for uncountable Paul Larson Miami University September 26, 2015 T - PowerPoint PPT Presentation

T RIVIALITY T URZANKI ’ S Q UESTION P ARTIAL T RIVIALITY F IXED P OINTS U NIFORMIZATION Automorphisms of P ( λ ) / I κ for λ uncountable Paul Larson Miami University September 26, 2015

T RIVIALITY T URZANKI ’ S Q UESTION P ARTIAL T RIVIALITY F IXED P OINTS U NIFORMIZATION joint with Paul McKenney

T RIVIALITY T URZANKI ’ S Q UESTION P ARTIAL T RIVIALITY F IXED P OINTS U NIFORMIZATION Given an infinite cardinal κ , we let I κ denote the ideal of sets of cardinality less than κ . Fin = I ℵ 0 ; Ctble = I ℵ 1 Given cardinals κ ≤ λ and A ⊆ λ , we let [ A ] λ,κ = { B ⊆ λ | | A △ B | < κ }

T RIVIALITY T URZANKI ’ S Q UESTION P ARTIAL T RIVIALITY F IXED P OINTS U NIFORMIZATION A function π : P ( λ ) / I κ → P ( χ ) / I ρ is said to be trivial on A ⊆ λ if there exist B ∈ [ λ ] <κ and f : A \ B → χ such that π ([ C ] λ,κ ) = [ f [ C \ B ]] χ,ρ for all C ⊆ A . The function π is trivial if it is trivial on λ .

T RIVIALITY T URZANKI ’ S Q UESTION P ARTIAL T RIVIALITY F IXED P OINTS U NIFORMIZATION Automorphisms of P ( ω ) / Fin Theorem. (W. Rudin, 1956) Assuming CH there exist 2 ℵ 1 many nontrivial automorphisms of P ( ω ) / Fin . Theorem. (Shelah, late 1970’s?) Consistently, all automorphisms of P ( ω ) / Fin are trivial. Theorem. (Velickovic, 1993) Assuming PFA (MA ℵ 1 + OCA for λ ≤ ℵ 1 ) all automorphisms of P ( λ ) / Fin are trivial, for all cardinals λ .

T RIVIALITY T URZANKI ’ S Q UESTION P ARTIAL T RIVIALITY F IXED P OINTS U NIFORMIZATION Question 1. Is every automorphism of P ( λ ) / Fin trivial on a cocountable set, for every uncountable cardinal λ ? Question 1a. Is every automorphism of P ( λ ) / Fin trivial on an uncountable set, for every uncountable cardinal λ ? Question 2. Is every automorphism of P ( λ ) / I κ trivial, whenever κ ≤ λ are uncountable? Question 3. Must an automorphism of P ( λ ) / Fin be trivial if it is trivial on all countable sets (or all sets of cardinality ℵ 1 ), for every uncountable cardinal λ ?

T RIVIALITY T URZANKI ’ S Q UESTION P ARTIAL T RIVIALITY F IXED P OINTS U NIFORMIZATION A partial result on Questions 1 and 1a ans) If λ > 2 ℵ 0 is less than the first Theorem. (Shelah-Stepr¯ strongly inaccessible cardinal, then every automorphism of P ( λ ) / Fin is trivial on a subset of λ with complement of cardinality 2 ℵ 0 . So : Question 1a has a positive answer for λ > 2 ℵ 0 .

T RIVIALITY T URZANKI ’ S Q UESTION P ARTIAL T RIVIALITY F IXED P OINTS U NIFORMIZATION Question 4. (Turzanski/Katowice) Is is consistent that P ( ω ) / Fin and P ( ω 1 ) / Fin are isomorphic? Theorem. (Balcar-Frankiewicz) If λ and κ are distinct cardinals such that P ( κ ) / Fin ≃ P ( λ ) / Fin, then { κ, λ } = { ω, ω 1 } . One step of the proof shows that (**) if P ( ω ) / Fin and P ( ω 1 ) / Fin are isomorphic then d = ℵ 1 (so MA ℵ 1 fails).

T RIVIALITY T URZANKI ’ S Q UESTION P ARTIAL T RIVIALITY F IXED P OINTS U NIFORMIZATION The same proof shows that if κ ≤ µ < λ are cardinals (with κ regular) such that P ( µ ) / I κ ≃ P ( λ ) / I κ then { µ, λ } = { κ, κ + } .

T RIVIALITY T URZANKI ’ S Q UESTION P ARTIAL T RIVIALITY F IXED P OINTS U NIFORMIZATION (Folklore) Suppose that P ( ω 1 ) / Fin and P ( ω ) / Fin are isomorphic, and consider the automorphism (call it π ) of P ( ω 1 ) / Fin conjugate to the shift on ω . It has no nontrivial fixed points, which shows that it is not cocountably trivial. Moreover, it has the property that for no (nontrivial) A ⊆ ∗ B is π ([ A ]) = [ B ] . (It has no nontrivial expanding points.)

T RIVIALITY T URZANKI ’ S Q UESTION P ARTIAL T RIVIALITY F IXED P OINTS U NIFORMIZATION Question 5. If π is an automorphism of P ( ω 1 ) / Fin, must there be an infinite, coinfinite A ⊂ ω 1 such that π ([ A ] Fin ) = [ A ] Fin ? That is, must π have a fixed point? Question 6. If π is an automorphism of P ( ω 1 ) / Fin, must there be infinite, coinfinite A ⊆ B ⊂ ω 1 such that π ([ A ] Fin ) = [ B ] Fin ? That is, must π have an expanding point?

T RIVIALITY T URZANKI ’ S Q UESTION P ARTIAL T RIVIALITY F IXED P OINTS U NIFORMIZATION Theorem. (Hart) If P ( ω ) / Fin and P ( ω 1 ) / Fin are isomorphic then there is a nontrivial automorphism of P ( ω ) / Fin. Proof: Break ω 1 into Z -chains and consider the automorphism of P ( ω ) / Fin induced by shifting the chains. It has uncountably many minimal disjoint fixed points.

T RIVIALITY T URZANKI ’ S Q UESTION P ARTIAL T RIVIALITY F IXED P OINTS U NIFORMIZATION Question 7. Can there be an isomorphism from P ( ω 1 ) / Fin to P ( ω ) / Fin which is trivial on all countable sets?

T RIVIALITY T URZANKI ’ S Q UESTION P ARTIAL T RIVIALITY F IXED P OINTS U NIFORMIZATION Preserving cardinalities A function π : P ( λ ) / I κ → P ( χ ) / I ρ is said to be cardinality preserving if for each A ⊆ λ there exists a B ⊆ χ such that | A | = | B | and π ([ A ] λ,κ ) = [ B ] χ,ρ . For any pairs of cardinals κ < λ with κ regular, the existence of an isomorphism between P ( κ + ) / I κ and P ( κ ) / I κ is equivalent to the existence of an automorphism of P ( λ ) / I κ which is not cardinality preserving.

T RIVIALITY T URZANKI ’ S Q UESTION P ARTIAL T RIVIALITY F IXED P OINTS U NIFORMIZATION Selectors A selector for a function π : P ( λ ) / I κ → P ( χ ) / I ρ is a function π : P ( λ ) → P ( χ ) ˆ π ( A )] χ,ρ for all A ⊆ λ . such that π ([ A ] λ,κ ) = [ˆ

T RIVIALITY T URZANKI ’ S Q UESTION P ARTIAL T RIVIALITY F IXED P OINTS U NIFORMIZATION Lemma 1. Let κ < µ ≤ λ be infinite cardinals, with κ regular, and let ˆ π be a selector for a cardinality preserving automorphism π of P ( λ ) / I κ . Define π µ on P ( λ ) / I µ by setting π µ ([ A ] λ,µ ) = [ˆ π ( A )] λ,µ . Then π µ is an automorphism of P ( λ ) / I µ . If π is not trivial on a set with compliment in I µ , then π µ is nontrivial.

T RIVIALITY T URZANKI ’ S Q UESTION P ARTIAL T RIVIALITY F IXED P OINTS U NIFORMIZATION Theorem 1. Let κ ≤ µ be infinite cardinals, with κ is regular. Then every automorphism of P ( 2 µ ) / I κ which is trivial on all sets of size µ + is trivial. In the case κ = µ = ω : every automorphism of P ( 2 ℵ 0 ) / Fin which is trivial on all sets of cardinality at most ℵ 1 is trivial.

T RIVIALITY T URZANKI ’ S Q UESTION P ARTIAL T RIVIALITY F IXED P OINTS U NIFORMIZATION So MA ℵ 1 + OCA, which implies 2 ℵ 0 = ℵ 2 and (by Velickovic) that all automorphisms of P ( ω 1 ) / Fin are trivial, implies (by Shelah-Stepr¯ ans) that all automorphisms of P ( λ ) / Fin are trivial, for all λ below the least strongly inaccessible cardinal.

T RIVIALITY T URZANKI ’ S Q UESTION P ARTIAL T RIVIALITY F IXED P OINTS U NIFORMIZATION Proof of Theorem 1. Let κ ≤ µ be infinite cardinals, and suppose that π is an automorphism of P ( 2 µ ) / I κ . π be a bijective selector for π , and let � x β : β < 2 µ � list P ( µ ) . Let ˆ For each γ < µ , let R γ = { β < 2 µ | γ ∈ x β } . For each α < 2 µ , let y α = { γ < µ | α ∈ ˆ π − 1 ( R γ ) } . Finally, for each α < 2 µ , let h ( α ) = β if y α = x β .

T RIVIALITY T URZANKI ’ S Q UESTION P ARTIAL T RIVIALITY F IXED P OINTS U NIFORMIZATION For each a ∈ [ 2 µ ] ≤ µ + , let f a be a trivializing function on a . Then | ( h ↾ a ) △ f a | ≤ µ.

T RIVIALITY T URZANKI ’ S Q UESTION P ARTIAL T RIVIALITY F IXED P OINTS U NIFORMIZATION Assuming that π is not trivial, there exist pairwise disjoint a α ( α < µ + ) of cardinality at most κ + such that, for each α , | ( h ↾ a α ) △ f a α | ≥ κ. Let a = � α<µ + a α . Then | ( h ↾ a ) △ f a | ≤ µ and, for all α < µ + , | ( f a ↾ a α ) △ f a α | < κ.

T RIVIALITY T URZANKI ’ S Q UESTION P ARTIAL T RIVIALITY F IXED P OINTS U NIFORMIZATION In fact, using the fact that every automorphism of P ( λ ) / I κ is determined by how it acts on sets of size κ , one can show Theorem 1’ . For any infinite cardinal µ , every automorphism of P ( 2 µ ) / I µ + which is trivial on all sets of size µ + is trivial.

T RIVIALITY T URZANKI ’ S Q UESTION P ARTIAL T RIVIALITY F IXED P OINTS U NIFORMIZATION What about automorphisms of P ( 2 ℵ 0 ) / Fin which are trivial on countable sets?

T RIVIALITY T URZANKI ’ S Q UESTION P ARTIAL T RIVIALITY F IXED P OINTS U NIFORMIZATION Given Γ ⊆ P ( 2 ω ) , we let CSN (Γ) be the smallest cardinality of a family F ⊆ ( 2 ω ) ω × ( 2 ω ) ω such that 1. for every ( f , g ) ∈ F , { f ( n ) : n < ω } ∪ { g ( n ) : n < ω } is dense in 2 ω , 2. for all pairs ( f , g ) , ( f ′ , g ′ ) from F , if g � = g ′ , then { g ( n ) : n < ω } ∩ { g ′ ( n ) : n < ω } = ∅ , 3. for every ( f , g ) ∈ F and n < ω , f ( n ) � = g ( n ) , and 4. for every set A ∈ Γ , the set { ( f , g ) ∈ F : ∃ ∞ n < ω | A ∩ { f ( n ) , g ( n ) }| = 1 } has cardinality smaller than that of F , if such a family F exists. If no such family exists, we set CSN (Γ) = ( 2 ℵ 0 ) + .

T RIVIALITY T URZANKI ’ S Q UESTION P ARTIAL T RIVIALITY F IXED P OINTS U NIFORMIZATION CSN ( open ) ≥ cov ( Meager ) Theorem 2 . If CSN ( Borel ) > ℵ 1 , then every cardinality preserving automorphism of P ( 2 ℵ 0 ) / Fin which is trivial on all countable sets is trivial. By Theorem 1, it suffices to show this for automorphisms of P ( ω 1 ) / Fin.