Introduction
1
Canonicalization for Σ1
1 and Π1
Canonicalization for More Equivalence Relations
Every Analytic Equivalence Relation with All Borel
Classes Is Borel Somewhere
William Chan
California Institute of Technology
Meeting of AAAS Pacific Division
Boise Extravaganza in Set Theory
University of San Diego
June 15, 2016
William Chan
Every Analytic Equivalence Relation with All Borel Classes Is Borel Somewhere
Introduction
1
Canonicalization for Σ1
1 and Π1
Canonicalization for More Equivalence Relations
Outlines
1
Introduction
2
Canonicalization for Σ11 and Π11
3
Canonicalization for More Equivalence Relations
William Chan
Every Analytic Equivalence Relation with All Borel Classes Is Borel Somewhere
Introduction
1
Canonicalization for Σ1
1 and Π1
Canonicalization for More Equivalence Relations
Introduction
Question (Vague Question)
Let E be an equivalence relation on ω ω . Is there some ∆11 set B so that
E B is a ∆11 equivalence relation?
Here, E B = E ∩ (B × B ).
Of course, if B is countable, then E B is ∆11 .
Any σ -ideal on ω ω will contain all the countable sets. This obvious triviality
will disappears if one asks that B be nontrivial according to some σ -ideal.
If I is a σ -ideal on ω ω , a set B is I + if and only if B ∈
/ I.
William Chan
Every Analytic Equivalence Relation with All Borel Classes Is Borel Somewhere
Introduction
1
Canonicalization for Σ1
1 and Π1
Canonicalization for More Equivalence Relations
Proper Forcing Ideals
However, it is unclear what property held in common by all σ -ideal could
possibly be useful for answering this question. Restricting to a certain very
large class of σ -ideals would allow for the tools from set theory to be applied:
Given a σ -ideal I, there is a natural forcing associated to I.
Definition
Suppose I is a σ -ideal on ω ω . Let PI be the forcing of I + ∆11 subsets,
ordered by ≤PI =⊆, with largest element 1PI = ω ω .
If M is a model of set theory and x ∈ ω ω , then x is a PI -generic over M real if
and only if {B ∈ ∆11 ∩ M : x ∈ B } is a PI -generic filter over M.
William Chan
Every Analytic Equivalence Relation with All Borel Classes Is Borel Somewhere
Introduction
1
Canonicalization for Σ1
1 and Π1
Canonicalization for More Equivalence Relations
Proper Forcing Ideals
Fact (Zapletal)
Let I be a σ -ideal on ω ω . The following are equivalent:
(i) PI is a proper forcing.
(ii) For all I + ∆11 set B and all countable M ≺ HΞ , where Ξ is some
appropriately large cardinal, with B , PI ∈ M, the set
{x ∈ B : x is PI -generic over M } is I + ∆11 .
There are many familiar σ -ideal I so that PI is proper.
Fact (Cohen Forcing)
PImeager is Cohen forcing.
Fact (Random Real Forcing)
PInull is random real forcing.
William Chan
Every Analytic Equivalence Relation with All Borel Classes Is Borel Somewhere
Introduction
1
Canonicalization for Σ1
1 and Π1
Canonicalization for More Equivalence Relations
Proper Forcing Ideals
Fact (Sacks Forcing)
PIcountable is Sacks forcing.
Let E0 be the equivalence relation on ω ω defined by eventual agreement of
reals: x E0 y if and only if (∃k )(∀n > k )(x (n) = y (n)).
Fact (Prikry-Silver Forcing)
(Zapletal) Let IE0 be the σ -ideal generated by the ∆11 sets B so that E B is
smooth. Then PIE is Prikry-Silver forcing.
0
William Chan
Every Analytic Equivalence Relation with All Borel Classes Is Borel Somewhere
Introduction
1
Canonicalization for Σ1
1 and Π1
Canonicalization for More Equivalence Relations
Better Question
With the above consideration in mind, we ask the following better test
question for the simplest class of equivalence relation in the projective
hierarchy just above ∆11 :
Question
Let E be an Σ11 equivalence relation on ω ω and let I be a σ -ideal on ω ω so
that PI is proper. Is there some I + ∆11 set C so that E C is ∆11 ?
Unfortunately, the answer is no.
Fact (Kanovei-Sabok-Zapletal)
There is an Σ11 equivalence relation E and a σ -ideal I so that for all I + ∆11 set
C, E C is not ∆11 .
However KSZ showed the question has a positive answer for the most
important classes of Σ11 equivalence relations.
William Chan
Every Analytic Equivalence Relation with All Borel Classes Is Borel Somewhere
Introduction
1
Canonicalization for Σ1
1 and Π1
Canonicalization for More Equivalence Relations
Positive Results
Fact (KSZ)
Let I be a σ -ideal so that PI is a proper forcing. If E is an Σ11 equivalence
relation with all countable classes or is ∆11 reducible to the orbit equivalence
relation of a ∆11 group action, then there is some I + ∆11 set C so that E C is
∆11 .
Given the negative answer to the previous question, to possibly get a positive
answer to any form of our question, the equivalence relation must bear at
least some resemblance to ∆11 equivalence relations. Of course, every ∆11
equivalence relation has all ∆11 classes. Note that if E has all countable
classes or is ∆11 reducible to orbit equivalence relation of a ∆11 group action,
then E has all ∆11 classes.
This gives evidence that perhaps a sufficient resemblance to ∆11 equivalence
relation that could give a positive answer is the property of having all ∆11
classes. KSZ then asked the following question:
William Chan
Every Analytic Equivalence Relation with All Borel Classes Is Borel Somewhere
Introduction
1
Canonicalization for Σ1
1 and Π1
Canonicalization for More Equivalence Relations
Main Question for Σ11 and Π11
Question
(KSZ) Let E be an Σ11 equivalence relation on ω ω with all ∆11 classes. Let I
be a σ -ideal on ω ω so that PI is proper. Is there an I + ∆11 set C so that
E C is ∆11 ?
And the same question when E is Π11 with all ∆11 classes.
First some partial results for more restricted classes of Σ11 or Π11 equivalence
relations:
William Chan
Every Analytic Equivalence Relation with All Borel Classes Is Borel Somewhere
Introduction
1
Canonicalization for Σ1
1 and Π1
Canonicalization for More Equivalence Relations
Thin Σ11 equivalence relations
An equivalence relation E is thin if and only if there is no perfect set of
E-inequivalent elements.
Example of thin Σ11 equivalence relation with all ∆11 classes and uncountably
many classes include the countable admissible ordinal equivalence relation
y
x Fω1 y ⇔ ω1x = ω1 or the isomorphism relation of counterexamples to
Vaught’s conjecture (if one exists).
Theorem
(ZFC) Let I is a σ -ideal on ω ω with PI -proper. Let E be a thin Σ11 equivalence
relation. Then there is an I + ∆11 set C so that C is contained in a single
E-class. (So E C is ∆11 as it has only one class.)
Note that there is no requirement that E has all ∆11 -classes.
William Chan
Every Analytic Equivalence Relation with All Borel Classes Is Borel Somewhere
Introduction
1
Canonicalization for Σ1
1 and Π1
Canonicalization for More Equivalence Relations
Null and Meager Ideal
Theorem
(ZFC + MA + ¬CH) Let E be an Σ11 equivalence relation with all ∆11 classes.
Then there is measure one (or comeager) ∆11 set C so that E C is ∆11 .
The key idea here is that under these assumptions, cov(I ) > ℵ1 and Π12 sets
have the Baire property and are Lebesgue measurable.
Theorem
Assume Π13 set are Lebesgue measureable (or have the Baire property). Let
E be an Σ11 equivalence relation with all ∆11 classes. Then there is an
measure one (respectively comeager) ∆11 set C so that E C is ∆11 .
The main idea is use the fact that a certain Π13 pre-well-ordering is Lebesgue
measurable or has the Baire property. This of course shows that the theorem
holds in a Coll(ω, < κ) extension, where κ is inaccessible.
William Chan
Every Analytic Equivalence Relation with All Borel Classes Is Borel Somewhere
Introduction
1
Canonicalization for Σ1
1 and Π1
Canonicalization for More Equivalence Relations
Π11 Equivalence Relation with All Countable Classes
Theorem
Assume ω1L < ω1 . Let E be a Π11 equivalence relation with all countable
classes. Then there is a comeager set C so that E C is ∆11 .
Theorem
Let κ be a remarkable cardinal. Let G ⊆ Coll(ω, < κ) be generic over L. In
L[G], if I is a σ -ideal with PI proper and E is a Π11 equivalence relation with
all countable classes, then there is I + ∆11 set C so that E C is ∆11 .
A remarkable cardinal (defined by Schlinder) is a relatively weak large
cardinal that can exist in L.
William Chan
Every Analytic Equivalence Relation with All Borel Classes Is Borel Somewhere
Introduction
1
Canonicalization for Σ1
1 and Π1
Canonicalization for More Equivalence Relations
Σ11 Equivalence Relation with all ∆11 classes
The following result was also obtained independently by Drucker.
Theorem
Assume there is a measurable cardinal (or sharps of certain sets exists). Let
E be an Σ11 equivalence relation with all Π11 classes (or Π11 equivalence
relation with all Σ11 classes). Let I be a σ -ideal on ω ω so that PI is proper.
Then there is an I + ∆11 set C so that E C is Π11 (or Σ11 , respectively).
Of course, an Σ11 equivalence with all Π11 classes has all ∆11 classes. This
phrasing emphasizes more clearly what is happening in the proof.
In the original proof of this result, the most important tool is a result of
Burgess which showed that every Σ11 equivalence relation E is the
T
intersection α<ω1 Eα for some definable decreasing sequence
{Eα : α < ω1 } of ∆11 eqivalence relations.
William Chan
Every Analytic Equivalence Relation with All Borel Classes Is Borel Somewhere
Introduction
1
Canonicalization for Σ1
1 and Π1
Canonicalization for More Equivalence Relations
New Question
Neeman then asked the following generalization of the previous result:
Question
If E is projective equivalence relation on ω ω with all ∆11 classes and I is a
σ -ideal on ω ω so that PI is proper, then is there some I + ∆11 set C so that
E C is ∆11 ?
The main tool in the proof of Σ11 canonicalization was Burgess’s result that if
T
E is Σ11 then E = α<ω1 Eα for a sequence of decreasing ∆11 equivalence
relation. For more complex equivalence relations, it is not clear what will play
the role of this ∆11 approximation.
Also it can be shown that canonicalization for ∆12 is not provable in ZFC.
William Chan
Every Analytic Equivalence Relation with All Borel Classes Is Borel Somewhere
Introduction
1
Canonicalization for Σ1
1 and Π1
Canonicalization for More Equivalence Relations
Failure of Canonicalization for ∆12 in L
The following result was also obtained independently by Drucker:
Fact
In L, there is a ∆12 equivalence relation EL which is thin and has all countable
classes.
So let I be any σ -ideal (possibly non-proper). Suppose there was a I + ∆11
subset C so that EL C is ∆11 . Since C is I + , EL C must have uncountably
many classes. Since EL is thin, EL C does not have a perfect set of
EL -inequivalent elements. But Silver’s dichotomy asserts that every ∆11
equivalence relation either has countably many classes or a perfect set of
inequivalent elements.
However it is not known if it is consistent that canonicalization fails for Σ11 or
Π11 equivalence relations with all ∆11 classes.
William Chan
Every Analytic Equivalence Relation with All Borel Classes Is Borel Somewhere
Introduction
1
Canonicalization for Σ1
1 and Π1
Canonicalization for More Equivalence Relations
General Framework for Canonicalization
(With Magidor) In order to answer the more general questions in presence of
stronger large cardinals, a different method is used in the following general
framework:
Let S be a homogeneous tree on ω × ω × γ . Roughly, these trees have a
coherent assignment of inner models to integer strings, so that f ∈ p[T ] if and
only if f determines a wellfounded direct limit. Fix a σ -ideal I with PI proper.
(Assumption A) Let E = p[S ]. E is an equivalence relation. In the PI -generic
extension, S is still a homogeneous tree and E is still an equivalence relation.
William Chan
Every Analytic Equivalence Relation with All Borel Classes Is Borel Somewhere
Introduction
1
Canonicalization for Σ1
1 and Π1
Canonicalization for More Equivalence Relations
Define a relation on reals by D (x , T ) ⇔ T is a tree on ω × ω and
(∀y )(y E x ⇔ T y is ill-founded).
D (x , T ) merely asserts that T gives the Σ11 definition of [x ]E .
(Assumption B) (∀x )(∃T )D (x , T ) and 1PI (∀x )(∃T )D (x , T ).
(Assumption C) There is a tree U on ω × ω × ε so that
p[U ] = {(x , T ) : D (x , T )} and 1PI p[Ǔ ] = {(x , T ) : D (x , T )}.
Assumption B asserts E has all Σ11 classes and continue to have all Σ11
classes in PI -extensions. Assumption C aserts that the formula D (x , T ) has a
tree representation that continues to represent D (x , T ) in the PI -extension.
William Chan
Every Analytic Equivalence Relation with All Borel Classes Is Borel Somewhere
Introduction
1
Canonicalization for Σ1
1 and Π1
Canonicalization for More Equivalence Relations
Let M ≺ HΞ be countable containing S and U. By Zapletal equivalence of
properness, the set C of PI -generic reals over M is an I + ∆11 set. Fix a g ∈ C.
Assumption B states that E has all Σ11 classes in M [g ]. So there is some
T ∈ M [g ] so that D (g , T ). Assumption C asserts that U continues to
represent the formula D (x , T ) in M [g ]. So M [g ] |= (g , T ) ∈ p[U ]. Therefore,
V |= (g , T ) ∈ p[U ]. V |= D (g , T ). This shows that in M [g ], there is a tree
T ∈ M [g ] giving the Σ11 definition of [g ]E and (importantly) still gives the Σ11
definition of [g ]E in V . Fix this tree T .
William Chan
Every Analytic Equivalence Relation with All Borel Classes Is Borel Somewhere
Introduction
1
Canonicalization for Σ1
1 and Π1
Canonicalization for More Equivalence Relations
Now consider the following closed game:
( m0 , n 0 )
(m1 , n1 )
α0
α1
... (mk −1 , nk −1 )
...
αk −1
where the rules are
(m0 ...mk −1 , n0 ...nk −1 ) ∈ T
(g k , m0 ...mk −1 , α0 ...αk −1 ) ∈ S
The first person to violate these rules loses. Player 2 wins if the game
continues forever.
Recall that assumption A asserts that S is homogeneous in the forcing
extension. Much like Martin’s proof of Σ11 determinacy, in M [g ], Player 2 has
a winning strategy, call it τ ∈ M [g ].
William Chan
Every Analytic Equivalence Relation with All Borel Classes Is Borel Somewhere
Introduction
1
Canonicalization for Σ1
1 and Π1
Canonicalization for More Equivalence Relations
τ is a winning strategy for Player 2 in V , as well: This is because since
τ ∈ M [g ] and Player 1 always plays natural numbers, every finite partial play
according to τ belongs to M [g ]. If Player 2 loses in V , the corresponding
finite play would witness that Player 2 lost in M [g ].
Let y ∈ (ω ω)V so that y E g. Since T gives the Σ11 definition of [g ]E in V ,
there is some f ∈ [T y ].
Consider the following play where Player 2 uses the its winning strategy τ
and Player 1 plays y and f :
(y (0), f (0))
(y (1), f (1))
α0
α1
... (y (k − 1), f (k − 1))
...
αk −1
Let L = (αi : i ∈ ω). Since τ is a winning strategy, L ∈ [S (g ,y ) ]. Since
τ ∈ M [g ], all finite partial plays belong to M [g ]. So L n ∈ M [g ]. Since
OnM = OnM [g ] , this shows that (S ∩ M )(g ,y ) is ill-founded.
William Chan
Every Analytic Equivalence Relation with All Borel Classes Is Borel Somewhere
Introduction
1
Canonicalization for Σ1
1 and Π1
Canonicalization for More Equivalence Relations
It has been shown that for all y ∈ (ω ω)V and g ∈ C
g E y ⇔ (S ∩ M )(g ,y ) is ill-founded
Since S ∩ M is a countable tree, this shows that E C is Σ11 .
Fact
Let I be a σ -ideal so that PI is proper. Let S be a homogeneous Suslin tree
on ω × ω × γ and let E = p[S ]. If Assumptions A, B, and C hold, then there
in I + ∆11 set C so that E C is Σ11 . (The Π11 case is similar.)
William Chan
Every Analytic Equivalence Relation with All Borel Classes Is Borel Somewhere
Introduction
1
Canonicalization for Σ1
1 and Π1
Canonicalization for More Equivalence Relations
L(R) Canonicalization
The Martin-Solovay tree of a weak-homogeneity system provides generically
correct tree representations for complements of weakly homogeneous sets in
small forcing extension. The presence of Woodin cardinals makes these
trees homogeneous, as well.
These generically correct tree representations can be used to show
assumption A, B, and C holds to prove the following:
Theorem (with Magidor)
Suppose there is a measurable cardinal with infinitely many Woodin cardinals
below it. Let E ∈ L(R) be an equivalence relation on ω ω with all Σ11 (or Π11 )
classes. Let I be a σ -ideal on ω ω with PI proper. Then there is an I + ∆11 set
C so that E C is Σ11 (Π11 , respectively).
William Chan
Every Analytic Equivalence Relation with All Borel Classes Is Borel Somewhere
Introduction
1
Canonicalization for Σ1
1 and Π1
Canonicalization for More Equivalence Relations
Canonicalization for All Equivalence Relations
Question
Is the following consistent? Let E be an equivalence relation with all ∆11
classes. Let I be a σ -ideal so that PI is proper. Is there an I + ∆11 set C so
that E C is ∆11 ?
Fact
(ZFC) There is a thin equivalence relation with classes of size at most 2.
Using the argument for ∆12 in L involving Silver’s Dichotomy, this shows that
this equivalence relation does not satisfy canonicalization. So a positive
answer is not consistent with the axiom of choice. However, many other
classical regularity property follow using some form of the axiom of
determinacy.
William Chan
Every Analytic Equivalence Relation with All Borel Classes Is Borel Somewhere
Introduction
1
Canonicalization for Σ1
1 and Π1
Canonicalization for More Equivalence Relations
Meager Ideal Canonicalization
Under ADR , every set of reals is homogeneously Suslin and every tree is
weakly homogeneous. Since PImeager is equivalent to Cohen forcing (which is
countable), homogeneity systems can be lifted to show that the
Martin-Solovay tree continues to represent complements in Cohen
extensions. Our general framework can then be used to show:
Theorem (with Magidor)
(ZF + DC + ADR ) Let E be an equivalence relation with all Σ11 (or Π11 ) classes.
Then there is a comeager ∆11 set C so that E C is Σ11 (or Π11 respectively).
William Chan
Every Analytic Equivalence Relation with All Borel Classes Is Borel Somewhere
Introduction
1
Canonicalization for Σ1
1 and Π1
Canonicalization for More Equivalence Relations
Proper Forcing Ideal Canonicalization
For the general ideal I, PI (for example Sacks forcing) is not well-ordered. It is
not clear how to lift homogenity system into forcing extension to ensure the
Martin-Solovay tree continue to represent complements in forcing extensions.
In this case, we will use the following strong form of absoluteness:
Fact (Neeman-Norwood)
Assume ZF + DC + ADR + V = L(P(R)). Let P be a proper forcing. Let
G ⊆ P be P-generic over V . Then there is an elementary embedding
j : L(P(R)) → L(P(R)V [G] ) which is identity on ordinals and reals.
Theorem (with Magidor)
Assume ZF + DC + ADR + V = L(P(R)). Let E be an equivalence on ω ω
with all classes Σ11 (or Π11 ). Let I be a σ -ideal so that PI is proper. Then there
is an I + ∆11 set C so that E C is Σ11 (or Π11 ), respectively.
William Chan
Every Analytic Equivalence Relation with All Borel Classes Is Borel Somewhere
Introduction
1
Canonicalization for Σ1
1 and Π1
Canonicalization for More Equivalence Relations
Thanks for listening.
William Chan
Every Analytic Equivalence Relation with All Borel Classes Is Borel Somewhere