MATH.22:1 - Problem frame

Use this pattern when a mathematical assumption excludes a construction you need, a theorem uses a stronger premise than its intended application supplies, or two accounts differ in what they assume. The work is to change the assumptions and establish what the changed theory permits.

An axiom is a statement taken as a premise of the theory being developed. A model of those axioms is a mathematical structure in which they hold under a stated interpretation. For example, an ordered set interprets an order relation; a collection of permutations interprets composition. This use of model concerns mathematical satisfaction. Applying the structure to an observed subject requires the additional correspondence work in C.29 and MMP.

First useful move: isolate one assumption and one construction or conclusion affected by it. Try to derive that conclusion from the remaining assumptions, or construct a case where they hold and the conclusion fails. The result identifies what can be retained and what must change.

The reader needs elementary reasoning about sets, operations, relations and their laws. The examples introduce the group, order and arithmetic assumptions they use. A change of logical inference rules needs the corresponding knowledge of that logic.

If a supplied theorem already works under the available assumptions, apply it. MATH.6 suffices for refuting one proposed consequence. Use the present method when the receiving work needs a revised theory, its changed constructions and its surviving results.