generalized descriptive set theory and the classiﬁcation ...€¦ · classical descriptive set...

Generalized descriptive set theoryand the classification of uncountable structures

and non-separable spaces

Luca Motto Ros

Department of Mathematics “G. Peano”

University of Turin, Italy

[email protected]

https://sites.google.com/site/lucamottoros/

Workshop on Generalized Baire spaces

Bonn, 21–22.09.2016

L. Motto Ros (Turin, Italy) Generalized DST and classification Bonn, 21–22.9.2016 1 / 45

Two premises... and a question

Premises

1 Most of the people in this room already know what generalized DSTis, its basics, and the most important results in this area.

2 Such results are not due to me, and I’m definitely not an expert in thefield.

The question

What am I doing here? What am I supposed to talk about in this “tutorial”?

(I’m not joking! I seriously asked this question to myself a hundred of times in

the past weeks...)


Part 1

Generalized DST and classification problems


Classical descriptive set theory

Descriptive set theory is the study of “definable sets” in Polish

(i.e., separable completely metrizable) spaces. In this theory, sets

are classified in hierarchies, according to the complexity of their

definitions, and the structure of the sets in each level of these

hierarchies is systematically analyzed.

Kechris, Classical descriptive set theory

Polish spaces were isolated as those spaces retaining the fundamentalproperties of the Euclidean spaces Rn that are needed to carry out a greatpart of the analysis of their definable sets/functions.

Among such spaces, a prominent role is occupied by the Baire space !!

and the Cantor space 2

!, both endowed with the product of the discretetopology.


Classical descriptive set theory

Classical DST has gained popularity also outside the logic communitybecause of its effectiveness in tackling (and often solving) problems fromother areas of mathematics. Part of this success is due to the combinationof the following facts:

Polish spaces are ubiquitous, most spaces considered in ordinarymathematics belong (up to coding) to such class;Polish spaces are quite manageable, and they avoid annoying(topological) pathologies;at least when the problem at hand is invariant under Borelisomorphism, it is not restrictive to work with well-understood spaceslike !! or 2!.


Classification problems

Let X be a set of mathematical objects and E be an equivalence relationon X . A solution to the classification problem for X up to E is just anassignment of complete invariants to elements of X up to E.

This means that we want to find:

a set I of invariants, andan assignment ' : X ! I

such that for all x0, x1 2 X

x0 E x1 () '(x0) = '(x1).

In order to have a meaningful solution, both I and ' must be as “natural”and “concrete” as possible.


Some examples

Classification up to similarity of all the n-by-n matrices over thecomplex numbers.Classification of linear orders up to isomorphism or equimorphism (=bi-embeddability).Classification of algebraic structures (groups, fields, ...) up toisomorphism or bi-embeddability.Classification of topological spaces/groups up to topologicalhomeomorphism or bi-embeddability.Classification of metric spaces up to isometry or isometricbi-embeddability.Classification of more complicated structures (Banach spaces, vonNeumann algebras, ...) up to the relevant notions ofisomorphism/bi-embeddability.


The DST point of view on classification: Borel reducibility

Suppose that the space X carries a natural Polish topological structure:then a concrete classification of the objects of X up to E consists of aPolish space I and a Borel map f : X ! I such that for all x0, x1 2 X

x0 E x1 () f(x0) = f(x1).

ExampleClassification of n-by-n complex matrices up to similarity by means of theassociated canonical Jordan form (up to permutation of the blocks).

When a concrete classification is not possible, one may still be satisfiedwith a classification map assigning

invariants up to countably many mistakes,invariants up to some natural equivalence (e.g. linear orders up toisomorphism),and so on.



Let X , Y be Polish spaces, and E, F be equivalence relations on X and Y.

Definition (Borel reducibility)E is Borel reducible to F (E

B

F ) if there is a Borel map f : X ! Ysuch that for all x0, x1 2 X

x0 E x1 () f(x0) F f(x1).

E B

F The objects in X may be classified up to E-equivalence

using the F -classes as invariants.

For example:there is a concrete classification for the elements of X up to Eequivalence if and only if E

B

id(R);(X , E) is classifiable by countable structures if E

B

⇠=

� Mod

!

L forL some countable first-order relational language (we can equivalentlyuse as invariants graphs, linear orders, ... up to isomorphim).



Borel reducibility can also be used as a tool for comparing the complexityof various classification problems.

Definition (Borel reducibility)E

B

F if there is a Borel map f : X ! Y such thatx0 E x1 () f(x0) F f(x1) for all x0, x1 2 X .E and F are Borel bi-reducible (E ⇠

B

F ) if E B

F B

E.

Intuitively, E B

F means: E is not more complicated than F .

E ⇠B

F means: E and F have the same complexity.

Usually, one concentrates on “definable” equivalence relations E on X ,most notably:

E is Borel if it is a Borel subset of the square X ⇥ X ;E is analytic if, as a subset of X ⇥ X , is a Borel image of !!.


First-order structures

Consider (for simplicity) the countable relational language L = (Rj

)

j2J ,where each R

j

has arity nj

. Up to isomorphism, all countable L-structurescan be assumed to have domain !, and thus they can be identified viacharacteristic functions of their predicates with the elements of the space

Mod

!

L =

Y

j2J2

(!nj ).

Endow Mod

!

L with the topology whose basic open sets are the collection ofstructures containing (as a substructure) a given finite L-structure on !:then Mod

!

L is homeomorphic to 2

! (thus it is Polish), and the isomorphismand bi-embeddability relations are analytic equivalence relations on it.

Fixing a first-order theory (or, more generally, an L!1!-sentence) T and

considering only the collection

Mod

!T = {x 2 Mod

!L | x |= T}

of models of it gives us a Borel subset of Mod

!L, and thus a standard Borel space.


First-order structures

Using this coding one may obtain many (anti-)classification results:the classification problem for countable graphs (trees, linear orders,non-abelian groups, ...) is as complex as possible, i.e. it is

B

aboveany equivalence relation which is classifiable by countable structures(H. Friedman-Stanley, Mekler);finitely generated groups are not concretely classifiable (but they canbe classified up to countably many mistakes) (Thomas?);the classification problem for torsion-free abelian countable groups offinite rank strictly increases in complexity with the rank(Hjorth-Thomas);countable graphs cannot be classified (in any reasonable sense) up tobi-embeddability (Louveau-Rosendal);the same is true for lattices, countable groups, ... but the classificationof e.g. countable linear orders up to bi-embeddability is simpler(Louveau-Rosendal, Laver).


Separable metric spaces

Every Polish (= separable complete) metric space is isometric to a closedsubspace of the separable Urysohn space U. Thus the space

F (U) = {C ✓ U | C closed in U}endowed with the Effros-Borel structure may be regarded as the standard

Borel space of all (up to isometry) Polish metric spaces.

The classification problem for compact Polish metric spaces is concretely

classifiable (Gromov).

The classification problem for arbitrary Polish metric spaces is Borel

bi-reducible with the most complex orbit equivalence relation (Gao-Kechris).

The classification problem for Polish ultrametric spaces is Borel bi-reducible

with graph isomorphism (Gao-Kechris).

The complexity does not drop if we consider only discrete Polish ultrametric

spaces (Camerlo-Marcone-M.).

It is not possible to classify (discrete) Polish (ultra)metric spaces up to

isometric bi-embeddability (Louveau-Rosendal, Camerlo-Marcone-M.).


Other separable spaces

In the same fashion one can consider more complicated separable spaces,as long as there is a “universal” space in the class of objects X :

compact metrizable spaces Hilbert cube;separable Banach spaces C[0; 1];...

This allows us to construe X as a Polish (or standard Borel) space and use

B

to classify its objects.

It is not possible to classify separable Banach spaces up to isomorphism or

bi-embeddability (Ferenczi-Louveau-Rosendal).

There is no reasonable classification of Polish groups up to (topological)

isomorphism (Ferenczi-Louveau-Rosendal).

The classification problem for separable nuclear C⇤-algebras up to

isomorphism is Borel bi-reducible with the most complex orbit equivalence

relation (Sabok).

Finitely generated operator systems are concretely classifiable

(Argerami-Coskey-Kalantar-Kennedy-Lupini-Sabok).L. Motto Ros (Turin, Italy) Generalized DST and classification Bonn, 21–22.9.2016 14 / 45

Limitations of classical DST

Despite its remarkable achievements, there are at least two seriouslimitations to this method.

1 So far, only analytic equivalence relations have been considered, butthere is a lot more out there.

For example, the conjugacy relation on the group of Borel automorphism of astandard Borel space is ⌃1

2-complete (Clemens). Borel bi-reducibility betweenanalytic equivalence relations is ⌃1

3 in the codes, and at least ⌃12-hard

(Adams-Kechris).

2 Polish (or standard Borel) spaces are not suitable for codinguncountable structures and non-separable spaces, which are clearlyrelevant in other areas of mathematics.

Think about the role of uncountable structures in model theory, or non-separableBanach/Hilbert spaces in analysis and operator theory.


Classical DST vs generalized DST

Descriptive set theory is the study of “definable sets” in Polish

spaces (such as !!

and 2

!

).

Kechris, Classical descriptive set theory

As it is conceived now,

Generalized descriptive set theory is the study of and 2

(endowed with a natural topology) and of their definable subsets

for an uncountable cardinal.

RemarkA large initial portion of classical DST (roughly speaking, up to Borel andanalytic sets) can be carried out in ZF+ DC. In contrast, the base theoryfor generalized DST is usually ZFC plus the cardinal assumption <

= .


Generalized Baire and Cantor spaces

Let be an uncountable cardinal.

DefinitionThe generalized Baire space is = {x | x : ! } endowed with the(bounded) topology generated by the sets of the form

N

s

= {x 2 | s ✓ x}

for s 2 <.

The generalized Cantor space is the closed subspace 2

of consistingof all binary sequences.

RemarkThe (bounded) topology described above is strictly finer than the producttopology. When is regular it coincides with the -product of the discretetopology.


A comparison with the classical context

and 2

are perfect, regular Hausdorff, zero-dimensional spaces.They are not metrizable (unless cof() = !), and their densitycharacter is < and 2

<; these are also the smallest sizes for theirtopological bases. In particular, has a base of size if and only if<

= .None of them is compact, but 2 is -compact if and only if is aweakly compact cardinal.The topology is closed under < cof()-intersections.


(+-)Borel sets and (-)analytic sets

DefinitionsA set A ✓ is (+-)Borel if it belongs to the smallest +-algebraon containing all open sets.A set A ✓ is (-)analytic if it is a projection of a closed subset of ⇥ .A set A ✓ is (-)bi-analytic if both A and \A are (-)analytic.

A comparison with the classical context:If <

= , Borel sets are the closure under complements and -unions of the basic open sets. Moreover, they form a propersubclass of P() that can be naturally stratified in a hierarchy withexactly +-many levels (the ⌃0

↵

’s and ⇧0↵

’s).All Borel sets are analytic, but there are bi-analytic sets which are notBorel (failure of the Souslin theorem).Analytic sets are closed under unions and intersections of size .


The central question

Why should one be interested in such spaces (and their subsets)?

The surly answer: Good mathematics needs not to be justified!

The pragmatic answer: Because it is a relatively unexplored area and onecan obtain nontrivial results (and a lot of publications!) in a reasonabletime.

The “good old days” answer: Because it helps in understanding whichproperties of ! are crucial to obtain the various results in classical DST.

The aestethic answer: Because the resulting theory is beautiful anddeeply connected with other areas of set theory (large cardinals, forcing,infinite combinatorics, ...).

The “working mathematician” answer: Because it is useful, e.g. whencoming to classification problems beyond the scope of classical DST...

Although none of these answers seems to be entirely wrong, I will

concentrate on the last one...


Analytic equivalence relations and Borel reducibility

Let X and Y be subsets of (or of a homeomorphic copy of it), and E,Fbe equivalence relations on X ,Y, respectively.

DefinitionsE is analytic (resp. Borel) if it is such as a subset of X ⇥ X .E is Borel reducible to F (E

B

F ) if there is a Borel mapf : X ! Y such that for every x0, x1 2 X

x0 E x1 () f(x0) F f(x1).

E and F are Borel bi-reducible (E ⇠

B

F ) if E

B

F and F

B

E.E is complete (for analytic eq. rel.) if F

B

E for every (-)analyticequivalence relation F .

The intended meaning of

B

and ⇠

B

is the same as in the classical case.


Uncountable first-order structures

By identifying the first-order L-structures with their characteristicfunctions, one can consider

Mod

L =

Yj2J

2

(nj )

as the space of all L-structures of size (up to isomorphism). There is acanonical bijection between Mod

L and 2

, thus Mod

L can be endowedwith the same (bounded) topology.

(When is regular, this topology coincides with the one whose basic open sets aredetermined by substructures of size < .)

It is easy to check that the isomorphism relation ⇠=

and thebi-embeddability relation ⇡ are analytic equivalence relations on Mod

L.



What happens if we restrict ourselves to the models of a first-order theory?

If <

= , then one obtains the following generalization of a theorem ofLopez-Escobar.

Theorem (Vaught, S. Friedman-Hyttinen-Kulikov, Andretta-M.)Assume that <

= . A subset of Mod

L is Borel and invariant underisomorphism if and only if it is of the form

Mod

� = {x 2 Mod

L | x |= �}

for some L

+

-sentence �.

RemarkThe isomorphism and bi-embeddability relations on the elementary classMod

� remain analytic equivalence relations.



Working in this setup, S. Friedman, Hyttinen, Kulikov, Moreno, and othersare developing a beautiful theory with remarkable and deep connectionswith Shelah’s stability theory.

Theorem (S. Friedman-Hyttinen-Kulikov)Let = < be an uncountable non weakly inaccessible cardinal, and T bea complete countable first-order theory. If the isomorphism relation onMod

T

is Borel, then T is classifiable and shallow. Conversely, if > 2

@0 ,then if T is classifiable and shallow, then the isomorphism relation onMod

T

is Borel.

Corollary (S. Friedman-Hyttinen-Kulikov)Under the same assumptions, if T is classifiable and shallow and T 0 is not,then isomorphism on Mod

T

0 does not Borel reduce to isomorphism onMod

T

.



Theorem (S. Friedman-Hyttinen-Kulikov)Assume that > !1 is such that <

= and �! < for all � < . Thenin L and in the forcing extension after adding +-many Cohen subsets of we have: for any theory T , T is classifiable if and only if the isomorphismrelation on Mod

T

is bi-analytic.

Corollary (S. Friedman-Hyttinen-Kulikov)Under the same assumptions, if T is classifiable and T 0 is not thenisomorphism on Mod

T

0 does not Borel reduce to isomorphism on Mod

T

.

Theorem (S. Friedman-Hyttinen-Kulikov)

Suppose = �+= 2

� > 2

@0 where �<�

= �. Let T be a first-order theory.Then T is classifiable if and only if for all regular µ < , E

µ

⇥

B

⇠=

� Mod

T

,where E

µ

is the equality on 2

modulo the ideal of not µ-stationary sets.



Theorem (Hyttinen-Kulikov)If V = L and is the successor of a regular cardinal, then the isomorphismrelation on dense linear orders of size is complete.

Theorem (Hyttinen-Kulikov)

Assume that V = L and that = �+ with �!

= �. Then there is a not toocomplicated (stable, NDOP, NOTOP) first-order theory T such that theisomorphism on Mod

T

is complete.

Under the same assumptions, isomorphism between trees of height! + ! + 2 and size is complete as well.



Theorem (Hyttinen-Moreno)Under suitable assumptions on = < > !, if T is classifiable and T 0 is astable theory with the OCP, then isomorphism on Mod

T

0 Borel reduces toisomorphism on Mod

T

.



In the same vein one can study the complexity of the bi-embeddabilityrelation.

Theorem (Mildemberger-M.)Assume that <

= . Then the bi-embeddability relation between graphsof size is complete.

Theorem? (Calderoni-M.)Assume that <

= . Then the bi-embeddability relation between groupsof size is complete.


Non-separable spaces

The striking connections between the

B

-complexity of the isomorphismrelation and the classification of first-order theories according to Shelah’sstability theory are certainly the most important achievements that havebeen obtained so far in the realm of generalized DST.

Unfortunately, not many non-logicians knows about model theory...

Can we tackle other kind of classification problems (metric spaces, Banachspaces, ...)?

Unlike the case of first-order structures, one cannot straightforwardlygeneralize the coding process used in the separable case:

universal objects may not exists in the relevant class;even when such an object exists, it is not clear that the collection ofits subspaces forms a well-behaved topological space (i.e. close to ).


Complete metric spaces of density character

Let Q+ be the set of positive rational numbers, and let X be the space2

⇥⇥Q+ : X is naturally isomorphic to 2

and can be identified with it.

Given a complete metric space (M,d) of density character and a densesubset D = {m

↵

| ↵ < } of it, we can identify M with the uniqueelement c

M

2 X such that for all ↵,� < and q 2 Q+

cM

(↵,�, q) = 1 () dM

(m↵

,m�

) < q.

In fact, M is isometric to the completion of the metric space (, dcM )

wheredcM (↵,�) := inf{q 2 Q+ | c

M

(↵,�, q) = 1},

so that dcM (↵,�) = d

M

(m↵

,m�

) for all ↵,� < .


Complete metric spaces of density character Consider now the space M

✓ X consisting of those c 2 2

⇥⇥Q+

satisfying the following conditions:

8↵,� < 8q, q0 2 Q+ ⇥q q0 ) c(↵,�, q) c(↵,�, q0)

⇤

8↵,� < 9q 2 Q+ [c(↵,�, q) = 1]

8↵ < 8q 2 Q+ [c(↵,↵, q) = 1]

8↵ < � < 9q 2 Q+ [c(↵,�, q) = 0]

8↵,� < 8q 2 Q+ [c(↵,�, q) = 1 () c(�,↵, q) = 1]

8↵,�, � < 8q, q0 2 Q+ ⇥c(↵,�, q) = 1 ^ c(�, �, q0) = 1 ) c(↵, �, q + q0) = 1

⇤

8↵ < 9� < 9q 2 Q+ 8� < ↵ [c(�,�, q) = 0] .

The first six conditions are designed so that given any c 2 M

, the

(well-defined) map dc

: ⇥ ! R given by

dc

(↵,�) := inf{q 2 Q+ | c(↵,�, q) = 1}

is a metric on ; we denote by Mc

the completion of (, dc

), and noticethat the last condition ensures that M

c

has density character .L. Motto Ros (Turin, Italy) Generalized DST and classification Bonn, 21–22.9.2016 31 / 45


It is straightforward to check thatthe code c

M

of any complete metric space M of density character belongs to M

, and is such that M is isometric to McM ;

conversely, for each c 2 M

the space Mc

is a complete metric spaceof density character .

Thus we can regard M

✓ 2

⇥⇥Q+ , endowed with the inherited(bounded) topology, as the space of (codes for) all complete metric

spaces of density character (up to isometry).



When = !, this procedure gives an alternative (but equivalent) way tocode Polish spaces.

FactThere are Borel maps � : M

!

! F (U) and : F (U) ! M!

such that forall c 2 M

!

Mc

and �(c) are isometric

and for all infinite M 2 M!

M and M (M) are isometric.

RemarkThis alternative coding has (essentially) been used e.g. by Vershik to showthat Polish metric spaces are not concretely classifiable up to isometry.



The explicit definition given by the previous six conditions shows that M

is a Borel subset of 2⇥⇥Q+ .

The isometry relation ⇠=

i on M

is given by

c ⇠=

i d () there is a metric-preserving bijection between Mc

and Md

,

while the isometric bi-embeddability relation ⇡i on M

is given by

c ⇡i d () Mc

isometrically embeds in Md

, and viceversa.

Using the Tarski-Kuratowski algorithm and some standard computations,one sees that the relations ⇠

=

i and ⇡i are analytic equivalence relations onM

.



We also consider some natural subclasses of M

, such as

D

= {c 2 M

| Mc

is discrete}

orU

= {c 2 M

| Mc

is ultrametric}.

RemarkNot all these subclasses are Borel: for example, U

is Borel (since it is aclosed subset of M

), while D

is not.



TheoremLet be any infinite cardinal. Then ⇠

=

i� D

(resp. ⇡i� D

) is Borelbi-reducible with isomorphism (resp. bi-embeddability) on graphs.

Proof.For one direction, associate to each graph G on the discrete space(, d

G

) given by dG

(↵,�) = 1 if ↵ G �, and dG

(↵,�) = 0 otherwise.

For the other direction, consider the language L = (Pq

)

q2Q+ with each Pq

binary, and to each c 2 D

associate the L-structure Ax

on given byA

x

|= Pq

[↵,�)] () c(↵,�, q) = 1. This works because Mc

coincideswith any of its dense subsets.



Corollary1 Assume that V = L and that = �+ with �!

= �. Then isometry onD

is complete for analytic equivalence relations.2 Assume that <

= . Then isometric bi-embeddability on D

iscomplete for analytic equivalence relations.

Thus, at least consistently, ⇠=

i and ⇡i may surprisingly have the samecomplexity already on D

.

RemarkObviously, we can replace “discrete” with locally compact,zero-dimensional, of size , and so on.



What about ultrametric spaces? An easy fact is the following.

PropositionLet be an infinite cardinal. Then isometry (resp. isometricbi-embeddability) on U

is Borel reducible to isomorphism (resp.bi-embeddability) between graphs of size .

Proof.We associate to each c 2 U

a structure whose points are the open balls Bin M

c

of rational radius and whose predicates are the following:for each q 2 Q+ a unary predicate P

q

which holds on B if and only ifdiam(B) < q;a binary predicate R such that B R B0 () B ✓ B0.

The fact that we are working with ultrametric spaces ensures that theresulting first-order structure has size and that the metric structure ofM

c

can be recovered from the above predicates.L. Motto Ros (Turin, Italy) Generalized DST and classification Bonn, 21–22.9.2016 38 / 45


NotationGiven an infinite cardinal and an ordinal ↵, let T

↵

be the collection of(set-theoretic) trees on of height ↵ and size .

Let ↵ be countable and fix any strictly decreasing sequence (r�

)

�<↵

ofpositive reals. Given T 2 T

↵

define a distance dT

on T by

dT

(u, v) =

(rlh(uuv) if u 6= v

0 otherwise.

Finally, let XT

be the completion of (T, dT

).

RemarkT ✓ X

T

is exactly the collection of all isolated point of XT

, so thatX

T

= T is discrete if inf{r�

| � < ↵} > 0.



TheoremFor every T, S 2 T

↵

T ⇠=

S () XT

⇠=

i XS

and T ⇡ S () XT

⇡i XS

.

Sketch of the proof.Any isomorphism (embedding) between T and S is an isometry (isometricembedding) between (T, d

T

) and (S, dS

), and the latter can be extendedto the whole spaces X

T

and XS

.

Conversely, an isometry ' between XT

and XS

maps T onto S (isolatedpoints!). Then one checks that ' may fail to be an isomorphism betweentrees just because it switches a terminal node with its immediatepredecessor (if such nodes exist): “straightening” the isometry gives thedesired isomorphism.

The case of (isometric) embeddings is a little bit more delicate.



CorollaryAssume that V = L and that = �+ with �!

= �. Then isometry onU

\D

is already complete for analytic equivalence relations.

Proof.By the result of Hyttinen-Kulikov, isomorphism on T

!+!+2 is complete, soit is enough to apply the previous theorem using a sequence of distances(r

�

)

�<!+!+2 bounded away from 0.


Banach spaces of density

In the same vein one can code Banach spaces of density character aselements of (a homeomorphic copy of) 2

, and check that the resulting setof codes B

is Borel. With this coding system, the relations of linearisometry ⇠

=

li and linear isometric embeddability ⇡li become analyticequivalence relations.

NotationLet c0 ✓ R be the collection of all sequences (x

↵

)

↵<

of real numberssuch that for all " > 0 we have |x

↵

| < " for all but finitely many ↵ < .

The Banach space c0 = (c0 , k · k1) is then obtained by endowing c0 withthe usual pointwise operations and the sup norm k · k1.


Banach spaces of density

Given a graph G on , let XG = (c0 , k · kG) be the Banach space withnorm

k(x↵

)

↵<

kG

= sup

⇢|x

i

|+ |xj

|3� �

G

(i, j)| i 6= j 2

�,

where �G

(i, j) = 1 if i G j and �G

(i, j) = 0 otherwise.

PropositionFor G,H graphs on , we have

G ⇠=

H () XG⇠=

li XH and G ⇡ H () XG ⇡li XH.

Corollary1 Assume that V = L and that = �+ with �!

= �. Then linearisometry on B

is complete for analytic equivalence relations.2 Assume that <

= . Then linear isometric bi-embeddability on B

is complete for analytic equivalence relations.


Projective equivalence relations on R

What about non-analytic equivalence relations on the real line? Using themethods of generalized DST one can e.g. obtain the following results.

Theorem (Andretta-M.)1 The elements of a Polish space can be classified up to an arbitrary ⌃1

2

equivalence relation by using as invariants graphs of size @1 up to ⇡.2 Assume that x# exists for all x 2 !!. Then the elements of a Polish

space can be classified up to an arbitrary ⌃13 equivalence relation by

using as invariants graphs of size @2 up to ⇡.3 Assume large cardinals (infinitely many Woodin’s cardinals with a

measurable above suffices). Then there is a monotone functionr : ! ! ! such that: The elements of a Polish space can be classifiedup to a ⌃1

n

equivalence relation by using as invariants graphs of size@r(n) up to ⇡.

(An upper bound for r(n) is given by 2k � 2, where k is the integer part of n+12 .)


End of part 1 of the tutorial

Thank you for your attention!


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