Decidability

How do we know if a problem is solvable or a theorem provable? This mode of reasoning takes us into the heart of mathematics. Obviously, if something cannot be solved or proved, we should not be wasting our time trying to find a solution or proof; that is, if it can be shown to be undecidable, that is the end of the matter. This is a central principle on which computer science is built. If something can be programmed to produce a certain output, then the algorithm used shows that it is decidable in the first place; if it does not produce an output, then it is undecidable. This chapter deals with the question of decidability and its basis in logic, culminating with Kurt Gödel’s remarkable proof that within any formal system of logic there are propositions (statements) that can be neither proved nor disproved. Since then, mathematicians have embraced a much more flexible way to do mathematics, without discarding the previous ways.