MATH.22:11 - SoTA-Echoing

The working question is how to change mathematical assumptions while retaining justified and useful consequences. The selected line combines proof-dependency analysis, construction of models and interpretations, and explicit separation of definitional change from added principles.

Adopt the model-and-consequence method. The Open Logic Project’s Models and Theories makes the relation between axioms and satisfying structures explicit. Logic and Proof, §§10.1-10.5 supplies interpretation and soundness for the first-order branch. These are method sources for :4.3/:4.5. Their use defeats an inference from unsuccessful proof search to non-consequence: a separating structure supplies the needed reason. A known theorem or short existing interpretation is cheaper when it already settles the same question. The chosen branch accepts the cost of constructing a model when that difference remains unresolved.

Adapt dependency analysis to the required return. Theorem Proving in Lean 4, Axioms and Computation explains how added principles affect proof and computational interpretation, and how axiom dependencies can be inspected. This changes :4.1/:4.4: trace the principle actually used and what a later construction can obtain. Reading a proof’s dependencies is a useful alternative to rebuilding it. That inspection alone does not settle whether another proof avoids the principle. A non-derivability claim needs a separate argument, such as a separating model under the stated soundness assumption. The source describes one formal system; Lean use is optional, and its evaluation mechanisms are not generalized to all mathematics.

Retain these approaches while their constructions answer the local question at acceptable effort. Reopen the comparison when the logic changes, a proposed interpretation fails, another proof removes the alleged dependency, or the receiving use needs computational content that the revised theory has not supplied.