introduction to classical descriptive set theoryintroduction to classical descriptive set theory...

Early Development Definable Sets of Reals Regularity Properties Metamathematical Issues References

Introduction toClassical Descriptive Set Theory

J.T. Chipman

Logic SeminarStanford University

13 October 2015

J.T. Chipman Logic Seminar Stanford University

Introduction to Classical Descriptive Set Theory


“The efforts that I exerted led me to the following totallyunexpected discovery: there exists a family [...]consisting of effective [i.e., definable] sets, such that onedoes not know and one will never know whether every setfrom this family, if uncountable, has the cardinality of thecontinuum, nor whether it [has the property of Baire],nor whether it is measurable. [...] This is the family ofthe projective sets of Mr. H. Lebesgue. It remains but torecognize the nature of this new development.” (Luzin(1925))




Table of contents

1 Early Development

2 Definable Sets of RealsBorel SetsProjective SetsL(R)

3 Regularity PropertiesLebesgue MeasurabilityProperty of BairePerfect Set PropertyDistribution of Regularity Properties

4 Metamathematical Issues

5 References




Kanamori summarizes the subject matter of Descriptive Set Theorysuccinctly:

“Descriptive Set Theory is the definability theory of thecontinuum.”

Concerning the origins of the theory, Moschovakis writes in theopening lines of his textbook:

“The roots of Descriptive Set Theory go back to the workof Borel, Baire, and Lebesgue around the turn of the20th century, when the young French analysts were tryingto come to grips with the abstract notion of a function. ”




In this section, we provide a brisk history of the relevant results ofBorel, Baire, Lebesgue, and, last, the Russians Luzin and Suslin.There are three goals of this presentation:

To provide a brief historical and technical introduction to thestudy of definable sets of reals and their regularity properties

To make explicit the relevant mathematical motivations ofthis theory (as opposed to logical or metamathematicalmotivations), and

To motivate the relationship between definability andprinciples of definable determinacy




Emile Borel (1871-1956)

Borel’s 1898 book, Lecons sur la theorie des fonctions (“Lessonson the theory of functions”), introduced the contemporary notionof a measure on the real line:

The development of analytical geometry motivated the needto define a general notion of measure for (subsets of) Rn

One of Borel’s key new ideas was employing countableadditivity and using this to define the Borel setsBeginning with a family of sets of an obvious measure, suchas open or closed intervals on [0, 1], and a function assigningto each interval its length, he recursively expands thefunction’s domain of definition in stages to sets whosecomplements are previously defined and sets that are theunions of previously defined disjoint sets




Rene-Louis Baire (1874-1932)

Baire’s 1899 thesis, Sur le fonctions de variables reelles (“On thefunctions of real variables”), analyzed real functions in terms of aproper transfinite hierarchy—the first such analysis since Cantor:

Baire class 0 consists of the continuous real functions

For countable ordinals α > 0, Baire class α consists offunctions f not in any previous class but are nonethelesspointwise limits of sequences of functions g0, g1, g2, ... fromprevious classes: i.e., f (x) = limn→∞gn(x).

Further, he proved two facts concerning the functions in theseclasses (i.e., the Baire functions):

They are closed under pointwise limits, andThere exist non-Baire real functions




Henri Lebesgue (1875-1941)

Lebesgue’s 1902 thesis, Integral, longueur, aire (“Integral, length,area”), is the source of modern measure theory. His concept ofmeasurable set subsumed the Borel sets and his analytic definitionof measurable function subsumed the Baire functions.

Further elaboration of these ideas in his 1905 Sur les fonctionsrepresentables analytiquement (“On the analytically representablefunctions”) yielded two major results (stated on the followingpage) of enduring interest. These results were enabled byestablishing a correlation between the Borel sets and the Bairefunctions: the former are exactly the pre-images {x |f (x) ∈ O} ofopen intervals O by Baire functions f .




Here are the theorems:

For every countable ordinal α, there is a Baire function ofclass α. I.e., the Baire (and hence Borel) hierarchy is proper.

There is a Lebesgue measurable function that is not in anyBaire class. Hence there are non-Borel Lebesgue measurablesets.

Between 1905 and 1915, the work of the French analysts alsoresulted in a codification of the regularity properties—propertiesindicative of “well-behaved” sets of reals. It would be, however, aRussian student in Paris, Nikolai Luzin, whose initiative led to thesubject of descriptive set theory as a distinct discipline.




Nikolai Luzin (1883-1950) & Mikhail Suslin (1894-1919)

While investigating the results of Lebesgue described above, one ofLuzin’s students, Suslin, made an important discovery concerningthe projections of Borel sets. I.e., for Y ⊆ Rk+1, the projection ofY is:

p[Y ] = {< x1, ..., xk >∈ Rk |∃y(< x1, ..., xk , y >∈ Y )}

Suslin discovered a mistaken assumption of Lebesgue’s to theeffect that projections of (Gδ) Borel subsets of the plane are alsoBorel by constructing a counterexample. Hence the Borel sets arenot closed under projection (it turns out that, more generally, theyare not closed under continuous images).




Suslin’s 1917 Sur une definition des ensembles mesurables B[orel]sans nombres transfinis (“On a definition of Borel measurable setswithout transfinite numbers”) uses something he calls “operationA” to directly define the sets used in his counterexample—theanalytic sets. We will, however, use the more modern definition ofthem:

X ⊆ R is analytic if there is a closed Y ⊆ R× ωω such that, for allx , x ∈ X ↔ ∃y ∈ ωω〈x , y〉 ∈ Y .

In effect, operation A had allowed Suslin to “factor out” andcollect the first coordinates of the ordered pairs in countableintersections of open sets in two dimensions by means ofuncountable unions.




Suslin’s work on analytic sets produced four important theorems:

Every Borel set is analytic

There is an analytic set that is not Borel

A set of reals is Borel iff both it and its complement areanalytic

Every analytic set is Lebesgue measurable, has the property ofBaire, and has the perfect set property

Note the analogy here, for the first three, to later results. I.e.,when “Borel” is replaced by “recursive” and “analytic” is replacedby “recursively enumerable.” We will return to this point later.




In the rest of the presentation, we use the preceding historicalsurvey to motivate central concerns of contemporary descriptiveset theory:

The hierarchy of definable sets of reals: Borel, Projective, andthose in L(R)

Regularity properties of definable sets of reals: Lebesguemeasurability, Baire property, and the perfect set property

Note that I omit the “structural properties,” such asUniformization, for the sake of time

The limitative results in the field and motivating principles ofdefinable determinacy




1 Early Development




5 References




Borel Sets

It is convenient to use the set of functions from ω to ω—ωω (the“Baire space”)—instead of R. This is for a few topological reasons:

Both ωω and R are perfect Polish spaces (no isolated points,completely metrizable, and contain a countable dense subset):

The Baire Category Theorem (by the second listed property)ensures that every non-empty open set in either space isnon-meagre (i.e., non-negligible in a sense we will makeprecise)Q is a countable dense subset of R and R\Q, the irrationals,form an uncountable subsetUnder the subspace topology inherited from R, the irrationalsare topologically equivalent to ωω

(continued on next page)




Borel Sets

So we picture ωω as the set of all sequences of non-negativeintegers

The basic open sets are formed from a countable subsetω<ω—the set of all finite strings of positive integers:

For s ∈ ω<ω, the set of sequences extending s form the openset Ns

The set of such Ns generate the product topology on ωω

Last, since each of ωω and R are perfect Polish spaces, so aren-dimensional versions for n < ω




Borel Sets

We define the Borel sets B as the smallest set of ωω with thefollowing properties:

Every open set is in BIf B ∈ B, then (ωω\B) ∈ BIf Bi ∈ B for all i ∈ ω, then

⋃i Bi ∈ B

I.e., the Borel sets are recursively generated from the open sets bythe operations of complementation and countable union—hence,they form a σ-algebra.




Borel Sets

We define the Borel hierarchy inductively as follows. SupposeA ⊆ ωω and α an ordinal with α < ℵ1, then:

A is Σ01 iff A is open

A is Π01 iff ωω\A is Σ0

1 (i.e., A is closed)

For α > 1, A is Σ0α iff, for each An ∈ Π0

β with β < α,A =

⋃n∈ω An

For α > 1, A is Π0α iff ωω\A is Σ0

α

A is ∆0α iff A is both Σ0

α and Π0α




Borel Sets

Hence, a set B ∈ B just in case, for α < ℵ1, B is either Σ0α or

Π0α—further, the least such α is the Borel rank of B. Recall that

Lebesgue’s 1905 showed that the Borel hierarchy is proper (i.e.,new sets appear at each level) via identification of each class of thehierarchy with a class of Baire functions. The classical namingconventions, beginning with Fσ and Gδ, correspond respectively toΣ0

2 and Π02.




Projective Sets

Recall now Suslin’s discovery that there are (lower dimensional)projections of Borel subsets of the plane that are not themselvesBorel and hence that B is not closed under projection. Further,that he introduced the study of the analytic sets by considerationof the projections of closed Π0

2 subsets of the plane along an axis.

We now use this operation to define a hierarchy in a manneranalogous to the use of countable union in defining the Borelhierarchy.




Projective Sets

We define the Projective hierarchy inductively as follows. SupposeA ⊆ ωω and n ∈ ω, then:

A is Σ10 iff A is open

A is Π10 iff A is closed

A is Π1n iff ωω\A is Σ1

n

A is Σ1n+1 iff A = p[A′] for A′ in Π1

n

A is ∆1n iff A is both Σ1

n and Π1n




Projective Sets

To define the analytic sets that emerged from Suslin’s discovery, itturns out that taking A to be closed is sufficient to ensurep[A] 6∈ B. In our hierarchy, Σ1

1 sets are the projections of closedsubsets of ωω—accordingly, for A ⊆ ωω, A is analytic just in caseA is Σ1

1.

Also recall Suslin’s claim that, for A ⊆ ωω, A is Borel iff both Aand ωω\A are analytic (i.e., A is both analytic and co-analytic).This can now be expressed as the claim that the sets in B areexactly the ∆1

1 sets.




L(R)

We have so far characterized the Borel and Projective hierarchiesin topological terms rather than in terms of definability.Nonetheless, the projective sets, for instance, are exactly those thatare definable with real parameters in second-order arithmetic:

The superscript in ‘�1n’ (where � ∈ {Σ,Π}) denotes

quantification over real numbers

For sets in classes denoted with ‘Σ’, defining formulas beginwith a block of existential quantifiers

For sets in classes denoted with ‘Π’, defining formulas beginwith a block of universal quantifiers

The subscript in ‘�1n’ (where � ∈ {Σ,Π}) denotes n

alternations of quantifiers




L(R)

When we do not allow real parameters, we generate thecorresponding familiar lightface hierachies of effective descriptiveset theory:

the Arithmetical hierarchy, which is analogous to the Borelhierarchy, and

The Analytical hierarchy, which is analogous to the Projectivehierarchy

In the effective case, by analogy, we take ω for R, recursivefunctions for continuous functions, hyperarithmetical sets for Borelsets, and analytical sets for projective sets.In either case, the heart of these hierarchies is that the objects innew classes are defined in terms of those objects defined in earlierclasses.




L(R)

Note a disanalogy, however, between our characterization of theBorel and Projective hierarchies: the former applies to all classesindexed by a countable ordinal whereas the latter applies only tothose classes indexed by finite ordinals.

Recall from the introduction to this seminar that, for X a transitiveset, a subset Y ⊆ X is “definable in (X ,∈) with parameters”when, for some first-order formula φ and parameters y1, ..., yn ∈ X ,we have Y = {y ∈ X |(X ,∈) |= φ(y , y1, ..., yn)}. The set of allsuch Y we denote here by Def (X ).




L(R)

Recall that, for some set X , L0(X ) = TC (X ) where TC (X ) is thesmallest transitive Y s.t. X ∈ Y . Note ω ⊆ Vω butω 6∈ Vω—taking P(Vω) gives us the set of all functions on ω × ωand so we begin with Vω+1. The hierarchy L(R) is accordinglygenerated as follows:

L0(R) = Vω+1

Lα+1(R) = Def (Lα(R))

For limit ordinals λ, Lλ(R) =⋃α<λ Lα(R)

L(R) denotes, for α ∈ On,⋃α Lα(R). The Projective sets are

exactly those in P(ωω) ∩ L1(R)—insofar as we’d like to remainagnostic about the truth of AC in L(R), we take the intersectionto avoid assuming that there is a (projectively definable)well-ordering of R in L(R).




L(R)

A quick preview of of the relevance of L(R):

Assuming ZFC +“there exists a proper class of Woodin cardinals,”it follows that the theory of L(R) (and so that of the projectivesets) is invariant under set forcing. We will discuss this strong formof absoluteness next week.




1 Early Development




5 References




Lebesgue Measurability

We define Lebesgue measurability together with a measurefunction µ that maps subsets of ωω to [0, 1]:

For s ∈ ω<ω a finite sequence with |s| = n,µ([s]) :=

∏ni=0

12s(i)+1

The standard measure-theoretic construction extends µ to allBorel sets via induction on negation and countable union

For B ∈ B, B is Lebesgue null if µ(B) = 0

For arbitrary X ⊆ ωω, X is Lebesgue null if X ⊆ B for someB ∈ B such that µ(B) = 0—then we define µ(X ) := 0

For X ⊆ ωω, X is Lebesgue measurable if there exists aB ∈ B s.t. X\B ∪ B\X (i.e., their symmetric difference) isLebesgue null—then we define µ(X ) := µ(B)

Hence, a set X has this property when it differs from a Borel set bya Lebesgue null set.




Property of Baire

For X ⊆ ωω,

X is meagre just in case X is the countable union of nowheredense sets

X has the property of Baire just in case there exists an openset O s.t. X\O ∪ O\X is meagre

Hence, a set X has this property when it differs from an open setby a meagre set—a type of set which has a measure-theoreticanalogue in the form of null sets. Both are, in their respectivecontexts, “negligible.” For instance, Q is a meagre subset of R.Last, recall that the Baire Category Theorem ensures us that everynon-empty open subset of ωω is non-meagre.




Perfect Set Property

For X ⊆ ωω,

X is perfect just in case: (i) X 6= ∅, (ii) X is closed, and (iii)X contains no isolated points (i.e., there is no x ∈ X s.t. {x}is open in X )

Every perfect set X in a Polish space contains a homeomorphiccopy of Cantor space 2ω and therefore has the cardinality ofthe Cantor set 2ω

X has the perfect set property just in case either: (i) X iscountable or (ii) there is a Y ⊆ X s.t. Y is perfect

The perfect set property is of interest insofar as the sets of realswith this property also satisfy CH (as can be seen from the secondclause).




Distribution of Regularity Properties

These three properties are called the regularity properties of setsand are diagnostic of the “well-behavedness” of subsets of ωω

studied in descriptive set theory.

Qualitatively speaking, we can view such sets as arrayed along aspectrum of “well-behavedness”—at one extreme, the (relatively)simple sets defined by application of countable union andcomplementation from open sets at the base of the Borel hierarchy.And, at the other extreme, the (relatively) complex sets defined byarbitrary first-order formulas from Vω+1 at the base of L(R).





The increasing (naive, qualitative) complexity of sets along thisspectrum derives from the two clauses invoked in the definitions ofthe hierarchies involved:

The complexity of the subsets of ωω used as the basis of theinductive definitions of each hierarchy. I.e.:

For Borel, we use the open setsFor Projective, we use the analytic sets, which are (relatively)more complex than the open setsFor L(R), we use Vω+1, which is (relatively) more complexthan the analytic sets

and...





The complexity of the permissible operations used in the“step” of the inductive definitions of each hierarchy. I.e.:

For Borel, we apply a simple generalization of a booleanoperation (i.e., countable union and complementation) topreviously-defined setsFor Projective, we additionally apply what is essentially anexistential quantification (i.e., projection)—an operation morecomplex than boolean—to previously-defined setsFor L(R), we additionally apply operations defined fromformulas of arbitrary first-order complexity—operations morecomplex than projection—to previously-defined sets





Some philosophical questions then emerge to motivatemetamathematical investigation of the material presented so far:

Is there an obvious distribution of the regularity propertiesacross this spectrum? If so, what is the distribution?

Is there relevance of this distribution? If so, what is therelevance?

Does this distribution seem to be strongly correlated with thelogical strength of the assumptions required to prove thoseproperties hold? If there seems to be a strong correlation, whydoes it exist?

What sorts of properties of (relatively simple) sets in ωω

should mathematicians expect to hold of (relatively complex)sets in ωω? Why should we expect them to hold in this way?




1 Early Development




5 References




Here is a timeline of the discovery of regularity properties ofdefinable sets of reals. Assume ZFC .

1883 (Cantor and Bendixson): Π01 perfect set property

1903 (Young): Π02 perfect set property

1916 (Aleksandrov): ∆11 perfect set property

1917 (Luzin and Suslin): ∆11 Lebesgue measurability, property

of Baire

1917 (Luzin and Suslin): Σ11 perfect set property, Lebesgue

measurability, property of Baire

1917 (Luzin and Suslin): Π11 Lebesgue measurability, property

of Baire

(continued on next page)




1938 (Godel): Suppose V = L, then there are Π11 without

perfect set property

1938 (Godel): Suppose V = L, then there are Σ12

non-Lebesgue measurable, without property of Baire

1965 (Solovay): Suppose there is a measurable cardinal, thenevery Σ1

2 perfect set property, Lebesgue measurability,property of Baire

Moschovakis writes of these last results that, “the logiciansentered the picture in their usual style, as spoilers.” In particular,the last two results demonstrate that the question of the whetherthe regularity properties hold of projective sets at Σ1

2 (and beyond)is independent of ZFC .




So, at the Σ12 level and beyond, the question of whether the rest of

the projective sets and the sets in L(R) have the regularityproperties is completely underdetermined by ZFC . We can nowrefine some of our philosophical questions:

Are there mathematically well-justified principles independentof ZFC that can decide whether Σ1

2 sets have the regularityproperties?

How about the sets in P(ωω) ∩ L1(R)?

And even L(R)?

If there are such principles, how much logical strength do theyadd to ZFC ? What kind of logical strength do they add?

Do we have reason to accept them? If so, are they axioms?




Starting in the 1950s and 1960s, descriptive set theorists began toinvestigate the consequences of asserting that there exist winningstrategies for infinite games of perfect information. These gamesare “played” over (representations of) of reals in the underlyingspace. It was discovered that the existence of winning strategiesfor these games, or determinacy, implies that their payoff sets haveall of the regularity properties.




The Axiom of Determinacy, AD, asserts that every set isdetermined. Because AD is itself inconsistent with AC , insteadrestrictions of AD that are consistent with AC are typicallyemployed—namely, restrictions to the definable sets in the Borelhierarchy, the Projective hierarchy, and those in L(R).

Moreover, many have attempted to address the metamathematicalbarriers that stymied the classical theory through an approach thatintegrates principles of definable determinacy and large cardinalhypotheses.




Next week:

Begin with infinite games of perfect information, strategies forthese games, and definable determinacy.

Then discuss metamathematical issues pertinent to ∆11

Determinacy, Projective Determinacy (PD), and ADL(R).

Last, examine the case supporting Steel and Martin’s 1985proof of PD from ZFC +“there exist infinitely-many Woodincardinals.”




References

Feferman, S. (1999), “Does mathematics need new axioms?,”American Mathematical Monthly 106: 99-111.

Ikegami, D. (2010), Games in Set Theory and Logic, PhDthesis, University of Amsterdam. ILLC DissertationsDS-2010-04.

Jech, T. (2003), Set theory, 3rd Edition, New York: Springer.

Kanamori, A. (1995), “The emergence of descriptive settheory,” in J. Hintikka (ed.), From Dedekind to Godel: Essayson the Development of the Foundations of Mathematics, v.251 of Synthese Library, Kluwer, Dordrecht, pp. 241-262.




References (cont.)

Khomskii, Y. (2012), Regularity Properties and Definability inthe Real Number Continuum, PhD thesis, University ofAmsterdam. ILLC Dissertations DS-2012-04.

Kechris, A. (1995), Classical Descriptive Set Theory, v. 156 ofGraduate Texts in Mathematics, Springer-Verlag, New York.

Koellner, P. (2013), “Large cardinals and determinacy,” TheStanford Encyclopedia of Philosophy (Spring 2014 Edition),Edward N. Zalta (ed.).




References (cont.)

Martin, D. (1998), “Mathematical evidence,” in H. G. Dalesand G. Oliveri (eds), Truth in Mathematics, Clarendon Press,pp. 215-231.

Moschovakis, Y. (1980), Descriptive Set Theory, Studies inLogic and the Foundations of Mathematics, North-HollandPub. Co.

Steel, J. (2000), “Mathematics needs new axioms,” Bulletinof Symbolic Logic 6(4): 422-433.

Steel, J. (2007), “What is a Woodin Cardinal?,” Notices ofthe American Mathematical Society, pp. 1146-1147. 291.



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