What problem did Alan Turing prove undecidable?

Alan Turing proved that the halting problem is undecidable. The problem asks whether an arbitrary computer program, given a particular input, will eventually stop running or continue forever.

Turing’s proof showed that no general algorithm can always answer that question correctly for every possible program and input. His argument uses a contradiction: if a perfect halting-checking program existed, one could construct another program that deliberately does the opposite of its prediction. That self-referential construction makes the supposed universal checker impossible.

Turing published his result in the 1930s while developing the mathematical theory of computation. The result does not mean that no individual program can ever be analyzed. For many specific programs, mathematicians and programmers can prove that execution will stop or loop forever. The impossibility concerns one algorithm that must decide every possible case. The halting problem is therefore a foundational example of an undecidable decision problem, not merely a difficult or slow calculation.