In the field of recursion theory

Recursion theory

Computability theory, also called recursion theory, is a branch of mathematical logic that originated in the 1930s with the study of computable functions and Turing degrees. The field has grown to include the study of generalized computability and definability...

, index sets describe classes of partial recursive functions, specifically they give all indices of functions in that class according to a fixed enumeration of partial recursive functions (a Gödel number

Gödel number

In mathematical logic, a Gödel numbering is a function that assigns to each symbol and well-formed formula of some formal language a unique natural number, called its Gödel number. The concept was famously used by Kurt Gödel for the proof of his incompleteness theorems...

ing).

Definition

Fix an enumeration of all partial recursive functions, or equivalently of recursively enumerable sets whereby the eth such set is and the eth such function (whose domain is ) is .

Let be a class of partial recursive functions. If then is the index set of . In general is an index set if for every with (i.e. they index the same function), we have . Intuitively, these are the sets of natural numbers that we describe only with reference to the functions they index.

Index sets and Rice's theorem

Rice's theorem

In computability theory, Rice's theorem states that, for any non-trivial property of partial functions, there is no general and effective method to decide whether an algorithm computes a partial function with that property...

Most index sets are incomputable (non-recursive) aside from two trivial exceptions. This is stated in Rice's theorem:

Let be a class of partial recursive functions with index set . Then is recursive if and only if is empty, or is all of .

where is the set of natural numbers, including zero.

Rice's theorem says "any nontrivial property of partial recursive functions is undecidable"