An index set is a set of program codes whose membership depends only on the partial function or enumerable language computed, rather than the code's syntax.
An extensional program property gives identical answers to all codes computing the same object.
Every nontrivial extensional property of partial computable functions or computably enumerable languages is undecidable.
For a nonempty set of words, the propertyis a nontrivial extensional property of unary partial computable functions: the nowhere-defined function fails it, while any total constant function with value in satisfies it. The Rice theorem therefore makes its index set undecidable.
The cofinite index set contains codes whose enumerable language has finite complement.
