Reducibility is how undecidability spreads. If we could solve B, and solving A reduces to solving B, then A is solvable too — so if A is known unsolvable, so is B.
Definition · Mapping reducibility
A ≤ₘ B if there is a computable function f with
w ∈ A ⟺ f(w) ∈ BIf A ≤ₘ B and B is decidable, then A is decidable. Contrapositive: if A is undecidable, so is B.
Almost every undecidability proof after the halting problem is a reduction from Aᴛᴍ: transform a machine-and-input question into an instance of the new problem so that yes maps to yes.
Same idea, new altitude
Polynomial-time reductions reuse this exact move in complexity theory to define NP-completeness — only with a running-time budget.