MATH.22:4 - Solution

Local mantra: isolate the assumption and wanted construction; choose the change; build or interpret the changed structures; trace the affected arguments; establish what the comparison warrants; return the revised theory to use.

MATH.22:4.1 - Identify the assumption and its mathematical job

State the objects, operations and laws currently used. Find the step where the disputed assumption enters. It may justify rearranging operations, comparing two objects, choosing an element, taking a limit, or introducing an object with a required property.

Distinguish a law about the objects from a logical rule used to infer statements about them. Removing commutativity changes the structures being studied. Restricting proof by contradiction can change the permitted arguments even when the question concerns the same objects.

State the desired gain. Examples include admitting noncommuting operations, retaining incomparable alternatives, obtaining a missing limit, or preserving the computational content of a proof. This gain guides which consequences to examine first.

For a local question, the relevant definitions and proof dependencies are enough. Reconstruct a whole formal foundation only when the conclusion being sought depends on it.

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.

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.

MATH.22:4.4 - Trace the consequences that the receiving work uses

For each required result, locate the affected proof step.

• Retain a proof when its steps use only assumptions that remain.
• Try another proof when the old one uses the removed assumption. A dependency of one proof need not be a necessary premise of the theorem.
• Keep a conditional result when the receiving use can supply the additional premise.
• Return a countermodel when the conclusion fails under the revised assumptions.
• Leave a particular consequence unresolved when neither an argument nor a countermodel has been obtained.

Apply the same reasoning to constructions. Check whether their inputs are still admitted, the operations remain defined, and the properties used by the next step still follow. For example, :5.1 retains cancellation after dropping commutativity but loses unrestricted rearrangement of factors.

When a proof is formalized, its recorded dependencies can help find affected steps. They report what that proof uses. A claim that no proof can avoid an axiom requires an additional argument.

If the result will generate data or a program, trace what the new principle supplies operationally. An existence argument and an effective construction support different next uses. MATH.12 handles extraction; the relevant computational method handles execution and cost.

MATH.22:4.5 - State what the model or proof establishes

In a sound interpretation of a deductive system, a proof preserves truth in every model of its premises. Therefore a model of T where A fails rules out a proof of A from T. If another model of T satisfies A, it likewise rules out a proof of not-A. Together these establish that A is independent of T, relative to the logic and semantics being used.

A model also supports consistency: a sound derivation of a contradiction from its axioms would have to make a contradiction true in that model. The construction of the model relies on background mathematics. Keep that dependence when reporting a consistency result, especially when comparing foundations. A failure to find a contradiction supplies no such construction.

For a definitional extension, expand the new symbols in the affected claims and arguments. When all new operations are definable in the old theory and the definitions can be eliminated, conclusions stated wholly in the old language retain their old justification. If an existence principle or inference rule has been added, examine its consequences separately.

The preceding model arguments use their stated semantics. A change to intuitionistic, dependent-type or another logic requires an interpretation sound for its own rules; an arbitrary transfer of classical model arguments could answer the wrong question.

MATH.22:4.6 - Use the revised theory

Return the changed assumptions together with the construction, theorem or obstruction needed by the receiving work. Carry the conditions under which it can be used. An unresolved consistency or consequence question can be handed to a collaborator without presenting it as settled.

Choose between revised theories by the work they enable and the costs they impose. One can admit more objects, another can make a needed construction available, and another can retain an effective procedure. FPF’s ordinary characterization and choice methods apply when these alternatives must be compared; the present method supplies their mathematical consequences.

Use the changed result to reformulate a model, modify a method, or pose the next mathematical problem. For a new conjecture, state which further relation might hold under the retained assumptions. A later failure reopens the assumption or inference on which the failed use depends.