Proof of Impossibility

A proof of impossibility, sometimes called a negative proof or negative result, is a proof demonstrating that a particular problem cannot be solved, or cannot be solved in general. Often proofs of impossibility have put to rest decades or centuries of work attempting to find a solution. Proofs of impossibility are usually expressible as universal propositions in logic (see universal quantification).

One of the oldest and most famous proofs of impossibility was the 1882 proof of Ferdinand von Lindemann showing that the ancient problem of squaring the circle cannot be solved, because the number π is transcendental and only algebraic numbers can be constructed by compass and straightedge. Another classical problem was that of creating a general formula using radicals expressing the solution of a polynomial equation of degree 5 or higher. Galois showed this impossible using concepts such as solvable groups from Galois theory, a new subfield of abstract algebra that he conceived.

Among the most important proofs of impossibility of the 20th century were those related to undecidability, which showed that there are problems that cannot be solved in general by any algorithm at all. The most famous is the halting problem.

In computational complexity theory, techniques like relativization (see oracle machine) provide "weak" proofs of impossibility excluding certain proof techniques. Other techniques like proofs of completeness for a complexity class provide evidence for the difficulty of problems by showing them to be just as hard to solve as other known problems that have proven intractable.