Editing Analytical hierarchy (section)

== Examples ==
* For a relation <math>\prec</math> on <math>\mathbb N^2</math>, the statement "<math>\prec</math> is a [[well-order]] on <math>\mathbb N</math>" is <math>\Pi_1^1</math>. (Not to be confused with the general case for [[well-founded]] relations on sets, see [[Lévy hierarchy]])
* The set of all natural numbers that are indices of computable [[ordinal (mathematics)|ordinal]]s is a <math>\Pi^1_1</math> set that is not <math>\Sigma^1_1</math>.
**These sets are exactly the [[Alpha recursion theory|<math>\omega_1^{CK}</math>-recursively-enumerable]] subsets of <math>\omega</math>. <nowiki>[</nowiki>Bar75, p.&nbsp;168]
* A function <math>f:\mathbb N\to\mathbb N</math> is definable by [[Jacques Herbrand|Herbrand]]'s 1931 formalism of systems of equations if and only if <math>f</math> is hyperarithmetical.<ref>P. Odifreddi, ''Classical Recursion Theory'' (1989), p.33. North-Holland, 0-444-87295-7</ref>
* The set of continuous functions <math>f:[0,1]\to\mathbb [0,1]</math> that have the [[Mean value theorem|mean value property]] is no lower than <math>\Delta_2^1</math> on the hierarchy.<ref>{{Cite arXiv|last=Quintanilla|first=M.|eprint=2206.10754|title=The realm numbers in inner models of set theory|date=2022|class=math.LO }}</ref>
* The set of elements of Cantor space that are the characteristic functions of well orderings of <math>\omega</math> is a <math>\Pi^1_1</math> set that is not <math>\Sigma^1_1</math>. In fact, this set is not <math>\Sigma^{1,Y}_1</math> for any element <math>Y</math> of Baire space.
* If the [[axiom of constructibility]] holds then there is a subset of the product of the Baire space with itself that is <math>\Delta^1_2</math> and is the graph of a [[well ordering]] of Baire space. If the axiom holds then there is also a <math>\Delta^1_2</math> well ordering of Cantor space.