Not every well-formed question has a computable answer. Some problems are undecidable, admitting no algorithm that settles every instance. The realm of the computable has a definite edge.
The principle
Undecidability is the property of a problem for which no algorithm can give the correct answer for all inputs. Such problems are not merely intractable but algorithmically unsolvable. No procedure suffices in general.
The mechanism
The mechanism of proof is reduction. By showing that solving a new problem would solve the halting problem, one inherits its impossibility. Undecidability spreads by such arguments.
An unexpected turn
The turn is that undecidability is common, not exotic. Many natural questions about programs, systems, and even mathematics turn out to be undecidable. The impossible is not rare.
The hidden cost
Undecidability coexists with partial methods. One can often decide many instances even when no method decides all, so practice proceeds despite the limit. Impossibility in general permits success in particular.
The limit
The implication is a boundary on formal power. There are truths no mechanical procedure can certify across the board, whatever resources it commands. Computation cannot answer everything.
The larger point
Undecidability names problems no algorithm can solve for all inputs, proved by reduction from halting and surprisingly widespread. Partial methods still handle many cases. It marks the definite edge of the computable. What makes the idea durable is not that it settles a question but that it reframes many. It teaches where to look and what to discount, which is often more valuable than any particular answer it yields. Understood in this spirit, it becomes a habit of attention rather than a doctrine, and habits of attention are what distinguish deep comprehension from mere knowledge.