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mathematical logic Mathematical logic is the study of Logic#Formal logic, formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory (also known as computability theory). Research in mathematical logic com ...
, the Borel hierarchy is a stratification of the Borel algebra generated by the open subsets of a Polish space; elements of this algebra are called Borel sets. Each Borel set is assigned a unique
countable In mathematics, a Set (mathematics), set is countable if either it is finite set, finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function fro ...
ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the leas ...
called the rank of the Borel set. The Borel hierarchy is of particular interest in
descriptive set theory In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" set (mathematics), subsets of the real line and other Polish spaces. As well as being one of the primary areas of research in set theory, it has a ...
. One common use of the Borel hierarchy is to prove facts about the Borel sets using transfinite induction on rank. Properties of sets of small finite ranks are important in
measure theory In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude (mathematics), magnitude, mass, and probability of events. These seemingl ...
and
analysis Analysis (: analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (38 ...
.


Borel sets

The Borel algebra in an arbitrary
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
is the smallest collection of subsets of the space that contains the open sets and is closed under countable unions and complementation. It can be shown that the Borel algebra is closed under countable intersections as well. A short proof that the Borel algebra is well-defined proceeds by showing that the entire powerset of the space is closed under complements and countable unions, and thus the Borel algebra is the intersection of all families of subsets of the space that have these closure properties. This proof does not give a simple procedure for determining whether a set is Borel. A motivation for the Borel hierarchy is to provide a more explicit characterization of the Borel sets.


Boldface Borel hierarchy

The Borel hierarchy or boldface Borel hierarchy on a space ''X'' consists of classes \mathbf^0_\alpha, \mathbf^0_\alpha, and \mathbf^0_\alpha for every countable ordinal \alpha greater than zero. Each of these classes consists of subsets of ''X''. The classes are defined inductively from the following rules: * A set is in \mathbf^0_1 if and only if it is open. * A set is in \mathbf^0_\alpha if and only if its complement is in \mathbf^0_\alpha. * A set A is in \mathbf^0_\alpha for \alpha > 1 if and only if there is a sequence of sets A_1,A_2,\ldots such that each A_i is in \mathbf^0_ for some \alpha_i < \alpha and A = \bigcup A_i. * A set is in \mathbf^0_\alpha if and only if it is both in \mathbf^0_\alpha and in \mathbf^0_\alpha. The motivation for the hierarchy is to follow the way in which a Borel set could be constructed from open sets using complementation and countable unions. A Borel set is said to have finite rank if it is in \mathbf^0_\alpha for some finite ordinal \alpha; otherwise it has infinite rank. If \mathbf^0_1 \subseteq \mathbf^0_2 then the hierarchy can be shown to have the following properties: * For every ''α'', \mathbf^0_\alpha \cup \mathbf^0_\alpha \subseteq \mathbf^0_. Thus, once a set is in \mathbf^0_\alpha or \mathbf^0_\alpha, that set will be in all classes in the hierarchy corresponding to ordinals greater than ''α'' * \bigcup_ \mathbf^0_\alpha = \bigcup_ \mathbf^0_\alpha = \bigcup_ \mathbf^0_\alpha. Moreover, a set is in this union if and only if it is Borel. * If X is an uncountable Polish space, it can be shown that \mathbf^0_\alpha is not contained in \mathbf^0_\alpha for any \alpha < \omega_1, and thus the hierarchy does not collapse.


Borel sets of small rank

The classes of small rank are known by alternate names in classical descriptive set theory. * The \mathbf^0_1 sets are the open sets. The \mathbf^0_1 sets are the closed sets. * The \mathbf^0_2 sets are countable unions of closed sets, and are called Fσ sets. The \mathbf^0_2 sets are the dual class, and can be written as a countable intersection of open sets. These sets are called Gδ sets.


Lightface hierarchy

The lightface Borel hierarchy (also called the effective Borel hierarchyP. G. Hinman, *Recursion-Theoretic Hierarchies*. Perspectives in Mathematical Logic, Springer-Verlag (1978). ISBN 3-540-07904-1.pp.163--164) is an effective version of the boldface Borel hierarchy. It is important in effective descriptive set theory and
recursion theory Computability theory, also known as recursion theory, is a branch of mathematical logic, computer science, and the theory of computation that originated in the 1930s with the study of computable functions and Turing degrees. The field has since ex ...
. The lightface Borel hierarchy extends the
arithmetical hierarchy In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy (after mathematicians Stephen Cole Kleene and Andrzej Mostowski) classifies certain sets based on the complexity of formulas that define th ...
of subsets of an effective Polish space. It is closely related to the hyperarithmetical hierarchy. The lightface Borel hierarchy can be defined on any effective Polish space. It consists of classes \Sigma^0_\alpha, \Pi^0_\alpha and \Delta^0_\alpha for each nonzero countable ordinal \alpha less than the Church–Kleene ordinal \omega^_1. Each class consists of subsets of the space. The classes, and codes for elements of the classes, are inductively defined as follows:D. Martin
Borel Determinacy
Annals of Mathematics vol. 102, pp.363--371 (1975)
* A set is \Sigma^0_1 if and only if it is effectively open, that is, an open set which is the union of a
computably enumerable In computability theory, a set ''S'' of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable, listable, provable or Turing-recognizable if: *There is an algorithm such that the ...
sequence of basic open sets. A code for such a set is a pair ''(0,e)'', where ''e'' is the index of a program enumerating the sequence of basic open sets. * A set is \Pi^0_\alpha if and only if its complement is \Sigma^0_\alpha. A code for one of these sets is a pair ''(1,c)'' where ''c'' is a code for the complementary set. * A set is \Sigma^0_ if there is a
computably enumerable In computability theory, a set ''S'' of natural numbers is called computably enumerable (c.e.), recursively enumerable (r.e.), semidecidable, partially decidable, listable, provable or Turing-recognizable if: *There is an algorithm such that the ...
sequence of codes for a sequence A_1,A_2,\ldots of sets such that each A_i is \Pi^0_ for some \alpha_i < \alpha and A = \bigcup A_i. A code for a \Sigma^0_\alpha set is a pair ''(2,e)'', where ''e'' is an index of a program enumerating the codes of the sequence A_i. A code for a lightface Borel set gives complete information about how to recover the set from sets of smaller rank. This contrasts with the boldface hierarchy, where no such effectivity is required. Each lightface Borel set has infinitely many distinct codes. Other coding systems are possible; the crucial idea is that a code must effectively distinguish between effectively open sets, complements of sets represented by previous codes, and computable enumerations of sequences of codes. It can be shown that for each \alpha < \omega^_1 there are sets in \Sigma^0_\alpha \setminus \Pi^0_, and thus the hierarchy does not collapse. No new sets would be added at stage \omega^_1, however. A famous theorem due to Spector and Kleene states that a set is in the lightface Borel hierarchy if and only if it is at level \Delta^1_1 of the analytical hierarchy. These sets are also called hyperarithmetic. Additionally, for all natural numbers n>0, the classes \Sigma^0_n and \Pi^0_n of the effective Borel hierarchy are the same as the classes \Sigma^0_n and \Pi^0_n of the
arithmetical hierarchy In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy (after mathematicians Stephen Cole Kleene and Andrzej Mostowski) classifies certain sets based on the complexity of formulas that define th ...
of the same name.p.168 The code for a lightface Borel set ''A'' can be used to inductively define a tree whose nodes are labeled by codes. The root of the tree is labeled by the code for ''A''. If a node is labeled by a code of the form ''(1,c)'' then it has a child node whose code is ''c''. If a node is labeled by a code of the form ''(2,e)'' then it has one child for each code enumerated by the program with index ''e''. If a node is labeled with a code of the form ''(0,e)'' then it has no children. This tree describes how ''A'' is built from sets of smaller rank. The ordinals used in the construction of ''A'' ensure that this tree has no infinite path, because any infinite path through the tree would have to include infinitely many codes starting with ''2'', and thus would give an infinite decreasing sequence of ordinals. Conversely, if an arbitrary subtree of \omega^\, has its nodes labeled by codes in a consistent way, and the tree has no infinite paths, then the code at the root of the tree is a code for a lightface Borel set. The rank of this set is bounded by the order type of the tree in the Kleene–Brouwer order. Because the tree is arithmetically definable, this rank must be less than \omega^_1. This is the origin of the Church–Kleene ordinal in the definition of the lightface hierarchy.


Relation to other hierarchies


See also

* Projective hierarchy * Wadge hierarchy * Veblen hierarchy


References


Sources

* Kechris, Alexander. ''Classical Descriptive Set Theory''. Graduate Texts in Mathematics v. 156, Springer-Verlag, 1995. . * Jech, Thomas. ''Set Theory'', 3rd edition. Springer, 2003. {{ISBN, 3-540-44085-2. Descriptive set theory Mathematical logic hierarchies