insolvable

Insovable is an adjective used to describe something that cannot be solved. In mathematics, logic, and related disciplines, an insolvable question or problem is one for which no solution procedure exists within a given formal system or under applicable rules. The term is often synonymous with unsolvable, though insolvable is less common in contemporary English and may appear mainly in older texts or specific contexts.

Etymology: insolvable comes from the prefix in- meaning not and solvable from Latin solvāre, “to loosen” or “to solve.” The form is older, and today unsolvable is more widely used. Insoluble, a distinct word, refers to substances that do not dissolve and should not be confused with insovable.

Contexts include mathematical problems that cannot be decided algorithmically, such as the Halting problem, and logical results like Gödel’s incompleteness theorems, which show that certain true statements cannot be proven within a given formal system. The concept also covers problems proven to have no general solution, as in Hilbert’s tenth problem, which established the nonexistence of a universal solving algorithm for Diophantine equations. In practice, a problem may be described as insolvable when no practical method exists to resolve it within accepted methods.

Usage notes: insolvable tends to be replaced by unsolvable in modern prose. In technical writing, the distinction between insolvable and undecidable or intractable may be important depending on whether a problem is provably non-resolvable or merely computationally difficult.