The analytical hierarchy of formulas includes formulas in the language of second-order arithmetic, which may have quantifiers over the the set of natural numbers, N undefined \mathbb undefined , and above functions from N undefined \mathbb undefined into N undefined \mathbb undefined . Sets’ hierarchy classifies sets it is the hierarchy’s edition.
This language doesn’t contain parameters that are set. The letters are symbols, which indicate this choice of language. Each boldface symbol denotes the class of formulas in the language using a parameter for each real; see hierarchy.
A formula in the language of second-order arithmetic is defined as Σ n + 1 1 undefined \Sigma _undefined^undefinedif it is logically equivalent to a formula of this form ∃ X 1 ⋯ ∃ X k ψ undefined \exists X_undefined\cdots \exists X_undefined\psi where ψ undefined \psi is Π n 1 undefined \Pi _undefined^undefined. A formula is defined as Π n + 1 1 undefined \Pi _undefined^undefinedif it is logically equivalent to a formula of this form ∀ X 1 ⋯ ∀ X k ψ undefined \forall X_undefined\cdots \forall X_undefined\psi where ψ undefined \psi is Σ n 1 undefined \Sigma _undefined^undefined. This inductive definition defines the courses Σ n 1 undefined \Sigma^1_nand Π n 1 undefined \Pi^1_nfor each natural number n undefined n.
Since every formula has a prenex normal form, each formula in the language of second-order arithmetic is Σ n 1 undefined \Sigma^1_nor Π n 1 undefined \Pi^1_nfor a few n undefined n. Because meaningless quantifiers can be added to any formula, after a formulation is given the classification Σ n 1 undefined \Sigma^1_nor Π n 1 undefined \Pi^1_nfor a few n undefined nit’ll be awarded the classifications Σ m 1 undefined \Sigma _undefined^undefinedand Π m 1 undefined \Pi _undefined^undefinedfor all m undefined mgreater than n undefined n.
When the set is equally Σ n 1 undefined \Sigma^1_nand Π n 1 undefined \Pi^1_nthen it is given the additional classification Δ n 1 undefined \Delta _undefined^undefined.
The Δ 1 1 undefined \Delta _undefined^undefinedsets are known as hyperarithmetical. Theory provides an classification of those sets by means of computable functionals.
The hierarchy can be defined on any space; because they match the terminology of arithmetic, the definition is simple for Cantor and space. Cantor space is the set of all strings of 1s and 0s . These are both spaces.
The axiomatization of arithmetic uses a language where the quantifiers that are set can be seen as measuring over Cantor space. A subset of Cantor space is assigned the classification Σ n 1 undefined \Sigma^1_nif it is definable by a Σ n 1 undefined \Sigma^1_nformula. The set is assigned the classification Π n 1 undefined \Pi^1_nif it is definable by a Π n 1 undefined \Pi^1_nformula. When the set is equally Σ n 1 undefined \Sigma^1_nand Π n 1 undefined \Pi^1_nthen it is given the additional classification Δ n 1 undefined \Delta _undefined^undefined.
A subset of Baire space has a corresponding subset of Cantor space beneath the map which takes each function from ω undefined \omega into ω undefined \omega into the characteristic function of its chart. A subset of Baire space is given the classification Σ n 1 undefined \Sigma^1_n, Π n 1 undefined \Pi^1_n, or Δ n 1 undefined \Delta _undefined^undefinedif and only if the corresponding subset of Cantor space has the same classification. Specifying the hierarchy of formulas using a version of arithmetic gives an equivalent definition of the hierarchy on Baire space the hierarchy on subsets of Cantor space could be defined in the hierarchy on Baire space. This definition gives the same classifications as the definition.
Since Cantor space is homeomorphic to some finite Cartesian power of itself, and Baire space is homeomorphic to some finite Cartesian power of itself, the analytic hierarchy applies equally well to finite Cartesian power of one of those spaces.A similar expansion is possible for countable powers and to products of powers of Cantor distance and forces of Baire space.
The collection of natural numbers that are indices of computable ordinals is a Π 1 1 undefined \Pi _undefined^undefinedset that isn’t Σ 1 1 undefined \Sigma _undefined^undefined.
The collection of components of Cantor space that are the characteristic functions of well orderings of ω undefined \omega is a Π 1 1 undefined \Pi _undefined^undefinedset that isn’t Σ 1 1 undefined \Sigma _undefined^undefined. Actually, this set isn’t Σ 1 1 , Y undefined \Sigma _undefined^undefinedfor any component Y undefined Yof Baire space.
When the axiom of constructibility holds then there is a subset of the item of the Baire space with itself that is Δ two 1 undefined \Delta _undefined^undefinedand is the chart of a well ordering of Baire space. If the axiom holds then there’s also a Δ two 1 undefined \Delta _undefined^undefinedwell ordering of Cantor space.
For every n undefined nwe have the following rigorous containments:
Π n 1 ⊂ Σ n + 1 1 undefined \Pi _undefined^undefined\subset \Sigma _undefined^undefined,
Π n 1 ⊂ Π n + 1 1 undefined \Pi _undefined^undefined\subset \Pi _undefined^undefined,
Σ n 1 ⊂ Π n + 1 1 undefined \Sigma _undefined^undefined\subset \Pi _undefined^undefined,
Σ n 1 ⊂ Σ n + 1 1 undefined \Sigma _undefined^undefined\subset \Sigma _undefined^undefined.
A set that’s in Σ n 1 undefined \Sigma^1_nfor a few nis stated to be analytical. Care is required to differentiate this usage.