Variants of the Borel Conjecture and Sacks dense ideals Wolfgang - - PowerPoint PPT Presentation

Variants of the Borel Conjecture and Sacks dense ideals

Wolfgang Wohofsky

Vienna University of Technology (TU Wien) and Kurt G¨

• del Research Center, Vienna (KGRC)

wolfgang.wohofsky@gmx.at

Trends in set theory Warsaw, Poland, July 08-11, 2012

Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 1 / 22

Outline

Outline of the talk

1 Special sets of real numbers, Borel Conjecture ◮ strong measure zero, strongly meager ◮ Borel Conjecture, dual Borel Conjecture, Con(BC + dBC) 2 Another variant of the Borel Conjecture ◮ Marczewski ideal s0, “Marczewski Borel Conjecture” ◮ . . . investigating “Sacks dense ideals” Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 2 / 22

Outline

Outline of the talk

1 Special sets of real numbers, Borel Conjecture ◮ strong measure zero, strongly meager ◮ Borel Conjecture, dual Borel Conjecture, Con(BC + dBC) 2 Another variant of the Borel Conjecture ◮ Marczewski ideal s0, “Marczewski Borel Conjecture” ◮ . . . investigating “Sacks dense ideals” Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 2 / 22

Special sets of real numbers, Borel Conjecture

Special sets of real numbers, Borel Conjecture

1 Special sets of real numbers, Borel Conjecture ◮ strong measure zero, strongly meager ◮ Borel Conjecture, dual Borel Conjecture, Con(BC + dBC) 2 Another variant of the Borel Conjecture ◮ Marczewski ideal s0, “Marczewski Borel Conjecture” ◮ . . . investigating “Sacks dense ideals” Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 3 / 22

Special sets of real numbers, Borel Conjecture “The reals” and their structure

The real numbers: topology, measure, algebraic structure

The real numbers (“the reals”)

R, the classical real line 2ω, the Cantor space (totally disconnected, compact) Structure on the reals: natural topology (intervals/basic clopen sets form a basis) standard (Lebesgue) measure group structure

◮ (2ω,+) is a topological group, with + bitwise modulo 2

Two translation-invariant σ-ideals

◮ meager sets M ◮ measure zero sets N Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 4 / 22

Special sets of real numbers, Borel Conjecture “The reals” and their structure

The real numbers: topology, measure, algebraic structure

The real numbers (“the reals”)

R, the classical real line 2ω, the Cantor space (totally disconnected, compact) Structure on the reals: natural topology (intervals/basic clopen sets form a basis) standard (Lebesgue) measure group structure

◮ (2ω,+) is a topological group, with + bitwise modulo 2

Two translation-invariant σ-ideals

◮ meager sets M ◮ measure zero sets N Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 4 / 22

Special sets of real numbers, Borel Conjecture “The reals” and their structure

The real numbers: topology, measure, algebraic structure

The real numbers (“the reals”)

R, the classical real line 2ω, the Cantor space (totally disconnected, compact) Structure on the reals: natural topology (intervals/basic clopen sets form a basis) standard (Lebesgue) measure group structure

◮ (2ω,+) is a topological group, with + bitwise modulo 2

Two translation-invariant σ-ideals

◮ meager sets M ◮ measure zero sets N Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 4 / 22

Special sets of real numbers, Borel Conjecture Strong measure zero sets

Strong measure zero sets

For an interval I ⊆ R, let λ(I) denote its length.

Definition (well-known)

A set X ⊆ R is (Lebesgue) measure zero (X ∈ N) iff for each positive real number ε > 0 there is a sequence of intervals (In)n<ω of total length

n<ω λ(In) ≤ ε

such that X ⊆

n<ω In.

Definition (Borel; 1919)

A set X ⊆ R is strong measure zero (X ∈ SN) iff for each sequence of positive real numbers (εn)n<ω there is a sequence of intervals (In)n<ω with ∀n ∈ ω λ(In) ≤ εn such that X ⊆

n<ω In.

Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 5 / 22

Special sets of real numbers, Borel Conjecture Strong measure zero sets

Strong measure zero sets

For an interval I ⊆ R, let λ(I) denote its length.

Definition (well-known)

A set X ⊆ R is (Lebesgue) measure zero (X ∈ N) iff for each positive real number ε > 0 there is a sequence of intervals (In)n<ω of total length

n<ω λ(In) ≤ ε

such that X ⊆

n<ω In.

Definition (Borel; 1919)

A set X ⊆ R is strong measure zero (X ∈ SN) iff for each sequence of positive real numbers (εn)n<ω there is a sequence of intervals (In)n<ω with ∀n ∈ ω λ(In) ≤ εn such that X ⊆

n<ω In.

Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 5 / 22

Special sets of real numbers, Borel Conjecture Equivalent characterization of strong measure zero sets

Equivalent characterization of strong measure zero sets

For Y , Z ⊆ 2ω, let Y + Z = {y + z : y ∈ Y , z ∈ Z}.

Key Theorem (Galvin,Mycielski,Solovay; 1973)

A set Y ⊆ 2ω is strong measure zero if and only if for every meager set M ∈ M, Y + M = 2ω. Note that Y + M = 2ω if and only if Y can be “translated away” from M, i.e., there exists a t ∈ 2ω such that (Y + t) ∩ M = ∅.

Key Definition

Let J ⊆ P(2ω) be arbitrary. Define J ⋆ := {Y ⊆ 2ω : Y + Z = 2ω for every set Z ∈ J }. J ⋆ is the collection of “J -shiftable sets”, i.e., Y ∈ J ⋆ iff Y can be translated away from every set in J .

Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 6 / 22

Special sets of real numbers, Borel Conjecture Equivalent characterization of strong measure zero sets

Equivalent characterization of strong measure zero sets

For Y , Z ⊆ 2ω, let Y + Z = {y + z : y ∈ Y , z ∈ Z}.

Key Theorem (Galvin,Mycielski,Solovay; 1973)

A set Y ⊆ 2ω is strong measure zero if and only if for every meager set M ∈ M, Y + M = 2ω. Note that Y + M = 2ω if and only if Y can be “translated away” from M, i.e., there exists a t ∈ 2ω such that (Y + t) ∩ M = ∅.

Key Definition

Let J ⊆ P(2ω) be arbitrary. Define J ⋆ := {Y ⊆ 2ω : Y + Z = 2ω for every set Z ∈ J }. J ⋆ is the collection of “J -shiftable sets”, i.e., Y ∈ J ⋆ iff Y can be translated away from every set in J .

Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 6 / 22

Special sets of real numbers, Borel Conjecture Strongly meager sets

Strongly meager sets

Key Definition (from previous slide)

Let J ⊆ P(2ω) be arbitrary. Define J ⋆ := {Y ⊆ 2ω : Y + Z = 2ω for every set Z ∈ J }.

Key Theorem (Galvin,Mycielski,Solovay; 1973)

A set Y is strong measure zero if and only if it is “M-shiftable”, i.e., SN = M⋆ Replacing M by N yields a notion dual to strong measure zero:

Definition

A set Y is strongly meager (Y ∈ SM) iff it is “N-shiftable”, i.e., SM := N ⋆

Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 7 / 22

Special sets of real numbers, Borel Conjecture Strongly meager sets

Strongly meager sets

Key Definition (from previous slide)

Let J ⊆ P(2ω) be arbitrary. Define J ⋆ := {Y ⊆ 2ω : Y + Z = 2ω for every set Z ∈ J }.

Key Theorem (Galvin,Mycielski,Solovay; 1973)

A set Y is strong measure zero if and only if it is “M-shiftable”, i.e., SN = M⋆ Replacing M by N yields a notion dual to strong measure zero:

Definition

A set Y is strongly meager (Y ∈ SM) iff it is “N-shiftable”, i.e., SM := N ⋆

Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 7 / 22

Special sets of real numbers, Borel Conjecture Strongly meager sets

Strongly meager sets

Key Definition (from previous slide)

Let J ⊆ P(2ω) be arbitrary. Define J ⋆ := {Y ⊆ 2ω : Y + Z = 2ω for every set Z ∈ J }.

Key Theorem (Galvin,Mycielski,Solovay; 1973)

A set Y is strong measure zero if and only if it is “M-shiftable”, i.e., SN = M⋆ Replacing M by N yields a notion dual to strong measure zero:

Definition

A set Y is strongly meager (Y ∈ SM) iff it is “N-shiftable”, i.e., SM := N ⋆

Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 7 / 22

Special sets of real numbers, Borel Conjecture Borel Conjecture, dual Borel Conjecture

Borel Conjecture + dual Borel Conjecture

Definition

The Borel Conjecture (BC) is the statement that there are no uncountable strong measure zero sets, i.e., SN = M⋆ = [2ω]≤ℵ0. Con(BC), actually BC holds in the Laver model (Laver, 1976)

Definition

The dual Borel Conjecture (dBC) is the statement that there are no uncountable strongly meager sets, i.e., SM = N ⋆ = [2ω]≤ℵ0. Con(dBC), actually dBC holds in the Cohen model (Carlson, 1993)

Theorem (Goldstern,Kellner,Shelah,W.; 2011)

There is a model of ZFC in which both the Borel Conjecture and the dual Borel Conjecture hold, i.e., Con(BC + dBC).

Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 8 / 22

Special sets of real numbers, Borel Conjecture Borel Conjecture, dual Borel Conjecture

Borel Conjecture + dual Borel Conjecture

Definition

The Borel Conjecture (BC) is the statement that there are no uncountable strong measure zero sets, i.e., SN = M⋆ = [2ω]≤ℵ0. Con(BC), actually BC holds in the Laver model (Laver, 1976)

Definition

The dual Borel Conjecture (dBC) is the statement that there are no uncountable strongly meager sets, i.e., SM = N ⋆ = [2ω]≤ℵ0. Con(dBC), actually dBC holds in the Cohen model (Carlson, 1993)

Theorem (Goldstern,Kellner,Shelah,W.; 2011)

There is a model of ZFC in which both the Borel Conjecture and the dual Borel Conjecture hold, i.e., Con(BC + dBC).

Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 8 / 22

Special sets of real numbers, Borel Conjecture Borel Conjecture, dual Borel Conjecture

Borel Conjecture + dual Borel Conjecture

Definition

The Borel Conjecture (BC) is the statement that there are no uncountable strong measure zero sets, i.e., SN = M⋆ = [2ω]≤ℵ0. Con(BC), actually BC holds in the Laver model (Laver, 1976)

Definition

The dual Borel Conjecture (dBC) is the statement that there are no uncountable strongly meager sets, i.e., SM = N ⋆ = [2ω]≤ℵ0. Con(dBC), actually dBC holds in the Cohen model (Carlson, 1993)

Theorem (Goldstern,Kellner,Shelah,W.; 2011)

There is a model of ZFC in which both the Borel Conjecture and the dual Borel Conjecture hold, i.e., Con(BC + dBC).

Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 8 / 22

Special sets of real numbers, Borel Conjecture Borel Conjecture, dual Borel Conjecture Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 9 / 22

Special sets of real numbers, Borel Conjecture Borel Conjecture, dual Borel Conjecture Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 10 / 22

Special sets of real numbers, Borel Conjecture Borel Conjecture, dual Borel Conjecture Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 11 / 22

Special sets of real numbers, Borel Conjecture Borel Conjecture, dual Borel Conjecture Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 12 / 22

Another variant of the Borel Conjecture

Another variant of the Borel Conjecture

1 Special sets of real numbers, Borel Conjecture ◮ strong measure zero, strongly meager ◮ Borel Conjecture, dual Borel Conjecture, Con(BC + dBC) 2 Another variant of the Borel Conjecture ◮ Marczewski ideal s0, “Marczewski Borel Conjecture” ◮ . . . investigating “Sacks dense ideals” Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 13 / 22

Another variant of the Borel Conjecture Marczewski Borel Conjecture (MBC)

Marczewski Borel Conjecture (MBC)

Assume that J ⊆ P(2ω) is a translation-invariant σ-ideal. Recall that J ⋆ := {Y ⊆ 2ω : Y + Z = 2ω for every set Z ∈ J }.

Definition

The J -Borel Conjecture (J -BC) the statement that there are no uncountable J -shiftable sets, i.e., J ⋆ = [2ω]≤ω. The Marczewski ideal s0 is the collection of all Z ⊆ 2ω such that for each perfect set P, there exists a perfect subset Q ⊆ P with Q ∩Z = ∅.

Definition

The Marczewski Borel Conjecture (MBC) is the statement that there are no uncountable s0-shiftable sets, i.e., s0⋆ = [2ω]≤ω. What about Con(MBC)? Can MBC be forced?

Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 14 / 22

Another variant of the Borel Conjecture Marczewski Borel Conjecture (MBC)

Marczewski Borel Conjecture (MBC)

Assume that J ⊆ P(2ω) is a translation-invariant σ-ideal. Recall that J ⋆ := {Y ⊆ 2ω : Y + Z = 2ω for every set Z ∈ J }.

Definition

The J -Borel Conjecture (J -BC) the statement that there are no uncountable J -shiftable sets, i.e., J ⋆ = [2ω]≤ω. The Marczewski ideal s0 is the collection of all Z ⊆ 2ω such that for each perfect set P, there exists a perfect subset Q ⊆ P with Q ∩Z = ∅.

Definition

The Marczewski Borel Conjecture (MBC) is the statement that there are no uncountable s0-shiftable sets, i.e., s0⋆ = [2ω]≤ω. What about Con(MBC)? Can MBC be forced?

Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 14 / 22

Another variant of the Borel Conjecture Marczewski Borel Conjecture (MBC)

Marczewski Borel Conjecture (MBC)

Assume that J ⊆ P(2ω) is a translation-invariant σ-ideal. Recall that J ⋆ := {Y ⊆ 2ω : Y + Z = 2ω for every set Z ∈ J }.

Definition

The J -Borel Conjecture (J -BC) the statement that there are no uncountable J -shiftable sets, i.e., J ⋆ = [2ω]≤ω. The Marczewski ideal s0 is the collection of all Z ⊆ 2ω such that for each perfect set P, there exists a perfect subset Q ⊆ P with Q ∩Z = ∅.

Definition

The Marczewski Borel Conjecture (MBC) is the statement that there are no uncountable s0-shiftable sets, i.e., s0⋆ = [2ω]≤ω. What about Con(MBC)? Can MBC be forced?

Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 14 / 22

Another variant of the Borel Conjecture Marczewski Borel Conjecture (MBC)

Marczewski Borel Conjecture (MBC)

Assume that J ⊆ P(2ω) is a translation-invariant σ-ideal. Recall that J ⋆ := {Y ⊆ 2ω : Y + Z = 2ω for every set Z ∈ J }.

Definition

The J -Borel Conjecture (J -BC) the statement that there are no uncountable J -shiftable sets, i.e., J ⋆ = [2ω]≤ω. The Marczewski ideal s0 is the collection of all Z ⊆ 2ω such that for each perfect set P, there exists a perfect subset Q ⊆ P with Q ∩Z = ∅.

Definition

The Marczewski Borel Conjecture (MBC) is the statement that there are no uncountable s0-shiftable sets, i.e., s0⋆ = [2ω]≤ω. What about Con(MBC)? Can MBC be forced?

Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 14 / 22

Another variant of the Borel Conjecture Marczewski Borel Conjecture (MBC)

Marczewski Borel Conjecture (MBC)

Assume that J ⊆ P(2ω) is a translation-invariant σ-ideal. Recall that J ⋆ := {Y ⊆ 2ω : Y + Z = 2ω for every set Z ∈ J }.

Definition

The J -Borel Conjecture (J -BC) the statement that there are no uncountable J -shiftable sets, i.e., J ⋆ = [2ω]≤ω. The Marczewski ideal s0 is the collection of all Z ⊆ 2ω such that for each perfect set P, there exists a perfect subset Q ⊆ P with Q ∩Z = ∅.

Definition

The Marczewski Borel Conjecture (MBC) is the statement that there are no uncountable s0-shiftable sets, i.e., s0⋆ = [2ω]≤ω. What about Con(MBC)? Can MBC be forced?

Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 14 / 22

Another variant of the Borel Conjecture Marczewski Borel Conjecture (MBC) Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 15 / 22

Another variant of the Borel Conjecture Sacks dense ideals

Sacks dense ideals

Unlike BC and dBC, the status of MBC under CH is unclear. . . Is MBC (i.e., s0⋆ = [2ω]≤ℵ0) consistent with CH? Or does CH even imply MBC? To investigate the situation under CH, I introduced the following notion:

Definition

A collection I ⊆ P(2ω) is a Sacks dense ideal (S.d.i.) iff I is a σ-ideal I is translation-invariant I is dense in Sacks forcing, more explicitly, for each perfect P ⊆ 2ω, there is a perfect subset Q in the ideal, i.e., ∃Q ⊆ P, Q ∈ I

Lemma (“Main Lemma”)

Assume CH. Let I be a Sacks dense ideal. Then s0⋆ ⊆ I.

Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 16 / 22

Another variant of the Borel Conjecture Sacks dense ideals

Sacks dense ideals

Unlike BC and dBC, the status of MBC under CH is unclear. . . Is MBC (i.e., s0⋆ = [2ω]≤ℵ0) consistent with CH? Or does CH even imply MBC? To investigate the situation under CH, I introduced the following notion:

Definition

A collection I ⊆ P(2ω) is a Sacks dense ideal (S.d.i.) iff I is a σ-ideal I is translation-invariant I is dense in Sacks forcing, more explicitly, for each perfect P ⊆ 2ω, there is a perfect subset Q in the ideal, i.e., ∃Q ⊆ P, Q ∈ I

Lemma (“Main Lemma”)

Assume CH. Let I be a Sacks dense ideal. Then s0⋆ ⊆ I.

Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 16 / 22

Another variant of the Borel Conjecture Sacks dense ideals

Sacks dense ideals

Unlike BC and dBC, the status of MBC under CH is unclear. . . Is MBC (i.e., s0⋆ = [2ω]≤ℵ0) consistent with CH? Or does CH even imply MBC? To investigate the situation under CH, I introduced the following notion:

Definition

A collection I ⊆ P(2ω) is a Sacks dense ideal (S.d.i.) iff I is a σ-ideal I is translation-invariant I is dense in Sacks forcing, more explicitly, for each perfect P ⊆ 2ω, there is a perfect subset Q in the ideal, i.e., ∃Q ⊆ P, Q ∈ I

Lemma (“Main Lemma”)

Assume CH. Let I be a Sacks dense ideal. Then s0⋆ ⊆ I.

Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 16 / 22

Another variant of the Borel Conjecture Sacks dense ideals

More and more Sacks dense ideals

Lemma (“Main Lemma”; from previous slide)

Assume CH. Let I be a Sacks dense ideal. Then s0⋆ ⊆ I. In other words: s0⋆ ⊆ {I : I is a Sacks dense ideal}. Can we (consistently) find many Sacks dense ideals under CH? M N are Sacks dense ideals

• M ∩ N
• E

SN is NOT a Sacks dense ideal, BUT. . .

Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 17 / 22

Another variant of the Borel Conjecture Sacks dense ideals

More and more Sacks dense ideals

Lemma (“Main Lemma”; from previous slide)

Assume CH. Let I be a Sacks dense ideal. Then s0⋆ ⊆ I. In other words: s0⋆ ⊆ {I : I is a Sacks dense ideal}. Can we (consistently) find many Sacks dense ideals under CH? M N are Sacks dense ideals

• M ∩ N
• E

SN is NOT a Sacks dense ideal, BUT. . .

Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 17 / 22

Another variant of the Borel Conjecture Sacks dense ideals

More and more Sacks dense ideals

Lemma (“Main Lemma”; from previous slide)

Assume CH. Let I be a Sacks dense ideal. Then s0⋆ ⊆ I. In other words: s0⋆ ⊆ {I : I is a Sacks dense ideal}. Can we (consistently) find many Sacks dense ideals under CH? M N are Sacks dense ideals

• M ∩ N
• E

SN is NOT a Sacks dense ideal, BUT. . .

Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 17 / 22

Another variant of the Borel Conjecture Sacks dense ideals

More and more Sacks dense ideals

Lemma (“Main Lemma”; from previous slide)

Assume CH. Let I be a Sacks dense ideal. Then s0⋆ ⊆ I. In other words: s0⋆ ⊆ {I : I is a Sacks dense ideal}. Can we (consistently) find many Sacks dense ideals under CH? M N are Sacks dense ideals

• M ∩ N
• E

SN is NOT a Sacks dense ideal, BUT. . .

Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 17 / 22

Another variant of the Borel Conjecture Sacks dense ideals

More and more Sacks dense ideals

Lemma (“Main Lemma”; from previous slide)

Assume CH. Let I be a Sacks dense ideal. Then s0⋆ ⊆ I. In other words: s0⋆ ⊆ {I : I is a Sacks dense ideal}. Can we (consistently) find many Sacks dense ideals under CH? M N are Sacks dense ideals

• M ∩ N
• E

SN is NOT a Sacks dense ideal, BUT. . .

Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 17 / 22

Another variant of the Borel Conjecture Sacks dense ideals

E

• {If : f ∈ ωω}

⊆ null-additive ⊆ SN ∩ SM

• {If : f ∈ ωω} ∩ E0

⊆ ∃ uncount. Y ∈ {Iα : α ∈ ω1}, for any ℵ1-sized family of Iα’s

• ←

− proved(?) 5 days ago (using strans ) {I : I is a Sacks dense ideal} ⊆ ← − “Main Lemma” s0⋆ ⊆ [2ω]≤ℵ0

Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 18 / 22

Another variant of the Borel Conjecture Sacks dense ideals

E

• {If : f ∈ ωω}

⊆ null-additive ⊆ SN ∩ SM

• {If : f ∈ ωω} ∩ E0

⊆ ∃ uncount. Y ∈ {Iα : α ∈ ω1}, for any ℵ1-sized family of Iα’s

• ←

− proved(?) 5 days ago (using strans ) {I : I is a Sacks dense ideal} ⊆ ← − “Main Lemma” s0⋆ ⊆ [2ω]≤ℵ0

Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 18 / 22

Another variant of the Borel Conjecture Sacks dense ideals

E

• {If : f ∈ ωω}

⊆ null-additive ⊆ SN ∩ SM

• {If : f ∈ ωω} ∩ E0

⊆ ∃ uncount. Y ∈ {Iα : α ∈ ω1}, for any ℵ1-sized family of Iα’s

• ←

− proved(?) 5 days ago (using strans ) {I : I is a Sacks dense ideal} ⊆ ← − “Main Lemma” s0⋆ ⊆ [2ω]≤ℵ0

Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 18 / 22

Another variant of the Borel Conjecture Sacks dense ideals

E

• {If : f ∈ ωω}

⊆ null-additive ⊆ SN ∩ SM

• {If : f ∈ ωω} ∩ E0

⊆ ∃ uncount. Y ∈ {Iα : α ∈ ω1}, for any ℵ1-sized family of Iα’s

• ←

− proved(?) 5 days ago (using strans ) {I : I is a Sacks dense ideal} ⊆ ← − “Main Lemma” s0⋆ ⊆ [2ω]≤ℵ0

Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 18 / 22

Another variant of the Borel Conjecture Sacks dense ideals

E

• {If : f ∈ ωω}

⊆ null-additive ⊆ SN ∩ SM

• {If : f ∈ ωω} ∩ E0

⊆ ∃ uncount. Y ∈ {Iα : α ∈ ω1}, for any ℵ1-sized family of Iα’s

• ←

− proved(?) 5 days ago (using strans ) {I : I is a Sacks dense ideal} ⊆ ← − “Main Lemma” s0⋆ ⊆ [2ω]≤ℵ0

Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 18 / 22

Another variant of the Borel Conjecture Sacks dense ideals

E

• {If : f ∈ ωω}

⊆ null-additive ⊆ SN ∩ SM

• {If : f ∈ ωω} ∩ E0

⊆ ∃ uncount. Y ∈ {Iα : α ∈ ω1}, for any ℵ1-sized family of Iα’s

• ←

− proved(?) 5 days ago (using strans ) {I : I is a Sacks dense ideal} ⊆ ← − “Main Lemma” s0⋆ ⊆ [2ω]≤ℵ0

Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 18 / 22

Another variant of the Borel Conjecture Sacks dense ideals

Question

Does [2ω]≤ℵ0 = {I : I is S.d.i.} (at least consistently) hold under CH? If yes, MBC (i.e., s0⋆ = [2ω]≤ℵ0) follows from CH (Con(MBC+CH), resp.).

Theorem

Let {Iα : α < ω1} be an ℵ1-sized family of Sacks dense ideals. Then there exists an uncountable set Y ∈

α∈ω1 Iα.

Moreover, we can construct the set Y in such a way that Y / ∈ J for some

• ther Sacks dense ideal J (proved(?) 5 days ago (using strans

)). Y ∈ s0 :⇐ ⇒ ∀p ∃q ≤ p |[q] ∩ Y | ≤ ℵ0

Definition

Y ∈ strans :⇐ ⇒ ∀p ∃q ≤ p ∀t ∈ 2ω |(t + [q]) ∩ Y | ≤ ℵ0

Question

What can we say about the family strans ? Any relation to null-additive?

Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 19 / 22

Another variant of the Borel Conjecture Sacks dense ideals

Question

Does [2ω]≤ℵ0 = {I : I is S.d.i.} (at least consistently) hold under CH? If yes, MBC (i.e., s0⋆ = [2ω]≤ℵ0) follows from CH (Con(MBC+CH), resp.).

Theorem

Let {Iα : α < ω1} be an ℵ1-sized family of Sacks dense ideals. Then there exists an uncountable set Y ∈

α∈ω1 Iα.

Moreover, we can construct the set Y in such a way that Y / ∈ J for some

• ther Sacks dense ideal J (proved(?) 5 days ago (using strans

)). Y ∈ s0 :⇐ ⇒ ∀p ∃q ≤ p |[q] ∩ Y | ≤ ℵ0

Definition

Y ∈ strans :⇐ ⇒ ∀p ∃q ≤ p ∀t ∈ 2ω |(t + [q]) ∩ Y | ≤ ℵ0

Question

What can we say about the family strans ? Any relation to null-additive?

Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 19 / 22

Another variant of the Borel Conjecture Sacks dense ideals

Question

Does [2ω]≤ℵ0 = {I : I is S.d.i.} (at least consistently) hold under CH? If yes, MBC (i.e., s0⋆ = [2ω]≤ℵ0) follows from CH (Con(MBC+CH), resp.).

Theorem

Let {Iα : α < ω1} be an ℵ1-sized family of Sacks dense ideals. Then there exists an uncountable set Y ∈

α∈ω1 Iα.

Moreover, we can construct the set Y in such a way that Y / ∈ J for some

• ther Sacks dense ideal J (proved(?) 5 days ago (using strans

)). Y ∈ s0 :⇐ ⇒ ∀p ∃q ≤ p |[q] ∩ Y | ≤ ℵ0

Definition

Y ∈ strans :⇐ ⇒ ∀p ∃q ≤ p ∀t ∈ 2ω |(t + [q]) ∩ Y | ≤ ℵ0

Question

What can we say about the family strans ? Any relation to null-additive?

Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 19 / 22

Another variant of the Borel Conjecture Sacks dense ideals

Question

Does [2ω]≤ℵ0 = {I : I is S.d.i.} (at least consistently) hold under CH? If yes, MBC (i.e., s0⋆ = [2ω]≤ℵ0) follows from CH (Con(MBC+CH), resp.).

Theorem

Let {Iα : α < ω1} be an ℵ1-sized family of Sacks dense ideals. Then there exists an uncountable set Y ∈

α∈ω1 Iα.

Moreover, we can construct the set Y in such a way that Y / ∈ J for some

• ther Sacks dense ideal J (proved(?) 5 days ago (using strans

)). Y ∈ s0 :⇐ ⇒ ∀p ∃q ≤ p |[q] ∩ Y | ≤ ℵ0

Definition

Y ∈ strans :⇐ ⇒ ∀p ∃q ≤ p ∀t ∈ 2ω |(t + [q]) ∩ Y | ≤ ℵ0

Question

What can we say about the family strans ? Any relation to null-additive?

Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 19 / 22

Another variant of the Borel Conjecture Sacks dense ideals

Question

Does [2ω]≤ℵ0 = {I : I is S.d.i.} (at least consistently) hold under CH? If yes, MBC (i.e., s0⋆ = [2ω]≤ℵ0) follows from CH (Con(MBC+CH), resp.).

Theorem

Let {Iα : α < ω1} be an ℵ1-sized family of Sacks dense ideals. Then there exists an uncountable set Y ∈

α∈ω1 Iα.

Moreover, we can construct the set Y in such a way that Y / ∈ J for some

• ther Sacks dense ideal J (proved(?) 5 days ago (using strans

)). Y ∈ s0 :⇐ ⇒ ∀p ∃q ≤ p |[q] ∩ Y | ≤ ℵ0

Definition

Y ∈ strans :⇐ ⇒ ∀p ∃q ≤ p ∀t ∈ 2ω |(t + [q]) ∩ Y | ≤ ℵ0

Question

What can we say about the family strans ? Any relation to null-additive?

Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 19 / 22

Another variant of the Borel Conjecture Sacks dense ideals

Question

Does [2ω]≤ℵ0 = {I : I is S.d.i.} (at least consistently) hold under CH? If yes, MBC (i.e., s0⋆ = [2ω]≤ℵ0) follows from CH (Con(MBC+CH), resp.).

Theorem

Let {Iα : α < ω1} be an ℵ1-sized family of Sacks dense ideals. Then there exists an uncountable set Y ∈

α∈ω1 Iα.

Moreover, we can construct the set Y in such a way that Y / ∈ J for some

• ther Sacks dense ideal J (proved(?) 5 days ago (using strans

)). Y ∈ s0 :⇐ ⇒ ∀p ∃q ≤ p |[q] ∩ Y | ≤ ℵ0

Definition

Y ∈ strans :⇐ ⇒ ∀p ∃q ≤ p ∀t ∈ 2ω |(t + [q]) ∩ Y | ≤ ℵ0

Question

What can we say about the family strans ? Any relation to null-additive?

Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 19 / 22

References

References

Timothy J. Carlson. Strong measure zero and strongly meager sets.

• Proc. Amer. Math. Soc., 118(2):577–586, 1993.

Martin Goldstern, Jakob Kellner, Saharon Shelah, and Wolfgang Wohofsky. Borel Conjecture and dual Borel Conjecture. Transactions of the American Mathematical Society, to appear. http://arxiv.org/abs/1105.0823 Richard Laver. On the consistency of Borel’s conjecture. Acta Math., 137:151–169, 1976. Janusz Pawlikowski. A characterization of strong measure zero sets. Israel J. Math., 93:171–183, 1996. My website: http://wohofsky.eu/math/

Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 20 / 22

Thank you

Thank you for your attention and enjoy Warsaw. . .

Myself in Wroc law

Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 21 / 22

Thank you

Thank you for your attention and enjoy Warsaw. . .

Danube in Vienna

Wolfgang Wohofsky (TU Wien & KGRC) Variants of the Borel Conjecture Warszawa, 2012 22 / 22