must remain true and what would defeat the conclusion. Then construct one such case. Return the case and its decisive calculation or argument. A familiar counterexample already satisfying the present assumptions can finish the work immediately.

The reader needs elementary sets, relations, functions and quantified statements. The constructions below use ordinary mathematical truth and explicit witnesses; symbolic logic notation is explained where it changes the construction. A proof of a true claim, an estimate of how often a procedure fails, and an observation about a physical system require their corresponding methods. Countermodel construction answers whether the stated mathematical assumptions force the conclusion.