Proof of impossibility

A proof of impossibility, also known as negative proof, proof of an impossibility theorem, 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. To prove that something is impossible is usually much harder than the opposite task; it is necessary to develop a theory. Impossibility theorems are usually expressible as universal propositions in logic (see universal quantification).

One of the 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 (non-algebraic) and only a subset of the algebraic numbers can be constructed by compass and straightedge. Two other classical problems—trisecting the general angle and doubling the cube—were also proved impossible in the nineteenth century.

A problem arising in the sixteenth century was that of creating a general formula using radicals expressing the solution of any polynomial equation of degree 5 or higher. In the 1820s, the Abel-Ruffini theorem showed this to be impossible using concepts such as solvable groups from Galois theory, a new subfield of abstract algebra.

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 proved intractable.

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