Definability of maximal families of reals in forcing
extensions
Jonathan Schilhan
Institute of Mathematics, University of Vienna

RIMS Workshop, 16.11.2020

generous support through FWF Project Y1012-N35
1/13
Jonathan Schilhan

Definability of maximal families of reals in forcing extensions

Hypergraphs

Definition
A hypergraph on a set X is a collection E (the edges) of finite non-empty
subsets of X , i.e. E ⊆ [X ]<ω \ {∅}. We say that Y ⊆ X is
E -independent if [Y ]<ω ∩ E = ∅. Y is maximal E -independent if Y is
maximal under inclusion as an E -independent subset of X .
If X is a Polish space, then [X ]<ω also has a natural Polish topology andwe can study definable hypergraphs E and definable maximal
E -independent sets.
FactIn L, every analytic hypergraph on a Polish space X has a ∆12 maximal
independent set.

2/13
Jonathan Schilhan

Definability of maximal families of reals in forcing extensions

Graphs

Theorem (Schrittesser 2016)
After forcing with a csi of Sacks forcing over L, every analytic(2-dimensional hyper)graph in a Polish space has a ∆12 maximalindependent set.
ExampleIn an extension of L by a csi of Sacks forcing, every analytic equivalence
relation has a ∆12 transversal.
This is wrong for the countable support product. There is no OD(R)transversal of E1 after an uncountably supported csp of Sacks forcing.

3/13
Jonathan Schilhan

Definability of maximal families of reals in forcing extensions

Examples

Example (Ultrafilter)
T
Let Eu := {A ∈ [P(ω)]<ω : A is finite}. Then an ultrafilter is amaximal Eu -independent set.
Question (S. 2019)
Is it consistent that u > ℵ1 , while there is a ∆12 ultrafilter?
Can we destroy all ground model ultrafilters while preserving a ∆12definition?

4/13
Jonathan Schilhan

Definability of maximal families of reals in forcing extensions

Examples
Example (Maximal independent families)
YT ⊆ P(ω)Tis an independent family if for all finite disjoint A, B ⊆ Y ,x∈A x ∩x∈B ω \ x is infinite. Letting˙ ∈ [P(ω)]<ω :Ei := {A∪B\x∩x∈A\ω \ x is finite}x∈B

an independent family is an Ei -independent set.
Question (Brendle, Fischer, Khomskii 2019)
Is it consistent that i > ℵ1 , while there is a Π11 maximal independentfamily?
Can we destroy all ground model maximal independent families while
preserving a Π11 definition?

5/13
Jonathan Schilhan

Definability of maximal families of reals in forcing extensions

Examples

Example (Hamel basis)
Let Eh := {A ∈ [R]<ω : A is linearly dependent over Q}. Then a Hamelbasis is a maximal Eh -independent set.
Every Hamel basis has size 2ℵ0 . This is reflected by the fact that adding
a single real destroys every ground model Hamel basis.
Question
Is it consistent that ¬ CH, while there is a ∆12 Hamel basis?
Can we destroy all ground model Hamel bases (i.e. add a new real) while
preserving a ∆12 definition?

6/13
Jonathan Schilhan

Definability of maximal families of reals in forcing extensions

Splitting reals

Definition
Let x, y ∈ [ω]ω . Then x splits y if |x ∩ y | = ω and |y \ x| = ω. x iscalled a splitting real over V if for every y ∈ V ∩ [ω]ω , x splits y .
Adding splitting reals over a model V , destroys all ultrafilters, maximal
independent family (and Hamel bases) in V and consequently increasesu, i.
Classical forcing notions adding splitting reals are Cohen, Random, Silverforcing and forcings adding dominating reals. But they all fail inpreserving ∆12 witnesses.

7/13
Jonathan Schilhan

Definability of maximal families of reals in forcing extensions

Splitting forcing

Definition (Splitting forcing)
A set A ⊆ 2<ω is called fat if there is m = m(A) ∈ ω so that for everyn ≥ m, i ∈ 2, there is s ∈ A so that s(n) = i.
Let T ⊆ 2<ω be a perfect tree. Then T is a splitting tree if for everys ∈ T , Ts is fat.
Splitting forcing SP consists of all splitting trees ordered by inclusion(T ≤ S iff T ⊆ S), as usual.
FactSP adds a generic splitting real xG ∈ 2ω (∼= P(ω)),
SP is proper (Axiom A),
SP is ω ω -bounding,
V SP is a minimal extension of V .

8/13
Jonathan Schilhan

Definability of maximal families of reals in forcing extensions

The main result
Theorem (S. 2020)
After forcing with a csi or a finite product of Sacks or splitting forcing
over L, every analytic hypergraph in a Polish space has a ∆12 maximal
independent set.
Corollary
It is consistent that there is a Π11 mif, a ∆12 ultrafilter and a ∆12 Hamelbasis while ℵ1 < i, u, c.
Note:
Theorem (Brendle, Khomski, Fischer 2019)
The existence of a Σ12 maximal independent family is equivalent to theexistence of a Π11 maximal independent family.

9/13
Jonathan Schilhan

Definability of maximal families of reals in forcing extensions

Borelized cardinal invariants
Definition
SuB = min{|B| : B ⊆ ∆11 and S B is an ultrafilter}iB = min{|B| : B ⊆ ∆11 and B is a max. ind. family}
similarly: ucl = min ..., icl = min ...
ObservationuB ≤ ucl ≤ u, iB ≤ icl ≤ i.
If there is a ∆12 ultrafilter/m.i.f then uB /iB = ω1 .
Corollary
It is consistent that uB , iB < u, i.
Theorem (S. 2020)(ZFC) d ≤ icl .
Recall: d ≤ i.
10/13
Jonathan Schilhan

Definability of maximal families of reals in forcing extensions

Other applications

Corollary of the constructionThere is a (∆12 ) P-point after iterating SP over L.
CorollaryZF+DC+“every analytic hypergraph has a maximal independentset”+“there is no well-order of the reals” is consistent relative to ZF.
Question (Pincus, Prikry 1975)(ZF) Does the existence of a Hamel basis imply a well-order of the reals?
This has been answered before by [Beriashvili, Schindler, Wu, Yu 2018](without LC) and by [Larson, Zapletal 2017] using an inaccessible.

11/13
Jonathan Schilhan

Definability of maximal families of reals in forcing extensions

QuestionsQuestion
Does the main result hold true for the csi of Miller forcing?
Remark: It fails for finite products of Miller forcing (M3 adds a Cohenreal).
This would yield a model of ω1 = iB < d = icl = i = ω2 .
Question (Brendle, Khomskii, Fischer 2019)
Is iB < icl consistent?

12/13
Jonathan Schilhan

Definability of maximal families of reals in forcing extensions

Thank you!

13/13
Jonathan Schilhan

Definability of maximal families of reals in forcing extensions