Borel equivalence relations

Greg Hjorth

March 30, 2006

This chapter is setting out to achieve an impossibility, namely to survey the rapidly explodingfield of Borel equivalence relations as found in descriptive set theory and the connections with areasentirely outside logic. The choice of content and emphasis is inevitably molded by this author’sown prejudices and research history, and indeed there are other researchers who could provide anentirely superior insight into many parts of the subject, and undoubtedly whatever I write will stirbitter disagreement from some who have a rather different vision of the subject. For instance I havesomewhat arbitrarily chosen to say nothing about the study of equivalence relations arising in Borelideals, as found in papers such as [57], [45], [14], or the parallel theory of Borel linear orderings asfound in say [42], [25], or the still unpublished work of Hugh Woodin’s and of Richard Ketchersid’son the cardinality of certain Borel equivalence relations under strong determinacy assumptions, thework on general Σ∼

11 equivalence relations as found in [3], or the topological Vaught conjecture as

discussed in [54], [39], [4], [31]. Moreoever the discussion of Borel equivalence relations is organizedaround the Borel reducibility order, ≤B, rather than notions such as orbit equivalence, as found insay [20], [18], [21], [53], [28], or notions of isomorphism, as discussed in [9]. Finally I should admitto being much more conversant with the mathematics of the subject than the history, and since mymain concern is to communicate the most vibrant ideas with a cerain immediacy, it is for certainthat I have frequently failed to give proper credit to results which lie at the prehistory of this subject.

Thus I stand impeached with prejudice, ignorance, arbitrariness, and discourtesy.But as unfortunate as these failings may be, they are inevitable, and I say all of this simply so the

reader will understand that this is a project doomed to at least partial failure and nevertheless worthpursuing in the hope of partial success. A similar but rather different point of view can be found[44]. The reader might also look at [5] for a closer examination of some of the issues surroundingactions induced by Polish group actions, or at [31] for a discussion of the Vaught conjecture, or [48]for orbit equivalence. Another survey is given in [29], but in fact I am unable to even point to anysmall set of papers which would be fully adequate.

I present a sketch and not a comprehensive treatment. There are crumbs of proofs so that thereader can obtain a flavor of the arguments, but I have not even come close to something resemblinga text book on the theory of Borel equivalence relations.

1 Definitions

Before moving onto the theory of Borel equivalence relations it would be helpful to discuss Borelsets in general. A thorough and more complete account can be found in [46].

Definition A Polish space is a separable topological space which admits a compatible completemetric. The Borel subsets of a Polish space are those appearing in the σ-algebra generated by theopen sets – that is to say, if we begin with the open sets and continue applying the operations

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of countable union, countable intersection, and complementation, then the Borel sets are thoseappearing in the collection which thereby arises.

Inside the Borel sets we make further distinctions. Thus a set is Σ∼01 if it is open, and after that

we define inductively a set to be Π∼0α if its complement is Σ∼

0α, and to be Σ∼

0α if it is a countable union

of Borel sets each of which are Π∼0β for some β < α. It is easily shown that Π∼

0α ⊂ Σ∼

0α+1 and every

Borel set will be Σ∼0α for some α < ω1. At the bottom level of this hierarchy there are alternate

notations used by analysts – for instance a Π∼02 set is also called Gδ and a Σ∼

02 set is also called Fσ .

For much of the time we are unconcerned with the topological structure of a Polish space, focusinginstead on its Borel structure. Accordingly a set X equippped with a σ-algebra B is a standard Borelspace if there is some Polish topology on X which gives rise to B as the collection of Borel sets.

Examples 1 R and C are Polish. Any compact metric space is Polish.2 There is a natural way to think of the collection of all subsets of the natural numbers as a

Polish space. By associating with each set its characteristic function, or indicator function, we canidentify the power set of N, denoted by P(N), with

{0, 1}N =df 2N,

which is a compact space in the product topology. Similarly P(N×N) or P(S) any countable set S.3 Any Borel subset of a Polish space is a standard Borel space in the inherited Borel structure

(see [50], 13.4 [46]).4 The Borel probability measures on a standard Borel space themselves again form a standard

Borel space (this ultimately follows from the Riesz representation theorem; compare 17.23 [46]).5 Consider the collection of subsets of N × N × N which form the graph of function

• : N × N → N

providing a finite rank torsion free abelian group structure on N. This collection in the natural Borelstructure is a standard Borel space, since it is a Borel subset of the Polish space N × N × N.

6 Given a countable language we can form two possible Polish topologies on the space of L-structures on N. For simplicity assume the language is relational, though the more general case ofL having function symbols is only slightly more complicated.

We let Mod(L) be the set of all L-structures with the natural numbers as their underlying set.The first of the topologies is τqf . For Ψ(~x) a formula in L and ~a = (a1, ..., an) a sequence in N,

we let U(ψ(~x),~a) be the collection of all N ∈ Mod(L) with

N |= ψ(~a),

and then τqf is the collection the topology with basis consisting of all the sets of the form U(ψ(~x),~a),as ψ ranges over the quantifier free formulas. If L consists of relations R1, R2, ..., each Ri and n(i)-

ary relation, then it is easily seen that (Mod(L), τqf ) is isomorphic to∏i∈N

NNn(i)

in the producttopology. Since the class of Polish spaces is closed under product (see §2.1 [32]) we have that(Mod(L), τqf ) is Polish.

The more subtle topology is τfo, with basis consisting of all U(ψ(~x),~a) as ψ ranges over firstorder formulas. This is a again a Polish topology, though the proof of this, say as found at 2.42.[32], is less obvious.

While these are divergent choices in topology, there is really one reasonable choice of Borelstructure on this space. Since τfo ⊃ τqf and both are Polish topologies, they give rise to the sameBorel structure. (This follows from 18.10, 18.14 [46]).

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Definition Given a Polish space X , we let F(X) be the collection of all closed subsets of X ,equipped with the σ-algebra generated by sets of the form {F ∈ F(X) : U ∩ F 6= ∅} for U open.This is known as the Effros Borel structure on the closed subsets of X . It is not hard to show thatF(X) equipped with this Borel structure is indeed a standard Borel space – see for instance §2 [32]or § [46]

The reader should definitely consult [46] for a more reasonable introduction to the theory ofBorel equivalence relations. This is the smallest sketch.

Definition A functionf : X → Y

is said to be Borel if f−1[B] is Borel for any Borel set B ⊂ Y . An equivalence relation E on X issaid to be Borel if it is Borel as subset of X ×X . We then use [x]E to denote the equivalence classof x for any x ∈ X – that is to say, the set {y ∈ X : xEy}.

Definition Given E and F Borel equivalence relations on standard Borel X and Y , we write

E ≤B F,

E is Borel reducible to F , if there is a Borel f : X → Y such that for all x1, x2 ∈ X

x1Ex2 ⇔ f(x1)Ff(x2);

in other words, f pushes down to injective map

f̄ : X/E → Y/F

between the quotient spaces.After this we naturally set E <B F if there is a Borel reduction from E to F but not F to E.

We set E ∼B F if each is Borel reducible to the other.

Note that the order ≤B is transitive, since the composition of two Borel functions is again Borel.

Remark This definition is only for Borel f , but one can of course consider more general classesof functions if this seems too stingy – for instance, C-measurable functions, projective functions,L(R) functions. In virtually all cases this makes no difference – the failures of reducibility for Borelfunctions persist to these wider classes. Indeed, appropriately understood most of the theoremsabout Borel equivalence relations turn into theorems about cardinality in L(R). See §4 below.

It should also be admitted that there are other ways in which we can compare Borel equivalencerelations, for instance asking that there be isomorphisms of the underlying spaces that conjugate therelations, or in the presence of a measure we can ask for measure preserving isomorphisms betweenthe spaces which conjugate the equivalence relations almost everywhere; indeed this second notionis the subject of extensive study in areas such as operator algebras (for instance [53]), geometricgroup theory (for instance [20], [52]), and the rigidity theory in the sense of Zimmer ([59]). For thepurposes of descriptive set theory, I incline to the view that ≤B is the central notion. Indeed somekind of defense of the philosophical significance of ≤B is given in §4.

Examples 1 For X a Polish space, we let id(X) be the identity relation on the space X . Since anytwo uncountable standard Borel spaces are isomorphic (15.6 [46]), it follows that for any uncountableX we have

id(X) ≤B id(R).

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2 We let E0 be the equivalence relation of eventual agreement on infinite binary sequences. Thusfor ~x = (x0, x1, ...), ~y = (y0, y1, ...) ∈ 2N, we set

~xE0~y

if and only if there exists some N ∈ N with

∀n > N(xn = yn).

Here we haveid(2N) <B E0.

To see that there is a Borel reduction from id(2N) to E0 is routine. It follows simply becausethere is perfect set in Cantor space consisting of mutually generic reals.

The failure of reducibility in the other direction is more subtle. First one observes that E0 isgiven by the continuous action of the countable group

⊕N

Z/2Z

with (g0, g2, ...)·(x0, x1, ...) = (g0+x0mod2, g1+x1mod2, ...). Then given a supposed Borel reductionf of E0 to the identity relation on 2N we can find a comeager set C on which f is continuous. Thenwe can take ⋂

~g∈L

Z/2Z

~g · C,

which will still be comeager and now invariant. Taking any ~x in this set we obtain that f will beconstant on [x]E0 . Since this equivalence class is dense, f will constant on the entire set

⋂~g ~g · C.

Since this set is uncountable, it contains many equivalence classes with a contradiction. (For a moredetailed and general argument, see 3.2 [32]).

3 We let E1 be the equivalence relation of eventual agreement on infinite sequences of reals. Hereone has

E0 <B E1.

(See [47], or even [32]).3 Given a countable group Γ, we let 2Γ be the collection of all functions

f : Γ → {0, 1}

with the product topology. We let Γ act on 2Γ with

γ · f(δ) = f(γ−1δ),

the left shift action. We then let EG be the resulting equivalence relation.In the case that we start with G = F2, the free group on two generators, one has

E0 <B EF2 .

(See forinstance appendix A of [36] for a survey of stronger and more general results one can provein this direction.)

4 An example of historical importance is the Vitali equivalence relation, Ev. For r, s ∈ R set

rEvs

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if and only ifr − s ∈ Q.

The classical argument that this equivalence relation has no measurable selector can be modified toshow that id(R) <B Ev ; alternatively one may appeal to a variation of the Baire category argumentat example 2 just above.

On the other hand, at the level of Borel reducibility there is no distinction between E0 and Ev .(See for instance the argument after 1.5 [33] for a short proof that E0 ∼B Ev .)

5 LetX2 be the space of all h ∈ 2N×N, where at each n 6= m there exists a k with h(n, k) 6= h(m, k).In other words, if we set hn ∈ 2N to be given by hn(k) = h(n, k) then for n 6= m we have hn 6= hm.

Then define T2 on X2 by h1T2h2 if and only if the corresponding countable sets in 2N are equal

– that is to say,{h1

n : n ∈ N} = {h2n : n ∈ N}.

Thus if we let S∞ be the group all permutations of N, acting on X2 by

σ · h(n, k) = h(σ−1(n), k),

then T2 is the resulting equivalence relation.It is well known that for any EG as above, arising from a countable group G acting on 2G, one

hasEG <B T2.

(See for instance 2.64 [32].)

Definition An equivalence relation E on standard Borel X is said to be smooth or tame if E ≤B

id(R).

Smoothness amounts to asserting the existence of a countable algebra {Bn : n ∈ N} of E-invariantBorel sets which separates points – which is to say

xEy ⇔ ∀n(x ∈ Bn ⇔ y ∈ Bn).

This in turn is equivalent to saying that X/E = {[x]E : x ∈ X} in the quotient Borel structureconsisting of all E-invariant Borel sets is a subset of a standard Borel space. (See [24], [41].)

Definition A Borel equivalence relation E on a standard Borel space X is said to be countable ifevery equivalence class is countable. It is essentially countable if there is some other countable Borelequivalence relation F with E ≤B F .

Example Let F2 be the free group on two generators and let E∞ arise from action of F2 on 2F2 .Then this is countable, since the responsible group is countable.

It is known from [41] that this equivalence relation is universal among the countable equivalencerelations, in the sense that for every countable Borel equivalence relation E one has

E ≤B E∞.

Theorem 1.1 (Feldman-Moore, [15]) If E is a countable Borel equivalence relation on X, thenthere is a Borel group of automorphisms whose orbits equal the E-equivalence classes.

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Proof Following the Lusin-Novikov uniformization theorem (18.10, 18.15 [46]) we can find a se-quence of Borel functions (fn)n∈N such that [x]E always equals {fn(x) : n ∈ N}. We can find Borelsets Bn consisting of exactly the points on which fn(x) 6= x. We can then find a partition of Bninto Borel sets {Cn,i : i ∈ N} with fn[Cn,i] disjoint from Cn,i. We then let gn,i be the Borel functionwhich is the identity off of fn[Cn,i] ∩ Cn,i, equal to fn on Cn,i, and to f−1

n on f [Cn,i].Letting G be the countable group of automorphisms generated by {gn,i : i, n ∈ N} we obtain

E = EG, the orbit equivalence relation induced by G. 2

Lemma 1.2 A countable Borel equivalence relation is smooth if and only if it has a Borel selector– that it so say, a Borel set which meets every equivalence class in exactly one point.

Proof This follows rapidly from the Lusin-Novikov uniformization theorem. See for instance [41],or [7] for far more general results. 2

This lemma fails for general equivalence relations. For instance if we take C ⊂ NN × NN withthe projection {x : ∃y((x, y) ∈ C)} Borel, and set (x, y)E(x′, y′) on C exactly when y = y′, then aBorel selector would result in the projection being Borel in light of that exact same Lusin-Novikovuniformization theorem.

Definition An equivalence relation E is hyperfinite if there is a sequence of Borel equivalencerelations, (Fn)n∈N, with each Fn having all its equivalence classes finite, each Fn ⊂ Fn+1, andE =

⋃n∈N

Fn.

Lemma 1.3 A countable Borel equivalence relation E on X is hyperfinite if and only if E ≤B E0.

Proof The if direction follows rather routinely from the observation that E0 is hyperfinite, sosuppose instead that E =

⋃n∈N

Fn, each Fn Borel with finite classes, Fn ⊂ Fn+1.It is well known that the collection of all closed subsets of X is a standard Borel space in the

obvious Borel structure (see 12.6 [46]), and so we can find at each n a Borel function

fn : X → 2N

with xFny if and only if fn(x) = fn(x), and then let

f(x) = (f0(x), f1(x), f2(x), ...).

This function witnesses E ≤B E0. 2

Definition An equivalence relation E on a standard Borel space X is treeable if its classes formthe components of a Borel treeing on X – that is to say, there is T ⊂ X ×X which is Borel withrespect to the product Borel structure, is symmetric, acyclic, loopless, and for which we have xEy ifand only if there is a finite chain x0 = x, x1, x2, ..., xn = y, each (xi, xi+1) ∈ T . (So here one has inmind the notion of tree prevelant in certain branches of combinatorics or computer science, ratherthan descriptive set theory – an acyclic graph, with no distinguished root in the various connectedcomponents).

Example Let F2 be the free group on two generators. Let F (2N) be the free part of the shift actionof F2 on 2F2 – that is to say, the set of h : F2 → {0, 1} such that for all σ ∈ F2 there exists τ withh(σ−1τ) 6= h(τ), equipped with the action (σ · h)(γ) = h(σ−1γ).

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In most cases treeable equivalence relations have been studied simply in the case that E is alreadycountable.

Here one has an analogue of 1.3.

Lemma 1.4 (See [41] or [26]). A countable equivalence relation is treeable if and only if it is Borelreducible to ET ∞.

Although this survey is primarily concerned with Borel equivalence relations, one must naturallyconnect this study with the analysis of those lying just outside this class.

Definition A subset of a standard Borel space is analytic or Σ∼11 if it is the image under a Borel

function of a Borel set.

Definition A Polish group is a topological group which is Polish as a space. Given a Polish groupG, a Polish G-space is a Polish space equipped with a continuous action of G; a standard BorelG-space is a standard Borel G-space equipped with a Borel action by G.

In either case, if X is the space we use EXG , or even just EG when there is no doubt to X , todenote the resulting orbit equivalence relation, and [x]G to denote the orbit of a point x.

The equivalence relations arising from the Borel action of a Polish group are always Σ∼11, but

frequently non-Borel. As noted in [5], for any such EXG we can write X as an ℵ1 union of invariantBorel sets,

X =⋃α<ω1

Xα,

and EXG restricted to each Xα Borel.

Examples 1 Let S∞ be the group of all infinite permutations of the natural numbers equippedwith the topology of pointwise convergence – that it so, a basic open set has the form {σ ∈ S∞ :σ(n0) = k0, ..., σ(n`) = k`}. For L a countable language we let S∞ act on Mod(L) in the naturalway –

(σ ·M) |= R(n0, ..., n`)

if and only ifM |= R(σ−1(n0), ..., σ

−1(n`).

The resulting equivalence relation is nothing than isomorphism on this class of structure, and forany reasonably rich one has ES∞

Σ∼11 but non-Borel.

2 Given L as above, and ϕ ∈ Lω1,ω, a countably infinitary formula, we let Mod(ϕ) be thestructures in Mod(L) which satisfy ϕ. Since this is a Borel subset of a Polish space, it is a standardBorel space in its own right, and in the induced action a standard Borel S∞-space.

Here one should see [5] for further details. The first paper to consider these examples seriouslyfrom the point of the view of the ≤B ordering is [17].

3 Given a Polish group G we can let it act on itself by conjugation –

σ · ρ = σ−1ρσ.

In many cases this corresponds naturally to some kind of classification problem.For instance, if M∞ is the group of all measure preserving transformations of the unit interval

considered up to agreement almost everywhere, then this group is indeed Polish in the naturaltopology and the equivalence relation arising from its conjugation action is the equivalence relation

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of isomorphism of measure preserving transformations. Or if we let U∞ be the unitary group oninfinite dimensional Hilbert space, we are find ourselves confronted with the classification up tounitary equivalence of unitary operators. Or one may consider Hom([0, 1]), the homeomorphismgroup of the unit interval, for the classification problem for homeomorphisms of the unit interval upto topological similarity.

All these examples are discussed at length in [32].

2 A survey of structure theorems

2.1 Structure

At the base level of the ≤B there is a sharp structure. The first of these results is due to Jack Silver.Although it was proved sometime prior to the general study of Borel equivalence relations, it maybe paraphrased in modern terminology as follows.

Theorem 2.1 (Silver, [56]). If E is a Borel equivalence relation, then exactly one of:(I) E ≤ id(N);(II) id(R) ≤B E.

In some ways this belongs to the prehistory of the subject. Momentum only began in ernestfollowing the seminal dichotomy theorem of Leo Harrington, Alexander Kechris, and Alain Louveau.This is sometimes known as the Glimm-Effros dichotomy, since the forerunners of this theorem wereproved by Glimm and then later Effros, who had the result in the case that E is an Fσ equivalencerelation induced by the continuous action of a Polish group.

Theorem 2.2 (Harrington-Kechris-Louveau, [24]). If E is a Borel equivalence relation, then exactlyone of:

(I) E ≤ id(R);(II) E0 ≤B E.

Proof (Sketch only) We describe some of the combinatorics under the drastic assumption that E isa countable Borel equivalence relation. This is misleading as to the difficult of the proof, since thenthe vast majority of the mathematical issues simply evaporate. It should also be pointed out thatthe theorem in this case was already known to Glimm and Effros, [11], [10].

First we can appeal to 1.1 to obtain a countable group G acting by Borel transformations on thePolish space X with EG = E. Applying a change of topology argument, as found in say §13 [46],we may assume G acts by homeomorphisms.

Now there are two possibilities.First of all we may have that for any x 6= y ∈ X we have the closures of their orbits, [x]G, [y]G,

distinct. In that case one defines a map

X → F(X),

x 7→ [x]G,

from X to the space of all closed subsets of X with the standard Borel structure. It is easily seenthat this map is Borel, and so we obtain a reduction of E to id(F(X)). Since any two uncountableBorel spaces are Borel isomorphic, this is as good as a reduction to id(R).

The other case is that there are distinct orbits with the same closure. We will now work entirelyinside some X0 ⊂ X consisting of all y with [y]G = C for some closed C. X0 is a Gδ subset of X ,

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and hence Polish in its own right. (See for instance §2.1 [32].) We are assuming that X0 containsmany orbits. We let {gn : n ∈ N} enumerate the group G.

Now note that EG is Fσ , and our assumptions on X0 imply that both it and its complement aredense in X0 ×X0.

We then construct non-empty open neighborhoods (Us)s∈2<N in X0 and (gs)s∈2<N

(i) Us ⊂ Ut for s a strict extension of t;(ii) with respect to a compatible complete metric on X0, the diameter of each Us is less than2−n for s ∈ 2n;(iii) gsa0 = gs; ga〈0,0,...0〉 = e;(iv) gs ·U〈0,0,..0〉 = Us, where 〈0, 0, ..., 0〉 is the constantly zero sequence of the same length ass;(v) if s, t ∈ 2n+1 and s(n) 6= t(n), then for all i ≤ n we have gn · Us ∩ Ut = ∅.

Assuming we can effect this elaborate arrangement, the conclusion is brief. (iv) will guaranteethat for each s, t ∈ 2n, w ∈ 2<N we have

gtg−1s · Usaw = Utaw.

Thus for any x ∈ 2N there will be a unique point

θ(x) ∈⋂Ux|n ,

and if x(n) = y(n) all n ≥ N , thengx|ng

−1y|n

(y) = x,

whilst if x(n) 6= y(n) for infinitely many n then (v) guarentees EG-inequivalence.So suppose we have done this up to some level. We have (Us)s∈2n , (gs)s∈2n . By the density of

the complement of EG we can find some x, y ∈ U〈0,0,...0〉 which are inequivalent. We can then letxs = gs · x, y = gs · y. We form small enough open sets W,V around x, y so that if Ws = gs ·W ,Vs = gs · V , then gi · Vs ∩Wt = ∅, g−1

i · Vs ∩Wt = ∅.We then find a point x′ ∈ [x]G ∩ V〈0,0,...0〉 by the density of the orbits. We let g〈0,0,...0〉a1 be a

group element moving x to x′. We let gsa1 be gsg〈0,0,...0〉a1. We then build a sufficiently small setU∗ ⊂W so that if we let Ut = gt · U∗ for each t ∈ 2n+1 then we have ensured (i), (ii), and (v). 2

Admittedly the short proof in the special case given above is misleading. The argument forgeneral Borel E is far harder, and uses the Gandy-Harrington topology. There is no known proof of2.2 which does not use ideas from logic.

Therefore at the base of the picture one obtains the Borel equivalence relations with finitely manyclasses, followed by any Borel equivalence relation with exactly ℵ0 many classes, then the identityrelation on the reals, and then E0, or, equivalently when considered up to Borel reducibility, theVitali equivalence relation.

It should be mentioned that Harrington-Kechris-Louveau implies Silver’s theorem. If f : 2N → Xwitnesses E0 ≤B E then we can find a comeager set on which this function is continuous, and theninside this comeager we can find a perfect set of E0-inequivalent reals. On the other hand if E ≤ id(R)then the equivalence classes of E can be identified with a Σ∼

11 subset of a Polish space, whereapon

Silver’s theorem reduces to the perfect set theorem for Σ∼11 sets.

Immediately after this one obtains a splintering. It is known from [47] that there are no otherBorel equivalence relations E such that for any other Borel F one has either E ≤B F or F ≤B E.Instead we have:

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Theorem 2.3 (Kechris-Louveau; [47]) Let E be a Borel equivalence relation with E ≤B E1.(I) E ≤ E0;(II) E1 ≤B E.

Definition Let EN0 be the equivalence relation on (2N)N given by

~xEN

0 ~y

if and only if∀n(xnE0yn);

that is to say, we have E0-equivalence at every coordinate.

Theorem 2.4 (Hjorth-Kechris; [37]) Let E be a Borel equivalence relation with E ≤B EN0 .

(I) E ≤ E0;(II) EN

0 ≤B E.

There are no other known immediate successors to E0, but Ilijas Farah in [12] has proposeda continuum of plausible examples which are inspired by ideas in Banach space theory. What hedoes show is that these examples are incomparable, and that any equivalence relation strictly belowany of them is essentially countable. The gnawing technical difficulty at the end of his argumentis our inability to determine which countable equivalence relations are hyperfinite. Very recently,in a spectacular piece of work and unpublished piece of work, David Boykin, Su Gao, and SteveJackson showed all equivalence relations arising from the Borel actions of countable abelian groupsare hyperfinite, and on this basis we might hope for further clarification in the coming years.

Although we are at a loss to provide further global dichotomy theorems, there is somethingmore that can be said if we restrict ourselves to the region of equivalence relations which are belowisomorphism of countable structures.

Definition For L a countable language, we let ∼=mod(L) be isomorphism on the standard Borelspace of L-structures with underlying set N. For ψ ∈ Lω1,ω we let ∼=mod(ψ) be the restriction of thisequivalence relation to models of ψ.

Lemma 2.5 (Becker-Kechris; [5]) Let E be Borel. Then the following are equivalent:(I) E ≤B∼=mod(L) some countable language L(II) E ≤B ES∞

, where ES∞arises from the continuous action of a S∞ on a Polish space.

Definition If either of these equivalent conditions hold, then we say that E admits classificationby countable structures.

In the still unpublished [27] a dichotomy theorem was announced for equivalence relations ad-mitting classification by countable structures. It should be emphasised that this proof is long andhas not been refereed, and accordingly the result has a provisional status.

Theorem 2.6 (Hjorth; [27])Let E be a Borel equivalence relation admitting classification by countable structures. Then

exactly one of(I) E ≤ E∞, that is to say, E is essentially countable;(II) EN0 ≤B E.

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2.2 Anti-structure

The last section represented the good news. Now for the evil.Refining an earlier result of Hugh Woodin’s, Alain Louveau and Boban Velickovic embedded

P(N)/Fin into the Borel equivalence relations considered up to Borel reducibility.

Theorem 2.7 (Louveau-Velickovic) There is an assignment

A 7→ EA

of Π∼03 equivalence relations to subset of N such that

EA1 ≤B EA2

if and only if A2 \A1 is finite.

Therefore there is no global structure theorem for the ≤B ordering.Given the dichotomy theorems of [47] and [37] we might at least hope for some kind of basis of

immediate successors to E0, but even this seems unlikely. In [13] Farah obtains an infinite descendingchain in ≤B which is unlikely to be above an immediate successor to E0. I say unlikely, though it ishard to make a fast guess at this point. Literally Farah obtains the existence of a sequence (Fn)n∈N

with each Fn+1 <B Fn, none of them essentially countable, and with the further property that anyequivalence relation E ≤B Fn at every n must be essentially countable. Since the Fn’s arise in a verysimple form, in particular the continuous action of an abelian Polish group, there is some groundsfor thinking the only countable equivalence relations reducible to one of these will be reducible toE0.

Farah’s work also disproves the existence of a dichotomy theorem for being classifiable by count-able structures in many dramatic ways. For instance he obtains an uncountable sequence of Borelequivalence relations, (Ex)x∈2N , which are not classifiable by countable structures, and such that forany x 6= y and Borel E with

E ≤B Ex, Ey

we have E classifiable by countable structures.

2.3 Beyond good and evil

There are slender few candidates for otright Borel dichotomy theorems in the style of 2.2, and 2.7and the later results of Farah would seem to rush hopes for a global analysis of the ≤B ordering.On the other hand there still seems that there are results which would help us understand whenequivalence relations fall on some side of a key divide.

A distinction of philosophical interest is when we can classify by countable structures, whichhas clear parallels in broader mathematical practise. Here one has in mind certain branches oftopology, where one seeks complete algebraic invariants, and in effect one is trying to classify acertain equivalence relation by a countable algebraic structure considered up to isomorphism. Theintroduction of [32] surveys several examples along these lines.

Definition Let G be a Polish group and X a Polish G-space. For U ⊂ G an open neighborhood ofthe identity, V ⊂ X an open neighborhood of a point x, the local U -V -orbit of x, written O(x, U, V ),is the set of all y such that there exists a finite sequence of points x0 = x, x1, x2, ..., xn = y in U witheach xi+1 ∈ V · xi – that is to say, we can move from xi to xi+1 using some group element in V .

Then we say that the action of G on X is turbulent if the following three things hold:

11

(I) every orbit [x]G is dense in X ;(II) every orbit [x]G is meager in X ;(III) given x, y ∈ X , U an open neighborhood of y, V an open neighborhood of the identityin G, there is some x′ ∈ [x]G ∩ U with y in the closure of the local orbit (x′, U, V ).

Theorem 2.8 (Hjorth; [32]) If EG arises from a turbulent action of G on the Polish G-space Xand L is a countable language, then EG is not Borel reducible to ∼=mod(L).

Proof (Sketch) We present the argument in a simple case with a number of simplifying assumptions.Consider the example of an S∞ action given before, where are space is X2, consisting of all h ∈ 2N×N,where at each n 6= m there exists a k with h(n, k) 6= h(m, k). Then define T2 on X2 by h1T2h

2 ifand only if the corresponding countable sets in 2N are equal. This is the orbit equivalence relationarising from the S∞ action

(σ · h)(m,n) = h(σ−1(m), n).

We assume for a contradiction we let θ : X → X2 witness EG ≤B T2.Every Borel function is continuous on a comeager set, so let us actually make the simplifying

assumption that θ is continuous everywhere. A relatively routine Baire category argument will showfor any n there is a comeager collection of x ∈ X which have some basic open neighborhood Vx,n ofthe identity in S∞ for which there is a comeager collection of σ ∈ Vx,n having

(θ(x))(n, ·) = (θ(σ · x))(n, ·).

Let us therefore make the additional assumption that

x 7→ Vx,n

is defined everywhere, and “continuous” in the sense that given any basic open V and n the collectionof x with Vx,n = V is open.

Let x, y ∈ X . Consider some n. It suffices to show that there is a representative of the orbit ofx,

x′ ∈ [x]G,

which has θ(x′)(n, ·) = θ(y)(n, ·). By the above simplifying assumptions we can find a basic openneighborhood of U of y and V a basic open neighborhood of the identity in the group which has

(θ(z))(n, ·) = (θ(σ · z))(n, ·)

all z ∈ U, σ ∈ V .Now if we take x′ ∈ [x]G with

y ∈ O(x′, U, V )

then we can find a sequence of points (zi)i∈N, each zi ∈ O(x′, U, V ) with

zi → y.

By the assumptions on U and V we have θ(zi)(n, ·) = θ(x′)(n, ·) at all n, and hence θ(x′)(n, ·) =θ(y)(n, ·), as required. 2

As a practical matter [32] suggests turbulence to be the key phenomena in determining whetheran equivalence relation is classifiable by countable structures. This has been supported in variousexamples and practical investigations, such as [22], [49], as well as some partial results given in [32].The correct theorem was not established until a few years later.

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Theorem 2.9 (Hjorth; [30]) Let G be a Polish group and X a Polish G-space. Suppose EXG is

Borel. Then exactly one of the following hold:(I) EXG admits classification by countable structures; or(II) there is a turbulent Polish G-space Y with EYG ≤ EXG .

3 Countable Borel equivalence relations

The subject of countable Borel equivalence relations is notable for its interactions with such diversefields as geometric group theory, the ergodic theory of non-amenable groups, operator algebras, andsuperrigidity in the sense of Zimmer. There is a sense in which this interaction has been largelyone way, since one finds logicians borrowing from these other fields rather than these areas beingserviced by logic.

In the course of this period it is perhaps unsurprising that logicians have made a few notablecontributions to the benefit of the other fields. However in almost every case the applications donot consist in applying deep ideas from logic, as found in say the work of Hrushovski, [40], butrather the natural result of mathematicians from one field rethinking the problems from anotherand approaching with a different point of view.

3.1 The global structure

In some form or another almost every argument which distinguishes countable Borel equivalencerelations in the ≤B ordering uses measure theory. At the very base level we can use Baire categoryarguments to show id(R) <B E0 but beyond this there is a fundamental obstruction.

Theorem 3.1 (Sullivan, Weiss, Wright; [58]) Let E be a countable Borel equivalence relation on aPolish space X. Then there is a comeager set on which E is hyperfinite.

Proof (Sketch; this draws from Segal’s thesis,[55]) E is induced by a countable group G acting byBorel automorphisms. Let us make the simplifying assumption that G acts by homeomorphisms –this argument is harmless, but we will skip over this point in the interest of keeping the argumentshort.

Let B be a countable basis for X . Let (gn)n∈N enumerate the countable group G.At each n let Yn be the collection of sequences

~A = (A0, A1, ..., An),

where each Ai is a finite subset of B. Given such a sequence we let R ~A be the graph on X given by

xR ~Ax′

if there is some i ≤ n, V, V ′ ∈ Ai, with x ∈ V, x′ ∈ V ′ and either

gi · x = x′

orgi · x

′ = x.

We let E ~A be the equivalence relation arising from the connected components of R ~A.

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We then let Y ∗n be the subset of Yn for which the induced equivalence relation E ~A has finite

classes. Given ~A = (A0, ..., An) ∈ Y ∗n , ~B = (B0, ..., Bm) ∈ Y ∗

n where m ≥ n, we say that ~B extends~A if every at every i ≤ n

Ai ⊂ Bi.

Claim: Given any x, y = gi · x, n ≥ i and

~A ∈ Y ∗n

we can find an extension ~B ∈ Y ∗n with

xR~By.

Proof of Claim: The point is that we can add to Ai a small enough open set V around x thatwill specify x with sufficient exactness to ensure that for any x′ ∈ V , zE ~Ax

′ we have either z = x′

or z /∈ V . Then if we were to take the simple extension ~B which has Bi = Ai ∪ {V } but Bj = Aj atj 6= i, then E ~A has index two in E ~B . (Claim�)

We then let Y be the space of all infinite sequences

( ~An)n∈N ∈∏n∈N

An

where each ~An+1 extends An. Any such ( ~An)n∈N ∈ Y gives rise to an increasing sequence of Borel

equivalence relations with finite classes, taking F( ~An)n∈N

n to be E ~An .∏n∈N

Y ∗n comes with a natural

product topology, under which it is Polish. Y is a closed subset of this product space, and hencePolish in its own right. The above claim shows that for all x ∈ X there is a comeager collection of( ~An)n∈N ∈ Y with

[x]E =⋃n∈N

[x]F

( ~An)n∈Nn

.

Thus by Kuratowski-Ulam (see §??? [46]), there is a comeager collection of ( ~An)n∈N ∈ Y for which

there is a comeager collection of x with [x]E =⋃n∈N [x]

F( ~An)n∈Nn

. Taking any such ( ~An)n∈N ∈ Y and

corresponding comeager set, we are done. 2

On the other hand the subject of countable Borel equivalence relations considered up to Borelreducibility might collapse into a kind of death by heat dispersal if all these examples were hyperfinite.It turns that in the presence of an invariant probability measure, non-amenable groups give rise toequivalence relations which are not hyperfinite.

Definition A countable group Γ is amenable if for any finite F ⊂ Γ and ε > 0 there is a finite,non-empty A ⊂ Γ such that for all σ ∈ A

|A∆σA|

|A|< ε.

As a word on the notation, we use |B| to denote the cardinality of the set B and B∆C todenote the symmetric difference of the sets B and C. Thus amenability amounts to the existence ofsomething like “almost invariant” subsets of the group – given any finite collection of group elements,any tolerance (ε > 0), we can find a finite set which when translated by any of the group elementsdiffers from its original position on a relatively small number of elements.

14

Examples 1 Z is amenable. Given F = {k1, ..., k`} in the group and ε > 0, we let k = max(|k1|, ..., |k`|),the max of the absolute values, and then let

N >2(k + 1)

ε.

The set A = {−N,−N + 1,−N + 2, ..., 0, 1, ..., N − 1, N} is as required.2 On the other hand, F2 = 〈a, b〉, the free group on two generators, is most famously non-

amenable. Just take ε = 1/5, F = {a, b, a−1, b−1}. If we think about dividing the non-identityelements of F2 into four regions, Ca, Cb, Ca−1 , Cb−1 , where a (reduced) word is in the region Cu if itbegins with u.

Consider some putative set A that is trying to witness amenability for 1/5 and {a, b, a−1, b−1}.For u = a, b, a−1, or b−1, n ∈ N, if at least n of A’s elements are not in to Ca−1 , Cb−1 , Ca, or Cb,respectively, then at least n of u · A’s elements are in Ca, Cb, Ca−1 , or Cb−1 respectively. Thereforewe can clearly find two distinct u1, u2 with at least three fifths of ui · A’s elements in Cui

. One ofthese sets must differ from A by at least 1/5|A|.

Note that there is nothing special here about the use of almost invariant sets in F2. We wouldobtain a similar contradiction if we considered almost invariant functions in `1(F2). Given f ∈ `1(F2)and u ∈ {a, b, a−1, b−1} we could look at the norm of the corresponding fu defined by fu(σ) = f(σ)if σinCu, fu(σ) = 0 otherwise.

Definition A standard Borel probability space is a standard Borel space X equipped with an atom-less σ-additive probability measure µ on its Borel sets.

Theorem 3.2 Let Γ be a countable non-amenable group. Suppose Γ acts by measure preservingtransformations on a standard Borel probability space (X,µ). Then EΓ is not hyperfinite.

Proof We sketch the argument just in the case that G = F2.For a contradiction suppose EF2 =

⋃n∈N

Fn, where each Fn is finite, Borel, and has Fn ⊂ Fn+1.At each x ∈ X we define

fn,x : F2 → R

with

fn,x(σ) =1

|[x]Fn|

if σ−1 · xFnx,fn,x(σ) = 0

otherwise. Let

fn(σ) =

∫X

fn,x(σ)dµ.

For any x we have ||fn,x||`1 = 1, and so certainly ||fn||`1 = 1, and in fact for any measurable setA ⊂ X ∫

A

fn,x(σ)dµ ≤ µ(A).

Claim: For any γ ∈ F2, as n→ ∞||fn − γ · fn||`1 → 0.

Proof of Claim: Note that if xFnγ · x then

γ · fn,x = fn,γ·x,

15

since for any σ we have

γ · fn,x(σ) = fn,x(γ−1σ) =

1

|[x]Fn|(=

1

|[γ · x]Fn|)

if and only if σ−1γ · xFnx, which, by the assumption xFnγ · x, amounts to saying if and only if

fn,γ·x(σ) =1

|[γ · x]Fn|.

Thus if we let An,γ = {γ ∈ Γ : xFnγ · x}, then

∫An,γ

γ · fn,x(σ)dµ =

∫γ·An,γ

fn,x(σ)dµ,

and hence

||fn − γ · fn||`1 ≤ 2

∫X\An,γ

fn,x ≤ 2µ(An,γ).

Since µ(An,γ) → 0 as n→ ∞, we are done. (Claim�)Now we have established the existence of almost invariant functions in `1(F2), and this can be

refuted by the same kind of argument as used to show the non-amenability of F2. 2

For the longest while there was only a small finite number of countable Borel equivalence relationsknown to be distinct in the ≤B ordering. It was a notorious open problem to establish even theexistence of ≤B-incomparable examples. This was finally settled by:

Theorem 3.3 (Adams-Kechris, [2]) There exists an assignment of countable Borel equivalence re-lations to Borel subsets of R,

B 7→ EB ,

such that EB ≤B EC if and only if B ⊂ C.

Formally their result relied on the superrigidity theory of Zimmer for lattices in higher rank liegroups, [59], and thus in turn had connections with earlier work of Margulis and Mostow. I will saynothing about Zimmer’s work as such, but instead try to describe some of the engine which drivesthe theory.

Definition For a compact metric space K we let M(K) denote the probability measures on K. Bythe Riesz representation theorem this is a closed subset of a the dual of C(K), and thus is a Polishspace in its own right. Note that the homeomorphism group of K acts on M(K) in a natural way:

(ψ · µ)(f) = µ(ψ−1 · f),

where ψ−1 · f is defined by (ψ−1 · f)(x) = f(ψ(x)).

Definition Given a group Γ acting on a space X and another group H we say that

α : G×X → H

is a cocycle if for all γ1, γ2 ∈ G, x ∈ X

α(γ2, γ1 · x)α(γ1, x) = α(γ1γ2, x).

16

Here is a typical situation in which a cocycle arises. Given Γ a group of Borel automorphismsof standard Borel space X , H a countable group acting freely and in a Borel manner on a standardBorel space Y , if

θ : X → Y

witnesses EXG ≤B EYH , then we obtain a Borel cocycle by letting α(γ, x) be the unique h ∈ H with

h · θ(x) = θ(γ · x).

Lemma 3.4 (Furstenberg, [19]) If ∆ is a countable amenable group acting in a Borel manner on astandard Borel probability space (X,µ) and

α : ∆ ×X → Hom(K)

is a cocyle into the homeomorphism group of a compact metric space K, then we can find a measurableassignment of measures

x 7→ νx,

X →M(K),

which is almost everywhere equivariant; that it is to say

∀µx∀γ(α(γ, x) · νx = νγ·x.

The applications of this lemma and its forerunners are much too involved to be discussed here.Some understanding of what is going on can be given by the following simple lemma. In fact, thehypotheses of the lemma can never be realized, and indeed the real theorem is that equivalencerelations of the form EXΓ×Z

are never treeable, but I simply want to give a short illustration of somekey ideas.

Lemma 3.5 Suppose Γ is a non-amenable countable group. Let X = {0, 1}Γ×Z and let µ be theproduct measure on this space. Let Γ × Z act in the natural way on this space:

σ · f(τ) = f(σ−1τ)

for any f ∈ X, σ, τ ∈ Γ × Z.Suppose F2 acts freely and by Borel automorphisms on standard Borel probability space Y . Sup-

poseθ : X → Y

witnesss EXΓ×Z≤B EY

F2.

Then there is homomorphism ρ : Γ → F2 and an alternative reduction

θ̂ : X → Y,

which is equivalent in the sense thatθ̂(x)EY

F2θ(x)

all x ∈ X and whose resulting cocycle accords with ρ almost everywhere, in the sense that

∀µx∀γ(ρ(γ) · θ̂(x) = θ̂(γ · x).

17

Proof (Sketch) Letα : (Γ × Z) ×X → F2

be the induced cocycle.Let ∂(F2) denote the infinite reduced words from {a, b, a−1, b−1}. This is a compact metric

space on which F2 acts by homeomorphisms. (See Appendix C [36].) Following 3.4 we can find ameasurable assignment

X →M(∂(F2)),

x 7→ µx,

withα(e, `) · µx = µ(e,`)·x

for all ` ∈ Z and a.e. x ∈ X . (We will use e for the identity in Γ and 0 for the identity in Z.)

Claim: We can choose this assignment so that the measures µx concentrate almost everywhere onmore than two points.

Proof of Claim: Suppose instead that every such Z-equivariant assignment of measures concen-trates on at most two points.

The key observation here is that ifx 7→ µx

is such an assignment of measures then for any γ ∈ Γ so is

x 7→ α(γ, 0)−1 · µ(γ,0)·x.

Thus to prevent a situation in which we could simply pile on more and more of these measures,passing from say x 7→ µx to

x 7→µx + α(γ, 0)−1 · µ(γ,0)·x

2

we must be able to obtain an assignment which is actually Γ-equivariant.Using certain strong ergodicity properties of the shift action of Γ on X (see Appendix A [36])

and the hyperfiniteness of the action of F2 on ∆(F2) (see Appendix C [36]) we obtain that there isa single measure ν0 such that for almost all x we have some σx ∈ F2 with

σx · µx = ν0.

Thus replacing θ with the reductionθ̂ : x 7→ σx · θ(x)

we obtain a reduction of EXΓ×Zto EXH where H is the subgroup of F2 corresponding which stabilizes

the measure ν0. This subgroup will be amenable (see Appendix C [36]), which in turn provides acontradiction to the non-amenability of Γ × Z (see Appendix A [36] again). (Claim�)

So now we obtain an assignment of measures that concentrates on at least three points almosteverywhere. Here an observation of Russ Lyons (the rough idea is that a measure on ∂(F2) concen-trating on more than three points can be assigned a center in an F2-equivariant manner – again, seeAppendix C [36]) gives us a measurable map

η : x 7→ F2

18

such thatη((e, `) · x) = α((e, `), x) · η(x)

almost everywhere. Replacing θ : X → Y by

θ̂ : X → Y,

x 7→ η(x)−1 · θ(x)

we obtain a reduction with an induced cocycle

α̂ : (Γ × Z) ×X → F2

with α̂((e, `), x) = e almost everwhere.Then, as at the proof of 2.2 [36], the ergodicity of the action of Z on X gives that for any γ ∈ Γ,

x 7→ α̂((γ, 0), x)

is a.e. invariant. From this it is easily seen that we obtain the required homomorphism into F2. 2

Arguments of this form can be found very clearly in papers by Scot Adams such as [1], thoughin truth the ideas trace back to Margulis and Mostow by the way of Zimmer’s [59].

[36] gives a self contained proof of the existence of many ≤B-incomparable countable Borelequivalence relations using arguments along these lines, but in truth something similar is implicitin the superrigidity results of [59] to which Adams and Kechris appeal in the course of proving 3.3.In that case one is dealing not with the compact space ∂(F2) but certain compact quotients of analgebraic group (see for instance page 88 [59]) or the measures on projective space over a locallycompact field (see 3.2.1 [59]). The appearance of product group actions is more subtle, but present;superrigidity typically on works for groups of matrices of rank greater than two, when we can hopeto find a subgroup which indeed has the form Γ × Z for Γ non-amenable.

In passing it should also be mentioned that there are applications of Furstenberg lemma to thetheory of the homeomorpism group of the circle. In [23] the combinatorial properties of M(S1) playa role in understanding what kinds of homomorphisms are possible into Homeo(S1).

3.2 Treeable equivalence relations

The situation with the countable Borel equivalence relations can in some sense be viewed as resolved.It is a giant mess, with many incomparable examples, but we know it to be the ghastly mess that itis.

The situation with treeable countable Borel equivalence relations is far different.It is well known that not all treeable equivalence relations are hyperfinite. If one take the free

part of the shift action of F2 on 2F2 then the resulting equivalence relation, ET ∞, is treeable – wedefine a Borel treeing by xT x′ if there is a generator of F2 which moves x to x′. The equivalencerelation is not hyperfinite, as shown for instance in § [36], since the product measure concentrateson the free part and is F2-invariant.

It is also known that there is a maximal countable treeable equivalence relation. [41] shows thatfor any treeable countable Borel equivalence relation E one has E ≤B ET ∞.

After that precious little is known. [28] gives the existence of a treeable equivalence relation Ewith E0 <B E <B ET ∞, but at the time of writing it is still open whether there are exactly twotreeable countable Borel equivalence relations up to Borel reducibility.

19

3.3 Hyperfiniteness

One of the enduring problems in this field is to determine which countable Borel equivalence relationsare hyperfinite. Nowadays this is almost always asked in the Borel context, since the measuretheoretic setting is completely understood.

Theorem 3.6 (Connes-Feldman-Weiss, [8]) Let G be a countable amenable group acting by measurepreserving transformations on a standard Borel probability space (X,µ). Then there is a conull seton which the orbit equivalence relation is hyperfinite.

Equivalently, we can find a measurable reduction of EG to E0.Even this result for very simple groups remains a inscrutiatingly difficult in the Borel setting.

Theorem 3.7 (Jackson-Kechris-Louveau, [41]) Let G be a finitely generated group with a nilpotentsubgroup H with [G : H ], the index of H in G, finite.

If G acts by Borel automorphisms on a standard Borel space X, then the resulting equivalencerelation EG is hyperfinite.

There was a considerable pause until quite recently it was shown:

Theorem 3.8 (Boykin-Gao-Jackson, [6]) Let G be a countable abelian group.If G acts by Borel automorphisms on a standard Borel space X, then the resulting equivalence

relation EG is hyperfinite.

To give an idea of how difficult these problems have proved, it was not until 3.8, despite consid-erable efforts, we even knew that the commensurability equivalence relation on R \ {0},

rEcs⇔ r/s ∈ Q

was hyperfinite. It would be natural to conjecture all EG’s arising from the Borel action of ancountable amenable group on a standard Borel space are hyperfinite. This conjecture is presentlyfar out of reach, and actually has little in the way of supporting evidence.

4 Effective cardinality

One way in which to look at the theory of Borel reducibility is as a kind of theory of Borel cardinality,and in this sense there are definitely roots in papers by Harvey Friedman such as [16]. If we wereto truly embrace a mathematical ontology consisting solely of Borel objects, then we would alsobe naturally led to consider certain kinds of quotients arising from Borel equivalence relations andto make any comparison along the lines of cardinality it seems we would use something like Borelreducibility.

In this sense I am more inclined to consider the theory of ≤B as something like a theory ofcardinality, as opposed to a theory of reducibility of information, as one finds in the theory of Turingreducibility. In fact there are close parallels between the structure of the ≤B-ordering on Borelequivalence relations and the cardinality theory of L(R) under suitable determinacy assumptions.

Definition For A,B ∈ L(R) write|A|L(R) ≤ |B|L(R),

the L(R) cardinality of A does not exceed that of B, if there is an injection

i : A ↪→ B

in L(R).

20

Lemma 4.1 (Assume ADL(R); this is folklore, but see [34].) Let E,F be Borel equivalence relationson R. If

|R/E|L(R) ≤ |R/F |L(R)

then there is a functionf : R → R

in L(R) with for all x1, x2 ∈ R

x1Ex2 ⇔ f(x1)Ff(x2).

Then in parallel to Silver’s theorem Woodin has shown:

Theorem 4.2 (ADL(R), Woodin) Let A ∈ L(R). Then exactly one of:(I) There is an ordinal α with |A|L(R) ≤ |α|L(R) – in other words, A is well orderable; or(II) |R|L(R) ≤ |A|L(R).

And for Harrington-Kechris-Louveau:

Theorem 4.3 (ADL(R), Hjorth, [35]) Let A ∈ L(R). Then exactly one of:(I) There is an ordinal α with |A|L(R) ≤ |(α)|L(R) – in other words, A has a well orderableseparating family; or(II) |R/Ev|L(R) ≤ |A|L(R).

In (II) we can equivalently say that P/Fin embeds into A.The theory of effective cardinality, whether we choose to explicate it using Borel functions and

objects or the more joyously playful world of L(R), can also be compared with the idea of classifi-cation difficulty. If the effective cardinality of A is below that of B, then any objects which suceedas complete invariants for B do as well for A. Here one could even begin certain kinds of wild spec-ulations, to the effect that subconsciously part of the mathematical activity of vaguely searchingfor some kind of ill defined classification theorem for a class of objects is in fact a query as to itseffective cardinality

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