A property of sets for which one can determine whether something is a member or not in a finite number of computational steps.
Decidability is an important concept in computability theory. A set (e.g. "all numbers with a 5 in them") is said to be "decidable" if I can write a program (usually for a Turing Machine) to determine whether a number is in the set and the program will always terminate with an answer YES or NO after a finite number of steps.
Most sets you can describe easily are decidable, but there are infinitely many sets so most sets are undecidable, assuming any finite limit on the size (number of instructions or number of states) of our programs. I.e. how ever big you allow your program to be there will always be sets which need a bigger program to decide membership.
One example of an undecidable set comes from the halting problem. It turns out that you can encode every program as a number: encode every symbol in the program as a number (001,
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