Assuming a total halting decider lets a program reverse the predicted behavior on its own code. I want the notation, the mechanism, and the failure mode visible at the same time.
Start locally
A decision problem is computable when a Turing machine \(M_e(x)\) halts on every input with the correct answer. A set \(A\subseteq\mathbf N\) is computably enumerable when a machine can list its members.
I keep the defining relation \(\mathsf D\) above the derived relation \(\mathsf C\). This exposes whether cancellation used \(x\ne0\) and whether the conclusion is canonical.
Compute before generalising
The following line is the smallest calculation that still exercises the mechanism. It keeps nested delimiters and the order of operations explicit.
The global view
What survives the example is not its particular numbers but the relation encoded by the two rows below. That relation is the part worth transporting to a new setting.
Edge conditions
Enumerability is weaker than decidability. A search may confirm membership eventually without ever certifying nonmembership.
The final box is a summary, not a new assumption; the proof still lives in the definitions and the intervening calculation. The source keeps each scope delimiter visible for later inspection.