Descriptive set theory

In mathematical logic, descriptive set theory (DST) is the study of certain classes of "well-behaved" 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 applications to other areas of mathematics such as functional analysis, ergodic theory, the study of operator algebras and group actions, and mathematical logic.

Polish spaces

Descriptive set theory begins with the study of Polish spaces and their Borel sets.

A Polish space is a second-countable topological space that is metrizable with a complete metric. Heuristically, it is a complete separable metric space whose metric has been "forgotten". Examples include the real line ${\displaystyle \mathbb {R} }$, the Baire space ${\displaystyle {\mathcal {N}}}$, the Cantor space ${\displaystyle {\mathcal {C}}}$, and the Hilbert cube ${\displaystyle I^{\mathbb {N} }}$.

Universality properties

The class of Polish spaces has several universality properties, which show that there is no loss of generality in considering Polish spaces of certain restricted forms.

• Every Polish space is homeomorphic to a G_δ subspace of the Hilbert cube, and every G_δ subspace of the Hilbert cube is Polish.
• Every Polish space is obtained as a continuous image of Baire space; in fact every Polish space is the image of a continuous bijection defined on a closed subset of Baire space. Similarly, every compact Polish space is a continuous image of Cantor space.

Because of these universality properties, and because the Baire space ${\displaystyle {\mathcal {N}}}$ has the convenient property that it is homeomorphic to ${\displaystyle {\mathcal {N}}^{\omega }}$, many results in descriptive set theory are proved in the context of Baire space alone.

Borel sets

The class of Borel sets of a topological space X consists of all sets in the smallest σ-algebra containing the open sets of X. This means that the Borel sets of X are the smallest collection of sets such that:

• Every open subset of X is a Borel set.
• If A is a Borel set, so is ${\displaystyle X\setminus A}$. That is, the class of Borel sets are closed under complementation.
• If A_n is a Borel set for each natural number n, then the union ${\displaystyle \bigcup A_{n}}$ is a Borel set. That is, the Borel sets are closed under countable unions.

A fundamental result shows that any two uncountable Polish spaces X and Y are Borel isomorphic: there is a bijection from X to Y such that the preimage of any Borel set is Borel, and the image of any Borel set is Borel. This gives additional justification to the practice of restricting attention to Baire space and Cantor space, since these and any other Polish spaces are all isomorphic at the level of Borel sets.

Borel hierarchy

Each Borel set of a Polish space is classified in the Borel hierarchy based on how many times the operations of countable union and complementation must be used to obtain the set, beginning from open sets. The classification is in terms of countable ordinal numbers. For each nonzero countable ordinal α there are classes ${\displaystyle \mathbf {\Sigma } _{\alpha }^{0}}$, ${\displaystyle \mathbf {\Pi } _{\alpha }^{0}}$, and ${\displaystyle \mathbf {\Delta } _{\alpha }^{0}}$.

• Every open set is declared to be ${\displaystyle \mathbf {\Sigma } _{1}^{0}}$.
• A set is declared to be ${\displaystyle \mathbf {\Pi } _{\alpha }^{0}}$ if and only if its complement is ${\displaystyle \mathbf {\Sigma } _{\alpha }^{0}}$.
• A set A is declared to be ${\displaystyle \mathbf {\Sigma } _{\delta }^{0}}$, δ > 1, if there is a sequence ⟨ A_i ⟩ of sets, each of which is ${\displaystyle \mathbf {\Pi } _{\lambda (i)}^{0}}$ for some λ(i) < δ, such that ${\displaystyle A=\bigcup A_{i}}$.
• A set is ${\displaystyle \mathbf {\Delta } _{\alpha }^{0}}$ if and only if it is both ${\displaystyle \mathbf {\Sigma } _{\alpha }^{0}}$ and ${\displaystyle \mathbf {\Pi } _{\alpha }^{0}}$.

A theorem shows that any set that is ${\displaystyle \mathbf {\Sigma } _{\alpha }^{0}}$ or ${\displaystyle \mathbf {\Pi } _{\alpha }^{0}}$ is ${\displaystyle \mathbf {\Delta } _{\alpha +1}^{0}}$, and any ${\displaystyle \mathbf {\Delta } _{\beta }^{0}}$ set is both ${\displaystyle \mathbf {\Sigma } _{\alpha }^{0}}$ and ${\displaystyle \mathbf {\Pi } _{\alpha }^{0}}$ for all α > β. Thus the hierarchy has the following structure, where arrows indicate inclusion.

${\displaystyle {\begin{matrix}&&\mathbf {\Sigma } _{1}^{0}&&&&\mathbf {\Sigma } _{2}^{0}&&\cdots \\&\nearrow &&\searrow &&\nearrow \\\mathbf {\Delta } _{1}^{0}&&&&\mathbf {\Delta } _{2}^{0}&&&&\cdots \\&\searrow &&\nearrow &&\searrow \\&&\mathbf {\Pi } _{1}^{0}&&&&\mathbf {\Pi } _{2}^{0}&&\cdots \end{matrix}}{\begin{matrix}&&\mathbf {\Sigma } _{\alpha }^{0}&&&\cdots \\&\nearrow &&\searrow \\\quad \mathbf {\Delta } _{\alpha }^{0}&&&&\mathbf {\Delta } _{\alpha +1}^{0}&\cdots \\&\searrow &&\nearrow \\&&\mathbf {\Pi } _{\alpha }^{0}&&&\cdots \end{matrix}}}$

보렐 집합의 정규성 특성

고전적 서술적 집합 이론은 보렐 집합의 규칙성 특성에 대한 연구를 포함한다. 예를 들어, 폴란드 공간의 모든 보렐 세트는 Baire의 특성과 완벽한 세트 특성을 가지고 있다. 현대적 서술적 집합 이론은 이러한 결과가 폴란드 공간의 다른 세분류 하위 집합에 일반화되거나 일반화되지 않는 방법에 대한 연구를 포함한다.

분석 및 공동 분석 세트

복잡성의 보렐 집합 바로 너머에는 분석 집합과 공동 분석 집합이 있다. 폴란드 공간 X의 부분 집합은 그것이 일부 다른 폴란드 공간의 보렐 부분 집합의 연속 이미지라면 분석적이다. 보렐 세트의 모든 연속적인 프리이미지는 보렐이지만, 모든 분석 세트가 보렐 세트는 아니다. 한 세트는 그것의 보완물이 분석적이라면 공동 분석적이다.

투영 세트 및 와지 도

서술 집합 이론의 많은 질문들은 궁극적으로 이론적 설정 고려사항과 서수 및 기수 숫자의 특성에 의존한다. 이 현상은 특히 투영적인 집합에서 뚜렷하게 나타난다. 이는 폴란드 공간 X의 투영적 계층 구조를 통해 정의된다.

• 분석적인 경우$$ 집합은 $\sigma {\displaystyle {\mathbf{}}_{}^{}}$ ${\displaystyle {\mathrm{}}_{}^{}}$ ${\$ }$_{1}^{1}:{$1}로 선언된다.
• 세트는$$ ${\displaystyle {\mathrm{}}_{}^{}}$ ${\displaystyle {\mathrm{}}_{}^{}}${\$displaystyle \mathbf${\$Pi$}$_{1}^{1$}이다.
• A set A is ${\displaystyle \mathbf {\Sigma } _{n+1}^{1}}$ if there is a ${\displaystyle \mathbf {\Pi } _{n}^{1}}$ subset B of ${\displaystyle X\times X}$ such that A is the projection of B to the first coordinate.
• A set A is ${\displaystyle \mathbf {\Pi } _{n+1}^{1}}$ if there is a ${\displaystyle \mathbf {\Sigma } _{n}^{1}}$ subset B of ${\displaystyle X\times X}$ such that A is the projection of B to the first coordinate.
• $\Delta {\displaystyle {\mathbf{}}_{}^{}}$ ${\displaystyle {}_{}^{}}$ ${\displaystyle {\mathrm{}}_{}^{}}$ ${\$} $_{$$n}^{1}{$$n1}{{n}^{$1}와$$ $andn$ ${\displaystyle {\mathrm{}}_{}^{}}$ ${\$

As with the Borel hierarchy, for each n, any ${\displaystyle \mathbf {\Delta } _{n}^{1}}$ set is both ${\displaystyle \mathbf {\Sigma } _{n+1}^{1}}$ and ${\displaystyle \mathbf {\Pi } _{n+1}^{1}.$$}$

The properties of the projective sets are not completely determined by ZFC. Under the assumption V = L, not all projective sets have the perfect set property or the property of Baire. However, under the assumption of projective determinacy, all projective sets have both the perfect set property and the property of Baire. This is related to the fact that ZFC proves Borel determinacy, but not projective determinacy.

More generally, the entire collection of sets of elements of a Polish space X can be grouped into equivalence classes, known as Wadge degrees, that generalize the projective hierarchy. These degrees are ordered in the Wadge hierarchy. The axiom of determinacy implies that the Wadge hierarchy on any Polish space is well-founded and of length Θ, with structure extending the projective hierarchy.

Borel equivalence relations

A contemporary area of research in descriptive set theory studies Borel equivalence relations. A Borel equivalence relation on a Polish space X is a Borel subset of ${\displaystyle X\times X}$ that is an equivalence relation on X.

Effective descriptive set theory

The area of effective descriptive set theory combines the methods of descriptive set theory with those of generalized recursion theory (especially hyperarithmetical theory). In particular, it focuses on lightface analogues of hierarchies of classical descriptive set theory. Thus the hyperarithmetic hierarchy is studied instead of the Borel hierarchy, and the analytical hierarchy instead of the projective hierarchy. This research is related to weaker versions of set theory such as Kripke–Platek set theory and second-order arithmetic.

Table

Lightface		Boldface
Σ0 0 = Π0 0 = Δ0 0 (sometimes the same as Δ0 1)		Σ0 0 = Π0 0 = Δ0 0 (if defined)
Δ0 1 = recursive		Δ0 1 = clopen
Σ0 1 = recursively enumerable	Π0 1 = co-recursively enumerable	Σ0 1 = G = open	Π0 1 = F = closed
Δ0 2		Δ0 2
Σ0 2	Π0 2	Σ0 2 = Fσ	Π0 2 = Gδ
Δ0 3		Δ0 3
Σ0 3	Π0 3	Σ0 3 = Gδσ	π0 3σδ = F
⋮		⋮
σ0 <ω = π0 <ω = Δ0 <ω = σ1 0 = σ1 0 = Δ = Δ1 0 = 산술적		σ0 <ω = π0 <ω = Δ0 <ω = σ1 0 = δ1 0 = Δ = Δ1 0 = 굵은 얼굴 산술적
⋮		⋮
Δ0 α(α 재귀)		Δ0 α (α 카운트 가능)
Σ0 α	Π0 α	Σ0 α	Π0 α
⋮		⋮
σ0 ωCK 1 = π0 ωCK 1 = Δ0 ωCK 1 = Δ = Δ1 1 = 초산술		σ0 ω1 = π0 ω1 = Δ0 ω1 = Δ1 1 = B = 보렐
σ1 1 = 라이트페이스 분석법	π1 1 = 경량형 코아날리틱	σ1 1 = A = 분석적	π1 1 = CA = 공분석
Δ1 2		Δ1 2
Σ1 2	Π1 2	σ1 2 = PCA	π1 2 = CPCA
Δ1 3		Δ1 3
Σ1 3	Π1 3	σ1 3 = PCPCA	π1 3 = CPCPCA
⋮		⋮
σ1 <ω = π1 <ω = Δ1 <ω = Δ2 0 = σ = Δ2 0 = Δ2 0 = 분석적		σ1 <ω = Δ1 <ω = Δ1 <ω = σ2 0 = σ2 0 = Δ2 0 = P = 투영적
⋮		⋮

참고 항목

• 포인트 클래스
• 프리웰오더링
• 척도 속성

참조

• Kechris, Alexander S. (1994). Classical Descriptive Set Theory. Springer-Verlag. ISBN 0-387-94374-9.
• Moschovakis, Yiannis N. (1980). Descriptive Set Theory. North Holland. p. 2. ISBN 0-444-70199-0.

외부 링크

• 기술 집합 이론, 데이비드 마커, 2002. 강의 노트.