Balanced forcing extensions Paul Larson Department of Mathematics - - PowerPoint PPT Presentation

Balanced forcing extensions

Paul Larson

Department of Mathematics Miami University Oxford, Ohio 45056 larsonpb@miamioh.edu

November 18, 2018

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joint work with Jindˇ rich Zapletal

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Solovay models

Suppose that κ is a strongly inaccessible cardinal, and that G ⊆ Col(ω, <κ) is a V -generic filter. The resulting model HODV [G]

V ,P(ω)

is a Solovay model (which we will call W ). We study forcing extensions of the Solovay model which recover forms of the Axiom of Choice.

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Suslin partial orders

Definition

A preorder P, ≤ is Suslin if there is a Polish space X such that

1 P is an analytic subset of X; 2 the ordering ≤ is an analytic subset of X 2; 3 the incompatibility relation is an analytic subset of X 2.

We are mostly (but not only) interested in the case where P is σ-closed. We do not require ≤ to be antisymmetric, so different elements

• f P can represent the same condition in the separative

quotient.

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Examples

• Countable partial functions from ωω to 2
• Countable subsets of P(ω) with the finite intersection

property, under the relation of generating a larger filter (P(ω)/Fin)

• Countable partial selectors for a Borel equivalence relation
• Countable subsets of R which are linearly independent
• ver Q (adds a Hamel basis)
• Countable almost disjoint families (of various kinds)
• Disjoint pairs (a, b) ∈ [R]<ω × [R]ω, under containment

(forces ¬DCR)

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More examples

• Countable subsets of C which are algebraically

independent over Q

• The poset of countable injections from the E-classes to

the F-classes, for Borel equivalence relations E and F on Polish spaces.

• The poset of countable linear orders of E-classes, for a

Borel equivalence relation E on a Polish space.

• The poset of countable acyclic subsets of a Borel graph on

a Polish space.

• The post of countable partial k-colorings of a Borel graph
• n a Polish space, for k ∈ ω + 1.

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Virtual conditions

A virtual condition for a Suslin partial order P is a pair (Q, τ) such that

• Q is a partial order,
• τ is a Q-name for an element of P, and
• τ realizes to an equivalent P-condition in every V -generic

Q-extension. There is a natural notion of equivalence for virtual conditions : realizing to equivalent conditions in all generic extensions.

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Balanced conditions

Given a Suslin forcing P, a balanced condition for P is a pair (Q, τ) such that

• (Q, τ) is a virtual condition,
• for any two V -generic filters G0 and G1 for Q existing

respectively in mutually generic extensions V [H0] and V [H1], and any two conditions p0 ≤ τG0 and p1 ≤ τG1 in V [H0] and V [H1] respectively, p0 and p1 are compatible in V [H0, H1].

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Balance

V V [H0] V [H1] p0 ≤ τG0 p1 ≤ τG1 V [H0, H1] p2 ≤ p0, p1

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Example

For the partial order of countably generated filters, the balanced pairs are (up to equivalence) the pairs (Q, τ) where Q = Col(ω, 2ℵ0) and τ is a Q-name for an enumeration of an ultrafilter on ω in V . Sketch : If U is an ultrafilter on ω, the union of two mutually generic filters containing U has the finite intersection property, since any name for a member of either of these filters must have U-many possible members below each condition.

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More examples

• For the partial order of countable partial selectors for a

pinned equivalence relation, the balanced pairs are the Col(ω, 2ℵ0)-names for enumerations of the total selectors.

• For the partial order of countable partial functions from X

to 2 (for X a Polish space), the balanced pairs are the Col(ω, 2ℵ0)-names for (codes for) total functions from X to 2 in V .

• For the partial order of countable linearly independent

subsets of R over Q, the balanced pairs are the Col(ω, 2ℵ0)-names for enumerations of Hamel bases.

• If E is a Borel equivalence relation, and P is the partial
• rder of countable partial tournaments on the E-classes,

the balanced conditions are classified by the total tournaments on the virtual E-classes.

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Even more examples

• For the partial order of disjoint finite/countable pairs of

reals, the balanced conditions are characterized by finite sets and their complements.

• The partial order of countably generated filters disjoint

from a given Fσ ideal. There may be ultrafilters disjoint from the ideal which do not give balanced conditions.

• Does the partial order of countable subsets of C

algebraically independent over Q have balanced conditions?

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Balanced partial orders

A Suslin partial order is said to be balanced if below each condition there is a balanced virtual condition. Balance is not in general absolute between V and its forcing extensions. We say that P is cofinally balanced below κ if every partial

• rder in Vκ is regularly embedded in one forcing that P is

balanced in Vκ.

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Henle-Mathias-Woodin for cofinally balanced partial orders

Theorem

If

• P is a Suslin order, cofinally balanced below a strongly

inaccessible cardinal κ (in V ),

• α is an ordinal,
• W is a Solovay model for κ and
• G ⊆ P is W -generic,

then P(α) ∩ W [G] ⊆ W .

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Proof.

Suppose that, in W , some condition p ∈ P forces some P-name σ to represent a subset of α. In the Levy extension, σ is definable from some z ⊆ ω and an element of V . Let V [K] be a forcing extension of V contained in W such that p, z ∈ V [K] and such that P is balanced in V [K]. Let (Q, τ) ∈ Vκ[K] be a balanced virtual condition below p. Fix β < α and suppose there exist Q × Col(ω, <κ)-names ρ0, ρ1 in V [K] for conditions below the realization of τ forcing different truth values to the statement ˇ β ∈ σ. Since (Q, τ) is balanced, there are mutually generic extensions V [K][H0] and V [K][H1] in which the realizations of ρ0 and ρ1 are compatible, giving a contradiction. It follows that, in W , any V [K]-generic realization of τ forces that σG ∈ V [K].

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Consequences of balance

Theorem

If P is cofinally balanced below a strongly inaccessible cardinal κ, W is a Solovay model for κ and G ⊆ P is W -generic, then the following hold in W [G]:

• every wellordered sequence of elements of W is in W ;
• if E and F are Borel equivalence relations such that E is

pinned and F is unpinned, then |F| ≤ |E|;

• uniformization fails for sets whose cross-sections are

equivalence classes of a fixed unpinned Borel equivalence relation;

• there are no infinite MAD families on ω;
• if P is σ-closed, there are no unbounded linearly ordered

subsets of (ωω, ≤∗) or the Turing degrees.

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Products

Balance is preserved under countable (support) products.

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Weak balance

We get a weaker notion of balance if we require compatibility for some pair of extensions V [H0], V [H1] existing in a common extension instead of requiring compatibility in all mutually generic extensions.

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Weak balance

V V [H0] V [H1] ˙ p0,H0 ≤ τG0 ˙ p1,H1 ≤ τG1 V [F] p2 ≤ ˙ p0,H0, ˙ p1,H1

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In weak balanced extensions there are no new sets of ordinals, but there may be new functions from the ordinals to W . Certain types of (“improved”) MAD families can be added by weakly balanced partial orders, while preserving uncountable chromatic numbers for all Borel hypergraphs.

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Trim balance

We get a stronger notion of balance if instead of mutual genericity, we require only that V = V [H0] ∩ V [H1] and that H0 and H1 exist in a common forcing extension of V .

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Trim balance

V V [H0] V [H1] p0 ≤ τG0 p1 ≤ τG1 V [F] p2 ≤ p0, p1

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Examples of trim balanced orders

• Adding a selector for a countable Borel equivalence

relation

• Adding a maximal acyclic subgraph to a Borel graph
• Adding a Hamel basis to a Polish vector space

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In trim balanced extensions of W :

• there are no nonprincipal ultrafilters on ω;
• no uncountable field has a transcendence basis over a

countable subfield.

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m, n-balance

Definition

A virtual condition (Q, τ) for P is m, n-balanced if in any generic extension, whenever {Hi : i ∈ m} are filters generic over V (adding V -generic filters for Q) such that for every set a ∈ [m]n the filters {Hi : i ∈ a} are mutually generic over V , if for each i, pi is a condition in V [Hi] below the common realization of τ, then the set {pi : i ∈ m} has a common lower bound in P.

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3, 2-balance

V [F] V V [H1] V [H2] V [H0] p1 ≤ τG1 p2 ≤ τG2 p0 ≤ τG0

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3, 2-balanced posets

• The poset of countable partial selectors for an analytic

equivalence relation.

• The poset of countable injections from the E-classes to

the F-classes, for Borel equivalence relations E and F on Polish spaces.

• The poset of countable linear orders of E-classes, for a

Borel equivalence relation E on a Polish space.

• The poset of countable acyclic subsets of a Borel graph on

a Polish space.

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If P is 3, 2-balanced, then in the P-extension of W

• there is no linear order of the E0-classes (so no

nonprincipal ultrafilters on ω);

• no uncountable Polish vector space over countable field

has a basis;

• there are no discontinuous homomorphisms between Polish

groups.

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Consequences of Hamel bases

Classical arguments show that the existence of a Hamel basis for R over Q implies both the existence of an E0-selector and the existence of a discontinuous homomorphism from (R, +) to (R, +). More recent arguments (2018) show that the existence of an E0-selector follows from the existence of a discontinuous homomorphism between separable Banach spaces (assuming CCR) but does not imply the existence of a discontinuous homomorphism.

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No discontinuous homomorphisms in (3,2)-balanced extensions

Let P be (3, 2)-balanced, and let G be a Polish group. Let C = {(x, y, z) ∈ G 3 : x · y = z} and let QC be the partial order of nonempty open subsets of C, under containment. Then PC adds a triple (x∗, y∗, z∗) such that each pair is mutually generic for Cohen forcing. Assuming the existence of a P-name τ for a discontinuous homomorphism on (G, ·), x∗, y∗ and z∗ can be extended to pairwise generic triples (Gx, Gy, Gz) and (Gx, Gy, G ′

z)

respectively deciding values of τ for x∗, y∗ and z∗, but differently for z∗ in the case of Gz and G ′

z.

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No selectors from discontinuous homomorphisms into compact groups

If (G, ·) and (H, ·) are Polish groups, with (H, ·) divisible and compact, then partial order of countable partial homomorphisms from (G, ·) into (H, ·) does not add an E0-selector.

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Compact Balance

Definition

A Suslin poset P is compactly balanced if there is a definable compact Hausdorff topology T on the set B of all equivalence classes of balanced virtual conditions such that

1 for every p ∈ P the set {¯

p ∈ B : ¯ p ≤ p} ⊂ B is nonempty and T-closed;

2 if V [H0] ⊆ V [H1] are generic extensions of V then

• a. for every balanced virtual condition ¯

p0 ∈ V [H0] there is a balanced virtual condition ¯ p1 ∈ V [H1] such that ¯ p1 ≤ ¯ p0 holds;

• b. the relation

{¯ p0, ¯ p1 ∈ BV [H0] × BV [H1] : ¯ p1 ≤ ¯ p0} is closed in T V [H0] × T V [H1].

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If (G, ·) and (H, ·) are Polish groups, with (H, ·) divisible and compact, then partial order of countable partial homomorphisms from (G, ·) into (H, ·) is compactly balanced. Compactly balanced partial orders preserve |E0| > R.

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Question

Does the existence of a discontinuous homomorphism from (R, +) to (R, +) imply the existence of a Hamel basis?

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Orbit charms

Definition

Let P be a Suslin forcing. An orbit charm below a condition p ∈ P is a “choice-coherent” sequence Vn : n ∈ ω of generic extensions of V and a sequence ¯ pn : n ≤ ω so that

1 2ω ∩ Vn 2ω ∩ Vn+1 for all n ∈ ω; 2 ¯

p0 ≤ ¯ p1 ≤ · · · ≤ ¯ pω ≤ p where ¯ pn for each n ∈ ω is a balanced virtual condition in Vn and ¯ pω is a balanced virtual condition in the intersection model

n Vn.

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Orbit charming

Definition

Let κ be a Suslin forcing and κ an inaccessible cardinal. The poset P is orbit charming below κ if Vκ satisfies that every

• rdinal can be collapsed by a poset which forces P to have an
• rbit charm below every condition.

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Preserving the orbit divide

Theorem

Let κ be an inaccessible cardinal and P a Suslin forcing. If P is

• rbit charming below κ then in the P-extension of the Solovay

model derived from κ, |E1| ≤ |E| for every orbit equivalence relation E holds.

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Examples

• Compactly balanced partial orders are orbit charming.
• Let X be a Polish space and let G be a Borel graph on X.

The partial order of finite acyclic subsets of G is

• rbit-charming.

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Hamel bases are not orbit charming

If B is a Hamel basis, there is a predicate B′ such that, in any ω-model M amenable to B′, M ∩ B is a Hamel basis. Given x ∈ (ω2)ω, the sets L[B′, x↾[n, ω)] ∩ 2ω stabilize (to a common set containing a member of the E1-class of x). It follows that if there is a Hamel basis, then |E1| ≤ |F2|.

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No Hamel basis from a discontinuous homomorphism?

To show that the partial order of countable partial homomorphisms from (R, +) to (R, +) does not add a Hamel basis it suffices to show that one can produce, in some forcing extension (below any ground model condition f ) a choice-coherent sequence Vn : n ∈ ω of generic extensions of V and a homomorphism F ⊃ f with domain RV0 so that

1 2ω ∩ Vn 2ω ∩ Vn+1 for all n ∈ ω; 2 for each n ∈ ω, F maps RVn into RVn.

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Modular pre-geometries

Definition

Let X be a set and f : [X]<ℵ0 → [X]ℵ0 be a function. The pair X, f is a pre-geometry if

1 a ⊂ f (a); 2 (idempotence) b ⊂ f (a) implies f (b) ⊂ f (a) for all finite

sets a ⊂ b ⊂ X;

3 (exchange principle) if x ∈ f (a ∪ {y}) \ f (a) then

y ∈ f (a ∪ {x}). A finite set a ⊂ X is f -free if for every x ∈ a, x / ∈ f (a \ {x}). The pre-geometry X, f is modular if for all finite sets a, b ⊂ X and every point x ∈ f (a ∪ b) there are points xa ∈ f (a) and xb ∈ f (b) such that x ∈ f ({xa, xb}).

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Examples of modular pre-geometries

• X, f , where E is an equivalence relation on X with all

classes countable and f (a) = [a]E

• X, f , where X is a vector space over a countable field F

and f (a) is the linear span of a

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L(R)-generics for pre-geometries

Theorem

Suppose that there exists a Woodin limit of Woodin cardinals. Let X, f be a modular pre-geometry, with f Borel. Then there is an L(R)-generic filter for the partial order of countable f -free sets.

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No ultrafilters from pregeometries

The following statements are each forceable :

• for every nonprincipal ultrafilter U there is a p-point in

L(R)[U];

• there are no p-points

It follows that partial order for which large cardinals imply the existence of L(R)-generic filters (below each condition) can add a nonprincipal ultrafilter over L(R).

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Questions

• Do there exist A, B in W such that |A| ≤ |B| in W [U]

but not in W ?

• Is weak balance preserved under products?
• Is there a Borel filter F on ω such that the existence of an

ultrafilter containing F implies (in ZF) that |E0| ≤ R?

• Does the existence of a discontinuous homomorphism from

(R, +) to (R, +) imply the existence of a Hamel basis?

• If φ is Σ2

1 and ZFC ⊢ φ ↔ CH, must

ZF + φ + ℵ1 ≤ R ⊢ ¬DCR?

• Can a Suslin partial order have proper class many virtual

conditions? If “analytic” is replaced with Borel, then there are less than ℵ1 many.

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The generic ultrafilter model

Theorem

In the generic (Ramsey) ultrafilter model W [U], the following statements hold:

1 |E0| ≤ |2ω|; 2 |E1| ≤ |F| for any orbit equivalence relation F; 3 there do not exist tournaments on the quotient spaces of

E2 and F2;

4 if E is a Borel equivalence relation and A is a subset of the

E-quotient space then either |A| ≤ ℵ0 or |2ω| ≤ |A|;

5 every nonprincipal ultrafilter on ω has nonempty

intersection with the summability ideal;

6 countable-to-one uniformization.

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The generic Hamel basis model

Theorem

In the generic Hamel basis model W [B], the following statements hold:

1 there is no transcendence basis for any uncountable Polish

field;

2 there is no nonprincipal ultrafilter on ω; 3 countable-to-one uniformization.

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The generic E0-selector model

Theorem

In the generic E0-selector model W [T], the following statements hold:

1 |E1| ≤ |F| for any orbit equivalence relation F; 2 |E| ≤ |F| for any equivalence relation E generated by a

turbulent group action and equivalence relation F classifiable by countable structures;

3 there is no discontinuous homomorphism of (R, +); 4 there is no nonprincipal ultrafilter on ω; 5 countable-to-one uniformization.

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The generic improved MAD family model

Theorem

In the generic “improved” MAD family model W [A], the following statements hold:

1 there are no ω1 sequences of distinct reals; 2 there are no nonatomic measures on ω; 3 every set of reals is Lebesgue measurable; 4 the E0 classes are not linearly orderable; 5 there are no total selectors for E0; 6 2ℵ0 ≤ |A|.

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Equivalence relations

1 E0 is the Vitali equivalence on 2ω, connecting x, y ∈ 2ω if

they differ at only finite set of entries.

2 E1 is the equivalence relation on (2ω)ω connecting x, y if

they differ at only finite number of entries.

3 E2 is the relation on 2ω connecting x, y if the sum

• {

1 n + 1 : x(n) = y(n)} is finite.

4 F2 is the equivalence relation on (2ω)ω connecting x, y if

rng(x) = rng(y).