Descriptive set theory

The descriptive set theory is a branch of set theory , which deals with properties definable quantities. The basic idea is to construct more complicated sets based on "simple" sets by certain laws of formation and to investigate their properties. The quantities occurring in mathematical practice can usually be obtained in this way. Here, subsets of real numbers such as open sets , G _δ sets , Borel sets and the set hierarchies derived from them are in the foreground; The set-theoretical, topological or measure-theoretical properties can just as well be examined in general Polish spaces , with the Baire space homeomorphic to the set of irrational numbers playing a special role.

Historical beginnings

An important question in set theory from the beginning was the problem of the thickness of the continuum, that is, the set of real numbers. The continuum hypothesis , according to which there are no further powers between the power of countably infinite sets and the power of the continuum, has turned out to be neither provable nor refutable through the work of Gödel and Cohen . Of course, this does not exclude that one can show for certain types of subsets of the continuum that in the uncountable case they automatically have the thickness of the continuum; one then says that this type of set satisfies the continuum hypothesis. This is particularly easy for open sets in , because these are associations of open intervals. An open set is therefore either empty or contains an open interval and is therefore equal to ; the open sets thus satisfy the continuum hypothesis. For closed sets , i.e. for the complements of the open sets, this is a bit more difficult. A very early result in this direction emerges from the Cantor-Bendixson theorem ; in fact, the closed sets also suffice for the continuum hypothesis. ${\ displaystyle \ mathbb {R}}$ ${\ displaystyle \ mathbb {R}}$

Baire had already introduced the so-called Baire functions in 1899 ; this is the smallest set of functions on or on other Polish spaces that contains all continuous functions and is closed under point-wise convergence . Lebesgue characterized this in 1905 as so-called analytically representable , that is, as the smallest set of functions that contains all constants and all projections and is closed under sums, products and point-by-point convergence. In this context he introduced the Borel sets and asserted in a lemma that projections of Borel sets are again such. However, Suslin noticed that this was wrong , from which the concept of the analytic set developed. It could also be shown for analytical sets that they fulfill the continuum hypothesis. The question remains open for larger classes, which can be derived from the analytical ones by means of certain educational laws and can be arranged in so-called hierarchies. ${\ displaystyle \ mathbb {R}}$ ${\ displaystyle (x_ {1}, \ ldots, x_ {n}) \ mapsto x_ {i}}$

The branch of effective descriptive set theory goes back largely to developments by Stephen Cole Kleene , such as the development of the arithmetic hierarchy , but the connections to classical descriptive set theory were only shown later.

Hierarchies

The following remarks are intended to give a first impression of the research area of descriptive set theory.

Borel hierarchy

The starting point of the Borel hierarchy is the class of open sets in or, more generally, in a perfect Polish space ; the class of open sets is denoted by. If the set of natural numbers has the discrete topology , then it is again a Polish space. is now defined as the set of all projections of complements from out onto the first component , i.e. consists of all sets of the form , where is a set, i.e. an open set, and is the projection onto the first component. This procedure can be iterated by defining it as the class of all sets of the form , where all subsets of whose complements are -Sets. ${\ displaystyle \ mathbb {R}}$ ${\ displaystyle X}$ ${\ displaystyle \ Sigma _ {1} ^ {0}}$ ${\ displaystyle \ omega}$ ${\ displaystyle X \ times \ omega}$ ${\ displaystyle \ Sigma _ {2} ^ {0}}$ ${\ displaystyle \ Sigma _ {1} ^ {0}}$ ${\ displaystyle X \ times \ omega}$ ${\ displaystyle X}$ ${\ displaystyle \ Sigma _ {2} ^ {0}}$ ${\ displaystyle p_ {1} (A)}$ ${\ displaystyle (X \ times \ omega) \ setminus A}$ ${\ displaystyle \ Sigma _ {1} ^ {0}}$ ${\ displaystyle p_ {1} \ colon X \ times \ omega \ rightarrow X}$ ${\ displaystyle \ Sigma _ {n + 1} ^ {0}}$ ${\ displaystyle p_ {1} (A)}$ ${\ displaystyle A}$ ${\ displaystyle X \ times \ omega}$ ${\ displaystyle \ Sigma _ {n} ^ {0}}$

The complements of form the class of -sets. The amounts are also known as amounts and their complements, i.e. the amounts, are known as amounts . Overall, the above method of formation gives you ascending classes ${\ displaystyle \ Sigma _ {n} ^ {0}}$ ${\ displaystyle \ Pi _ {n} ^ {0}}$ ${\ displaystyle \ Sigma _ {2} ^ {0}}$ ${\ displaystyle F _ {\ sigma}}$ ${\ displaystyle \ Pi _ {2} ^ {0}}$ ${\ displaystyle G _ {\ delta}}$

{\ displaystyle \ Sigma _ {1} ^ {0} \ subset \ Sigma _ {2} ^ {0} \ subset \ Sigma _ {3} ^ {0} \ subset \ dotsb}

{\ displaystyle \ Pi _ {1} ^ {0} \ subset \ Pi _ {2} ^ {0} \ subset \ Pi _ {3} ^ {0} \ subset \ dotsb}

and one can show that this construction does not lead out of the Borel sets and that in addition

{\ displaystyle \ Sigma _ {n} ^ {0} \ subset \ Pi _ {n + 1} ^ {0}}

and

{\ displaystyle \ Pi _ {n} ^ {0} \ subset \ Sigma _ {n + 1} ^ {0}}

applies. The question therefore arises whether all Borel sets agree with the class. The answer is no, one must continue the above formation process transfinitely , which can be carried out with the concept of the ordinal number . It then turns out that this process has to be carried out several times, whereby the smallest uncountable ordinal number is (see also the Aleph function ) in order to obtain all Borel sets in this way. ${\ displaystyle \ textstyle \ bigcup _ {n \ in \ mathbb {N}} \ Sigma _ {n} ^ {0} = \ bigcup _ {n \ in \ mathbb {N}} \ Pi _ {n} ^ { 0}}$ ${\ displaystyle \ aleph _ {1}}$ ${\ displaystyle \ aleph _ {1}}$

Projective hierarchy

The projective hierarchy arises from the class of the open sets according to the same pattern, only the space is replaced by the Baire space , whereby the set of all functions is what is identified as usual with the -fold Cartesian product of itself and then the Product topology considered. This space is homeomorphic to the space of irrational numbers with the relative topology of , which is why the Baire space is often called the space of irrational numbers in descriptive set theory. The names of the hierarchies are ${\ displaystyle \ omega}$ ${\ displaystyle {\ mathcal {N}} = \ omega ^ {\ omega}}$ ${\ displaystyle \ omega ^ {\ omega}}$ ${\ displaystyle \ omega \ rightarrow \ omega}$ ${\ displaystyle \ omega}$ ${\ displaystyle \ omega}$ ${\ displaystyle \ mathbb {R}}$

{\ displaystyle \ Sigma _ {1} ^ {1} \ subset \ Sigma _ {2} ^ {1} \ subset \ Sigma _ {3} ^ {1} \ subset \ dotsb}

{\ displaystyle \ Pi _ {1} ^ {1} \ subset \ Pi _ {2} ^ {1} \ subset \ Pi _ {3} ^ {1} \ subset \ dotsb}

.

Note that the top index is a 1. is the class of all sets of the form , where all closed subsets of passes through and is a Polish space, these sets are also called analytical. is again the class of the complements of such sets, which is therefore also called co-analytic. ${\ displaystyle \ Sigma _ {1} ^ {1}}$ ${\ displaystyle p_ {1} (A)}$ ${\ displaystyle A}$ ${\ displaystyle X \ times {\ mathcal {N}}}$ ${\ displaystyle X}$ ${\ displaystyle \ Pi _ {1} ^ {1}}$

Suslin had already shown that exactly matches the Borel quantities. One can show that the sets fulfill the continuum hypothesis and that they are all Lebesgue measurable . These statements are lost for; Gödel had shown that, assuming the axiom of constructibility, there is a lot in that is not Lebesgue measurable. According to Sierpiński's theorem , every union of -many Borel sets. ${\ displaystyle \ Sigma _ {1} ^ {1} \ cap \ Pi _ {1} ^ {1}}$ ${\ displaystyle \ Sigma _ {1} ^ {1}}$ ${\ displaystyle \ Sigma _ {n} ^ {1}, n> 1}$ ${\ displaystyle \ Sigma _ {1} ^ {1} \ cap \ Pi _ {1} ^ {1}}$ ${\ displaystyle \ Sigma _ {2} ^ {1}}$ ${\ displaystyle \ aleph _ {1}}$

κ-Suslin sets

Replacing in the construction of the hierarchy Lusin® Baire the room through , wherein a cardinal number with the discrete space is, so to get to the concept of -Suslin amount. A subset of a Polish space is a -Suslin set if it has the form for a closed set . The class of all such sets is denoted by. ${\ displaystyle {\ mathcal {N}} = \ omega ^ {\ omega}}$ ${\ displaystyle \ kappa ^ {\ omega}}$ ${\ displaystyle \ kappa}$ ${\ displaystyle \ kappa}$ ${\ displaystyle X}$ ${\ displaystyle \ kappa}$ ${\ displaystyle p_ {1} (A)}$ ${\ displaystyle A \ subset X \ times \ kappa ^ {\ omega}}$ ${\ displaystyle S (\ kappa)}$

${\ displaystyle S (\ aleph _ {0})}$ obviously agrees with the , that is, with the class of all analytic sets. According to Shoenfield's theorem , each is a -Suslin set. Statements about these set classes require deeper methods of set theory, which often raises the question of sufficiently strong axioms of set theory. ${\ displaystyle \ Sigma _ {1} ^ {1}}$ ${\ displaystyle \ Sigma _ {2} ^ {1}}$ ${\ displaystyle \ aleph _ {1}}$

Regularity properties

In addition to such sets resulting from certain operations, one considers certain regularity properties of subsets of Polish spaces and their relationships to the sets obtained by such constructions. Examples of such properties are:

A set is baire if it differs only a meager amount from an open set.

A set is said to be universally measurable if it is measurable with respect to every complete, finite measure that is defined for all Borel sets.

A set has the perfect set property if it is countable or if it contains a non-empty perfect set .

Further questions

Other important questions of descriptive set theory also concern the functions between Polish spaces, in particular their measurability properties, as well as equivalence relations and algebraic structures in Polish spaces. Furthermore, the educational processes described above can be examined for their predictability ; this happens in the sub-area of effective descriptive set theory , which is closely interlinked with recursion theory .

Areas of application

Descriptive set theory is used in the following areas:

literature

Alexander S. Kechris : Classical Descriptive Set Theory . Springer , Berlin 1994, ISBN 0-387-94374-9 , pp. 341 .
YN Moschovakis: Descriptive Set Theory . North Holland, 1987, ISBN 0-444-70199-0 .

Web links

AG El'kin, VI Ponomarev: Descriptive set theory . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

Individual evidence

↑ Moschovakis.
↑ Akihiro Kanamori : The Emergence of Descriptive Set Theory . S. 256 ( online [PDF; 1000 kB ; accessed on November 30, 2012]).
^ Donald L. Cohn: Measure Theory , Birkhäuser, Boston (1980), ISBN 3-7643-3003-1 , Chapter 8.2, Corollary 8.3.3
^ Thomas Jech : Set Theory. 3rd millennium edition, revised and expanded. Springer, Berlin et al. 2003, ISBN 3-540-44085-2 , Corollary 25.28
^ YN Moschovakis: Descriptive Set Theory , North Holland 1987, ISBN 0-444-70199-0 , Theorem 2B.2

[1] Moschovakis.

[2] Akihiro Kanamori : The Emergence of Descriptive Set Theory . S. 256 ( online [PDF; 1000 kB ; accessed on November 30, 2012]).

[3] Donald L. Cohn: Measure Theory , Birkhäuser, Boston (1980), ISBN 3-7643-3003-1 , Chapter 8.2, Corollary 8.3.3

[4] Thomas Jech : Set Theory. 3rd millennium edition, revised and expanded. Springer, Berlin et al. 2003, ISBN 3-540-44085-2 , Corollary 25.28

[5] YN Moschovakis: Descriptive Set Theory , North Holland 1987, ISBN 0-444-70199-0 , Theorem 2B.2