In mathematical logic and descriptive set theory, the analytical hierarchy is an extension of the arithmetical hierarchy.
The analytical hierarchy of formulas includes formulas in the language of second-order arithmetic, which can have quantifiers over both the set of natural numbers,
The analytical hierarchy of sets classifies sets by the formulas that can be used to define them; it is the lightface version of the projective hierarchy.
indicates the class of formulas in the language of second-order arithmetic with number quantifiers but no set quantifiers.
This language does not contain set parameters.
The Greek letters here are lightface symbols, indicating the language choice.
Each corresponding boldface symbol denotes the corresponding class of formulas in the extended language with a parameter for each real; see projective hierarchy for details.
A formula in the language of second-order arithmetic is defined to be
if it is logically equivalent to a formula of the form
if it is logically equivalent to a formula of the form
This inductive definition defines the classes
Kuratowski and Tarski showed in 1931 that every formula in the language of second-order arithmetic has a prenex normal form,[1] and therefore is
A set of natural numbers is assigned the classification
formula (with one free number variable and no free set variables).
An alternate classification of these sets by way of iterated computable functionals is provided by the hyperarithmetical theory.
The analytical hierarchy can be defined on any effective Polish space; the definition is particularly simple for Cantor and Baire space because they fit with the language of ordinary second-order arithmetic.
Cantor space is the set of all infinite sequences of 0s and 1s; Baire space is the set of all infinite sequences of natural numbers.
The ordinary axiomatization of second-order arithmetic uses a set-based language in which the set quantifiers can naturally be viewed as quantifying over Cantor space.
A subset of Cantor space is assigned the classification
formula (with one free set variable and no free number variables).
A subset of Baire space has a corresponding subset of Cantor space under the map that takes each function from
A subset of Baire space is given the classification
if and only if the corresponding subset of Cantor space has the same classification.
An equivalent definition of the analytical hierarchy on Baire space is given by defining the analytical hierarchy of formulas using a functional version of second-order arithmetic; then the analytical hierarchy on subsets of Cantor space can be defined from the hierarchy on Baire space.
Because Cantor space is homeomorphic to any finite Cartesian power of itself, and Baire space is homeomorphic to any finite Cartesian power of itself, the analytical hierarchy applies equally well to finite Cartesian powers of one of these spaces.
As is the case with the arithmetical hierarchy, a relativized version of the analytical hierarchy can be defined.
The language is extended to add a constant set symbol A.
A formula in the extended language is inductively defined to be
, for any parameter Y, are classified in the projective hierarchy, and often denoted by boldface Greek letters to indicate the use of parameters.
Care is required to distinguish this usage from the term analytic set, which has a different meaning, namely