Does there exist an algorithm for deciding whether or not a specific mathematical assertion does or does not have a proof? The decision problem is also known as the Entscheidungsproblem (which, not so coincidentally, is German for "decision problem").

Using the concept of the Turing machine, Turing showed the answer to be negative for elementary number theory. J. Robinson and Tarski showed the decision problem is undecidable for arbitrary fields.

Church, A. "A Note on the Entscheidungsproblem." J. Symb. Logic 1, 1936.

Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, pp. 1100 and 1136, 2002.

CITE THIS AS:

Weisstein, Eric W. "Decision Problem." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/DecisionProblem.html