There are no large sets which can be translated away from every - PowerPoint PPT Presentation

There are no large sets which can be translated away from every Marczewski null set Wolfgang Wohofsky joint work with J¨ org Brendle Universit¨ at Hamburg wolfgang.wohofsky@gmx.at TOPOSYM 2016 Prague, Czech Republic 26th July 2016 Wolfgang Wohofsky (Universit¨ at Hamburg) No s 0 -shiftable sets TOPOSYM 2016 1 / 21

Theorem (Brendle-W., 2015) (ZFC) No set of reals of size continuum is “ s 0 -shiftable”. Definition A set Y ⊆ 2 ω is Marczewski null ( Y ∈ s 0 ) : ⇐ ⇒ for any perfect set P ⊆ 2 ω there is a perfect set Q ⊆ P with Q ∩ Y = ∅ . ⇐ ⇒ ∀ p ∈ S ∃ q ≤ p [ q ] ∩ Y = ∅ Definition A set X ⊆ 2 ω is s 0 -shiftable : ⇐ X + Y � = 2 ω ⇒ ∀ Y ∈ s 0 ∃ t ∈ 2 ω ⇐ ⇒ ∀ Y ∈ s 0 ( X + t ) ∩ Y = ∅ . Theorem (Brendle-W., 2015, restated more explicitly) (ZFC) Let X ⊆ 2 ω with | X | = c . Then there is a Y ∈ s 0 with X + Y = 2 ω . Wolfgang Wohofsky (Universit¨ at Hamburg) No s 0 -shiftable sets TOPOSYM 2016 2 / 21

Theorem (Brendle-W., 2015) (ZFC) No set of reals of size continuum is “ s 0 -shiftable”. Definition A set Y ⊆ 2 ω is Marczewski null ( Y ∈ s 0 ) : ⇐ ⇒ for any perfect set P ⊆ 2 ω there is a perfect set Q ⊆ P with Q ∩ Y = ∅ . ⇐ ⇒ ∀ p ∈ S ∃ q ≤ p [ q ] ∩ Y = ∅ Definition A set X ⊆ 2 ω is s 0 -shiftable : ⇐ X + Y � = 2 ω ⇒ ∀ Y ∈ s 0 ∃ t ∈ 2 ω ⇐ ⇒ ∀ Y ∈ s 0 ( X + t ) ∩ Y = ∅ . Theorem (Brendle-W., 2015, restated more explicitly) (ZFC) Let X ⊆ 2 ω with | X | = c . Then there is a Y ∈ s 0 with X + Y = 2 ω . Wolfgang Wohofsky (Universit¨ at Hamburg) No s 0 -shiftable sets TOPOSYM 2016 2 / 21

Theorem (Brendle-W., 2015) (ZFC) No set of reals of size continuum is “ s 0 -shiftable”. Definition A set Y ⊆ 2 ω is Marczewski null ( Y ∈ s 0 ) : ⇐ ⇒ for any perfect set P ⊆ 2 ω there is a perfect set Q ⊆ P with Q ∩ Y = ∅ . ⇐ ⇒ ∀ p ∈ S ∃ q ≤ p [ q ] ∩ Y = ∅ Definition A set X ⊆ 2 ω is s 0 -shiftable : ⇐ X + Y � = 2 ω ⇒ ∀ Y ∈ s 0 ∃ t ∈ 2 ω ⇐ ⇒ ∀ Y ∈ s 0 ( X + t ) ∩ Y = ∅ . Theorem (Brendle-W., 2015, restated more explicitly) (ZFC) Let X ⊆ 2 ω with | X | = c . Then there is a Y ∈ s 0 with X + Y = 2 ω . Wolfgang Wohofsky (Universit¨ at Hamburg) No s 0 -shiftable sets TOPOSYM 2016 2 / 21

Theorem (Brendle-W., 2015) (ZFC) No set of reals of size continuum is “ s 0 -shiftable”. Definition A set Y ⊆ 2 ω is Marczewski null ( Y ∈ s 0 ) : ⇐ ⇒ for any perfect set P ⊆ 2 ω there is a perfect set Q ⊆ P with Q ∩ Y = ∅ . ⇐ ⇒ ∀ p ∈ S ∃ q ≤ p [ q ] ∩ Y = ∅ Definition A set X ⊆ 2 ω is s 0 -shiftable : ⇐ X + Y � = 2 ω ⇒ ∀ Y ∈ s 0 ∃ t ∈ 2 ω ⇐ ⇒ ∀ Y ∈ s 0 ( X + t ) ∩ Y = ∅ . Theorem (Brendle-W., 2015, restated more explicitly) (ZFC) Let X ⊆ 2 ω with | X | = c . Then there is a Y ∈ s 0 with X + Y = 2 ω . Wolfgang Wohofsky (Universit¨ at Hamburg) No s 0 -shiftable sets TOPOSYM 2016 2 / 21

Theorem (Brendle-W., 2015) (ZFC) No set of reals of size continuum is “ s 0 -shiftable”. Definition A set Y ⊆ 2 ω is Marczewski null ( Y ∈ s 0 ) : ⇐ ⇒ for any perfect set P ⊆ 2 ω there is a perfect set Q ⊆ P with Q ∩ Y = ∅ . ⇐ ⇒ ∀ p ∈ S ∃ q ≤ p [ q ] ∩ Y = ∅ Definition A set X ⊆ 2 ω is s 0 -shiftable : ⇐ X + Y � = 2 ω ⇒ ∀ Y ∈ s 0 ∃ t ∈ 2 ω ⇐ ⇒ ∀ Y ∈ s 0 ( X + t ) ∩ Y = ∅ . Theorem (Brendle-W., 2015, restated more explicitly) (ZFC) Let X ⊆ 2 ω with | X | = c . Then there is a Y ∈ s 0 with X + Y = 2 ω . Wolfgang Wohofsky (Universit¨ at Hamburg) No s 0 -shiftable sets TOPOSYM 2016 2 / 21

Theorem (Brendle-W., 2015) (ZFC) No set of reals of size continuum is “ s 0 -shiftable”. Definition A set Y ⊆ 2 ω is Marczewski null ( Y ∈ s 0 ) : ⇐ ⇒ for any perfect set P ⊆ 2 ω there is a perfect set Q ⊆ P with Q ∩ Y = ∅ . ⇐ ⇒ ∀ p ∈ S ∃ q ≤ p [ q ] ∩ Y = ∅ Definition A set X ⊆ 2 ω is s 0 -shiftable : ⇐ X + Y � = 2 ω ⇒ ∀ Y ∈ s 0 ∃ t ∈ 2 ω ⇐ ⇒ ∀ Y ∈ s 0 ( X + t ) ∩ Y = ∅ . Theorem (Brendle-W., 2015, restated more explicitly) (ZFC) Let X ⊆ 2 ω with | X | = c . Then there is a Y ∈ s 0 with X + Y = 2 ω . Wolfgang Wohofsky (Universit¨ at Hamburg) No s 0 -shiftable sets TOPOSYM 2016 2 / 21

Strong measure zero For an interval I ⊆ R , let λ ( I ) denote its length. Definition (well-known) A set X ⊆ R is (Lebesgue) measure zero if for each positive real number ε > 0 there is a sequence of intervals ( I n ) n <ω of total length � n <ω λ ( I n ) ≤ ε such that X ⊆ � n <ω I n . Definition (Borel; 1919) A set X ⊆ R is strong measure zero if for each sequence of positive real numbers ( ε n ) n <ω there is a sequence of intervals ( I n ) n <ω with ∀ n ∈ ω λ ( I n ) ≤ ε n such that X ⊆ � n <ω I n . Wolfgang Wohofsky (Universit¨ at Hamburg) No s 0 -shiftable sets TOPOSYM 2016 3 / 21

Strong measure zero For an interval I ⊆ R , let λ ( I ) denote its length. Definition (well-known) A set X ⊆ R is (Lebesgue) measure zero if for each positive real number ε > 0 there is a sequence of intervals ( I n ) n <ω of total length � n <ω λ ( I n ) ≤ ε such that X ⊆ � n <ω I n . Definition (Borel; 1919) A set X ⊆ R is strong measure zero if for each sequence of positive real numbers ( ε n ) n <ω there is a sequence of intervals ( I n ) n <ω with ∀ n ∈ ω λ ( I n ) ≤ ε n such that X ⊆ � n <ω I n . Wolfgang Wohofsky (Universit¨ at Hamburg) No s 0 -shiftable sets TOPOSYM 2016 3 / 21

. . . -shiftables M σ -ideal of meager sets N σ -ideal of Lebesgue measure zero (“null”) sets σ -ideal of Marczewski null sets s 0 M -shiftable ⇐ ⇒ strong measure zero N -shiftable ⇐ ⇒ : strongly meager s 0 -shiftable only the countable sets are M -shiftable ⇐ ⇒ : BC only the countable sets are N -shiftable ⇐ ⇒ : dBC Thilo Weinert only the countable sets are s 0 -shiftable ⇐ ⇒ : MBC Wolfgang Wohofsky (Universit¨ at Hamburg) No s 0 -shiftable sets TOPOSYM 2016 4 / 21

. . . -shiftables M σ -ideal of meager sets N σ -ideal of Lebesgue measure zero (“null”) sets σ -ideal of Marczewski null sets s 0 M -shiftable ⇐ ⇒ strong measure zero N -shiftable ⇐ ⇒ : strongly meager s 0 -shiftable only the countable sets are M -shiftable ⇐ ⇒ : BC only the countable sets are N -shiftable ⇐ ⇒ : dBC Thilo Weinert only the countable sets are s 0 -shiftable ⇐ ⇒ : MBC Wolfgang Wohofsky (Universit¨ at Hamburg) No s 0 -shiftable sets TOPOSYM 2016 4 / 21

. . . -shiftables M σ -ideal of meager sets N σ -ideal of Lebesgue measure zero (“null”) sets σ -ideal of Marczewski null sets s 0 M -shiftable ⇐ ⇒ strong measure zero N -shiftable ⇐ ⇒ : strongly meager s 0 -shiftable only the countable sets are M -shiftable ⇐ ⇒ : BC only the countable sets are N -shiftable ⇐ ⇒ : dBC Thilo Weinert only the countable sets are s 0 -shiftable ⇐ ⇒ : MBC Wolfgang Wohofsky (Universit¨ at Hamburg) No s 0 -shiftable sets TOPOSYM 2016 4 / 21

. . . -shiftables M σ -ideal of meager sets N σ -ideal of Lebesgue measure zero (“null”) sets σ -ideal of Marczewski null sets s 0 M -shiftable ⇐ ⇒ strong measure zero N -shiftable ⇐ ⇒ : strongly meager s 0 -shiftable only the countable sets are M -shiftable ⇐ ⇒ : BC only the countable sets are N -shiftable ⇐ ⇒ : dBC Thilo Weinert only the countable sets are s 0 -shiftable ⇐ ⇒ : MBC Wolfgang Wohofsky (Universit¨ at Hamburg) No s 0 -shiftable sets TOPOSYM 2016 4 / 21

. . . -shiftables M σ -ideal of meager sets N σ -ideal of Lebesgue measure zero (“null”) sets σ -ideal of Marczewski null sets s 0 M -shiftable ⇐ ⇒ strong measure zero N -shiftable ⇐ ⇒ : strongly meager s 0 -shiftable only the countable sets are M -shiftable ⇐ ⇒ : BC only the countable sets are N -shiftable ⇐ ⇒ : dBC Thilo Weinert only the countable sets are s 0 -shiftable ⇐ ⇒ : MBC Wolfgang Wohofsky (Universit¨ at Hamburg) No s 0 -shiftable sets TOPOSYM 2016 4 / 21

. . . -shiftables M σ -ideal of meager sets N σ -ideal of Lebesgue measure zero (“null”) sets σ -ideal of Marczewski null sets s 0 M -shiftable ⇐ ⇒ strong measure zero N -shiftable ⇐ ⇒ : strongly meager s 0 -shiftable only the countable sets are M -shiftable ⇐ ⇒ : BC only the countable sets are N -shiftable ⇐ ⇒ : dBC Thilo Weinert only the countable sets are s 0 -shiftable ⇐ ⇒ : MBC Wolfgang Wohofsky (Universit¨ at Hamburg) No s 0 -shiftable sets TOPOSYM 2016 4 / 21

Wolfgang Wohofsky (Universit¨ at Hamburg) No s 0 -shiftable sets TOPOSYM 2016 5 / 21

Consistency of MBC Theorem (Brendle-W., 2015) (ZFC) No set of reals of size continuum is “ s 0 -shiftable”. Corollary CH implies MBC (i.e., s 0 -shiftables = [2 ω ] ≤ℵ 0 ). So what about larger continuum? Theorem (Brendle-W., 2015) In the Cohen model, MBC holds. Wolfgang Wohofsky (Universit¨ at Hamburg) No s 0 -shiftable sets TOPOSYM 2016 6 / 21

Consistency of MBC Theorem (Brendle-W., 2015) (ZFC) No set of reals of size continuum is “ s 0 -shiftable”. Corollary CH implies MBC (i.e., s 0 -shiftables = [2 ω ] ≤ℵ 0 ). So what about larger continuum? Theorem (Brendle-W., 2015) In the Cohen model, MBC holds. Wolfgang Wohofsky (Universit¨ at Hamburg) No s 0 -shiftable sets TOPOSYM 2016 6 / 21