Projective hierarchy

The projective hierarchy is examined in the mathematical branch of descriptive set theory; it is a hierarchy of sets built up in stages according to a certain formation law, the lowest level of which begins with the Borel sets . The original interest lay in the investigation of the subsets of the continuum , i.e. the set of real numbers , but it has been shown that one can just as easily develop the theory for Polish spaces , in particular the Baire space in the one presented below Use Education Act. The projective hierarchy was introduced in 1925 by Lusin and Sierpiński .

definition

The following recursive definition is structurally based on the Borel hierarchy ; a one is used here as the upper index to distinguish it. stand for a Polish space, let it be the Baire space, that is to say the -fold Cartesian product of , provided with the product topology , whereby the term used in set theory is for the set of natural numbers. ${\displaystyle X}$ ${\displaystyle {\mathcal {N}}=\omega ^{\omega }}$ ${\displaystyle \omega }$ ${\displaystyle \omega }$ ${\displaystyle \omega }$

• ${\displaystyle \Sigma _{1}^{1}:={\mathcal {A}}}$, the class of all analytic sets .
• ${\displaystyle \Pi _{n}^{1}:=\left\{S\subseteq X|X\setminus S\in \Sigma _{n}^{1}\right\}}$for , that is, the -sets are the complements of the -sets. ${\displaystyle n\in \mathbb {N} }$ ${\displaystyle \Pi _{n}^{1}}$ ${\displaystyle \Sigma _{n}^{1}}$
• ${\displaystyle \Sigma _{n+1}^{1}}$be the class of all sets of the form , where the projection on the first component is and a -set. ${\displaystyle p_{1}(A)}$ ${\displaystyle p_{1}:X\times {\mathcal {N}}\rightarrow X}$ ${\displaystyle A}$ ${\displaystyle \Pi _{n}^{1}}$
• ${\displaystyle \Delta _{n}^{1}:=\Sigma _{n}^{1}\cap \Pi _{n}^{1}}$ For ${\displaystyle n\in \mathbb {N} }$

A set that is in one of or equivalently or is called a projective set . ${\displaystyle \Delta _{n}^{1}}$ ${\displaystyle \Sigma _{n}^{1}}$ ${\displaystyle \Pi _{n}^{1}}$

Remarks

• The above definition is recursive. One begins with the class of analytical sets, thus declaring it as the class of complements and from this using the projection technique given . This then again explains the class of complements, which then results again using the above projection technique, and so on. The projection method for the definition of is the namesake for the projective hierarchy. is explained as soon as and are explained. ${\displaystyle \Sigma _{1}^{1}}$ ${\displaystyle \Pi _{1}^{1}}$ ${\displaystyle \Sigma _{2}^{1}}$ ${\displaystyle \Pi _{2}^{1}}$ ${\displaystyle \Sigma _{3}^{1}}$ ${\displaystyle \Sigma _{n+1}^{1}}$ ${\displaystyle \Delta _{n}^{1}}$ ${\displaystyle \Sigma _{n}^{1}}$ ${\displaystyle \Pi _{n}^{1}}$

• The use of the Baire space can in principle be avoided, because this is homeomorphic to the space of irrational numbers , which can be provided with a complete metric . The proof that the irrational numbers can be fully metrised is essentially the proof that sets in Polish spaces are again Polish spaces. The metric to be constructed is not the Euclidean metric ; therefore, using the baire space is more natural. ${\displaystyle G_{\delta }}$

properties

The sets in are by definition the complements of analytical sets; they are therefore also called co-analytical. The sets from are analytic sets whose complements are also analytic. According to one of Suslin's theorem, these are exactly the Borel sets. ${\displaystyle \Pi _{1}^{1}}$ ${\displaystyle \Delta _{1}^{1}}$

The classes given above meet the following inclusions

In an uncountable Polish space, all specified inclusions are real. For a Polish space that can at most be counted, all sets are equal to the power set of the space.

All classes and are closed with regard to countable averages and countable unions, in particular is a σ-algebra . The projective sets as a whole, however, do not form an algebra for uncountable Polish spaces . The projective hierarchy can, however, analogously to the Borel hierarchy, be continued to form a hierarchy of for any countable ordinal numbers (which is considered less often than the projective hierarchy) . The union of all these sets forms the -algebra of -projective sets . ${\displaystyle \Sigma _{n}^{1},\Pi _{n}^{1}}$ ${\displaystyle \Delta _{n}^{1}}$ ${\displaystyle \Delta _{n}^{1}}$ ${\displaystyle \sigma }$ ${\displaystyle \Delta _{\alpha }^{1},\Sigma _{\alpha }^{1},\Pi _{\alpha }^{1}}$ ${\displaystyle \alpha }$ ${\displaystyle \sigma }$ ${\displaystyle \sigma }$

Is a Borel function between Polish spaces and belongs to one of the classes or , so too . ${\displaystyle f\colon X\rightarrow Y}$ ${\displaystyle A\subset Y}$ ${\displaystyle \Sigma _{n}^{1},\Pi _{n}^{1}}$ ${\displaystyle \Delta _{n}^{1}}$ ${\displaystyle f^{-1}(A)}$

Every amount in is Lebesgue measurable and every amount has the Baire property . Since these properties are transferred to complements, this also applies to quantities. Furthermore, every uncountable set has a perfect subset and therefore the width of the continuum. For higher levels of the projective hierarchy, these properties can no longer be proven in the Zermelo-Fraenkel set theory with the axiom of choice. Gödel had shown that, assuming the axiom of constructibility, there is a set in that is not Lebesgue-measurable and an uncountable set in that does not contain a perfect subset. Further statements sometimes require stronger axioms that go beyond the Zermelo-Fraenkel set theory, as explained in Chapters 25 ( Descriptive Set Theory ) and 32 ( More Descriptive Set Theory ) of the textbook by Thomas Jech given below. These properties are closely related to the determinacy of certain games. In fact, they can be deduced for Borel sets from the Borel determinacy , which is valid in ZFC. If one assumes the axiom of determinacy in addition to ZF , whose relative consistency to ZF cannot be proven in ZFC and which contradicts the axiom of choice, even all subsets of the real numbers are Lebesgue measurable, contain a non-empty perfect subset and have the Baire property. The requirement of these properties is possible for the class of projective sets together with the axiom of choice, in that one demands the determinateness of every game whose winning set is a projective subset of Baire space ( axiom of projective determinacy ). This in turn follows from certain axioms about the existence of large cardinal numbers . However, the determinacy of every game with an analytical winning amount cannot be proven in ZFC. ${\displaystyle \Sigma _{1}^{1}}$ ${\displaystyle \mathbb {R} }$ ${\displaystyle \Sigma _{1}^{1}}$ ${\displaystyle \Pi _{1}^{1}}$ ${\displaystyle \Sigma _{1}^{1}}$ ${\displaystyle \Delta _{2}^{1}}$ ${\displaystyle \Sigma _{2}^{1}}$

Individual evidence

1. ^ N. Lusin: Sur unproblem`eme de M. Emile Borel et les ensembles projectifs de M. Henri Lebesgue: les ensembles analytiques , Comptes Rendus Acad. Sci. Paris (1925), volume 180, pages 1318-1320
2. ^ W. Sierpinski Sur une classe d'ensembles , Fundamenta Methematicae (1925), Volume 7, pages 237-243
3. ^ YN Moschovakis: Descriptive Set Theory , North Holland 1987, ISBN 0-444-70199-0 , Chapter 1E: "The projective sets"
4. ^ Thomas Jech : Set Theory. 3rd millennium edition, revised and expanded. Springer, Berlin et al. 2003, ISBN 3-540-44085-2 , (11.12): "The Hierarchy of Projective Sets"
5. ^ YN Moschovakis: Descriptive Set Theory , North Holland 1987, ISBN 0-444-70199-0 , Theorem 1E.1 and 1E.3
6. ^ YN Moschovakis: Descriptive Set Theory , North Holland 1987, ISBN 0-444-70199-0 , Corollary 1F.2
7. ↑ Alexander S. Kechris : Classical Descriptive Set Theory . Springer , Berlin 1994, ISBN 0-387-94374-9 , pp. 341 .
8. ^ YN Moschovakis: Descriptive Set Theory , North Holland 1987, ISBN 0-444-70199-0 , Theorem 1G.1
9. ^ Thomas Jech : Set Theory. 3rd millennium edition, revised and expanded. Springer, Berlin a. a. 2003, ISBN 3-540-44085-2 , Corollary 25.28, Corollary 25.28
10. ^ Thomas Jech : Set Theory. 3rd millennium edition, revised and expanded. Springer, Berlin et al. 2003, ISBN 3-540-44085-2 , chapters 25, 32
11. ↑ Kechris, pp. 205 ff.