# Analytical hierarchy

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, ${\displaystyle \mathbb {N} }$, and over functions from ${\displaystyle \mathbb {N} }$ to ${\displaystyle \mathbb {N} }$. 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.

## The analytical hierarchy of formulas

The notation ${\displaystyle \Sigma _{0}^{1}=\Pi _{0}^{1}=\Delta _{0}^{1}}$  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, which indicate this choice of language. 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 ${\displaystyle \Sigma _{n+1}^{1}}$  if it is logically equivalent to a formula of the form ${\displaystyle \exists X_{1}\cdots \exists X_{k}\psi }$  where ${\displaystyle \psi }$  is ${\displaystyle \Pi _{n}^{1}}$ . A formula is defined to be ${\displaystyle \Pi _{n+1}^{1}}$  if it is logically equivalent to a formula of the form ${\displaystyle \forall X_{1}\cdots \forall X_{k}\psi }$  where ${\displaystyle \psi }$  is ${\displaystyle \Sigma _{n}^{1}}$ . This inductive definition defines the classes ${\displaystyle \Sigma _{n}^{1}}$  and ${\displaystyle \Pi _{n}^{1}}$  for every natural number ${\displaystyle n}$ .

Because every formula has a prenex normal form, every formula in the language of second-order arithmetic is ${\displaystyle \Sigma _{n}^{1}}$  or ${\displaystyle \Pi _{n}^{1}}$  for some ${\displaystyle n}$ . Because meaningless quantifiers can be added to any formula, once a formula is given the classification ${\displaystyle \Sigma _{n}^{1}}$  or ${\displaystyle \Pi _{n}^{1}}$  for some ${\displaystyle n}$  it will be given the classifications ${\displaystyle \Sigma _{m}^{1}}$  and ${\displaystyle \Pi _{m}^{1}}$  for all ${\displaystyle m}$  greater than ${\displaystyle n}$ .

## The analytical hierarchy of sets of natural numbers

A set of natural numbers is assigned the classification ${\displaystyle \Sigma _{n}^{1}}$  if it is definable by a ${\displaystyle \Sigma _{n}^{1}}$  formula. The set is assigned the classification ${\displaystyle \Pi _{n}^{1}}$  if it is definable by a ${\displaystyle \Pi _{n}^{1}}$  formula. If the set is both ${\displaystyle \Sigma _{n}^{1}}$  and ${\displaystyle \Pi _{n}^{1}}$  then it is given the additional classification ${\displaystyle \Delta _{n}^{1}}$ .

The ${\displaystyle \Delta _{1}^{1}}$  sets are called hyperarithmetical. An alternate classification of these sets by way of iterated computable functionals is provided by the hyperarithmetical theory.

## The analytical hierarchy on subsets of Cantor and Baire space

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. These are both Polish spaces.

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 ${\displaystyle \Sigma _{n}^{1}}$  if it is definable by a ${\displaystyle \Sigma _{n}^{1}}$  formula. The set is assigned the classification ${\displaystyle \Pi _{n}^{1}}$  if it is definable by a ${\displaystyle \Pi _{n}^{1}}$  formula. If the set is both ${\displaystyle \Sigma _{n}^{1}}$  and ${\displaystyle \Pi _{n}^{1}}$  then it is given the additional classification ${\displaystyle \Delta _{n}^{1}}$ .

A subset of Baire space has a corresponding subset of Cantor space under the map that takes each function from ${\displaystyle \omega }$  to ${\displaystyle \omega }$  to the characteristic function of its graph. A subset of Baire space is given the classification ${\displaystyle \Sigma _{n}^{1}}$ , ${\displaystyle \Pi _{n}^{1}}$ , or ${\displaystyle \Delta _{n}^{1}}$  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. This alternate definition gives exactly the same classifications as the first definition.

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 power of one of these spaces. A similar extension is possible for countable powers and to products of powers of Cantor space and powers of Baire space.

## Extensions

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 ${\displaystyle \Sigma _{n}^{1,A}}$  or ${\displaystyle \Pi _{n}^{1,A}}$  using the same inductive definition as above. Given a set ${\displaystyle Y}$ , a set is defined to be ${\displaystyle \Sigma _{n}^{1,Y}}$  if it is definable by a ${\displaystyle \Sigma _{n}^{1,A}}$  formula in which the symbol ${\displaystyle A}$  is interpreted as ${\displaystyle Y}$ ; similar definitions for ${\displaystyle \Pi _{n}^{1,Y}}$  and ${\displaystyle \Delta _{n}^{1,Y}}$  apply. The sets that are ${\displaystyle \Sigma _{n}^{1,Y}}$  or ${\displaystyle \Pi _{n}^{1,Y}}$ , for any parameter Y, are classified in the projective hierarchy.

## Examples

• The set of all natural numbers which are indices of computable ordinals is a ${\displaystyle \Pi _{1}^{1}}$  set which is not ${\displaystyle \Sigma _{1}^{1}}$ .
• The set of elements of Cantor space which are the characteristic functions of well orderings of ${\displaystyle \omega }$  is a ${\displaystyle \Pi _{1}^{1}}$  set which is not ${\displaystyle \Sigma _{1}^{1}}$ . In fact, this set is not ${\displaystyle \Sigma _{1}^{1,Y}}$  for any element ${\displaystyle Y}$  of Baire space.
• If the axiom of constructibility holds then there is a subset of the product of the Baire space with itself which is ${\displaystyle \Delta _{2}^{1}}$  and is the graph of a well ordering of Baire space. If the axiom holds then there is also a ${\displaystyle \Delta _{2}^{1}}$  well ordering of Cantor space.

## Properties

For each ${\displaystyle n}$  we have the following strict containments:

${\displaystyle \Pi _{n}^{1}\subset \Sigma _{n+1}^{1}}$ ,
${\displaystyle \Pi _{n}^{1}\subset \Pi _{n+1}^{1}}$ ,
${\displaystyle \Sigma _{n}^{1}\subset \Pi _{n+1}^{1}}$ ,
${\displaystyle \Sigma _{n}^{1}\subset \Sigma _{n+1}^{1}}$ .

A set that is in ${\displaystyle \Sigma _{n}^{1}}$  for some n is said to be analytical. Care is required to distinguish this usage from the term analytic set which has a different meaning.

## Table

 Lightface Boldface Σ00 = Π00 = Δ00 (sometimes the same as Δ01) Σ00 = Π00 = Δ00 (if defined) Δ01 = recursive Δ01 = clopen Σ01 = recursively enumerable Π01 = co-recursively enumerable Σ01 = G = open Π01 = F = closed Δ02 Δ02 Σ02 Π02 Σ02 = Fσ Π02 = Gδ Δ03 Δ03 Σ03 Π03 Σ03 = Gδσ Π03 = Fσδ ⋮ ⋮ Σ0<ω = Π0<ω = Δ0<ω = Σ10 = Π10 = Δ10 = arithmetical Σ0<ω = Π0<ω = Δ0<ω = Σ10 = Π10 = Δ10 = boldface arithmetical ⋮ ⋮ Δ0α (α recursive) Δ0α (α countable) Σ0α Π0α Σ0α Π0α ⋮ ⋮ Σ0ωCK1 = Π0ωCK1 = Δ0ωCK1 = Δ11 = hyperarithmetical Σ0ω1 = Π0ω1 = Δ0ω1 = Δ11 = B = Borel Σ11 = lightface analytic Π11 = lightface coanalytic Σ11 = A = analytic Π11 = CA = coanalytic Δ12 Δ12 Σ12 Π12 Σ12 = PCA Π12 = CPCA Δ13 Δ13 Σ13 Π13 Σ13 = PCPCA Π13 = CPCPCA ⋮ ⋮ Σ1<ω = Π1<ω = Δ1<ω = Σ20 = Π20 = Δ20 = analytical Σ1<ω = Π1<ω = Δ1<ω = Σ20 = Π20 = Δ20 = P = projective ⋮ ⋮