In computability theory, index sets describe classes of computable functions; specifically, they give all indices of functions in a certain class, according to a fixed Gödel numbering of partial computable functions.
φ
be a computable enumeration of all partial computable functions, and
be a computable enumeration of all c.e.
be a class of partial computable functions.
φ
is the index set of
is an index set if for every
φ
φ
(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.
Most index sets are non-computable, aside from two trivial exceptions.
This is stated in Rice's theorem: Let
be a class of partial computable functions with its index set
.Rice's theorem says "any nontrivial property of partial computable functions is undecidable".
[1] Index sets provide many examples of sets which are complete at some level of the arithmetical hierarchy.
-completeness is defined similarly.
Here are some examples:[2] Empirically, if the "most obvious" definition of a set