Some topics related to bounding by canonical functions Sean Cox - PowerPoint PPT Presentation

Some topics related to bounding by canonical functions Sean Cox Institute for mathematical logic and foundational research University of M¨ unster (Germany) sean.cox@uni-muenster.de wwwmath.uni-muenster.de/logik/Personen/Cox April 30, 2012 Sean Cox Bounding by canonical functions

Outline The partial order ( κ ORD , ≤ I ) and canonical functions 1 Self-generic structures (“antichain catching”) 2 How antichain catching is related to bounding by canonical 3 functions Forcing Axioms vs. nice ideals on ω 2 4 Sean Cox Bounding by canonical functions

The partial order ≤ I on κ ORD Let κ be regular, uncountable and I ⊂ ℘ ( κ ) a normal ideal. e.g. I := NS κ ; or I := NS ↾ S for some stationary S ⊂ κ . Define ≤ I on κ ORD by: f ≤ I g ⇐ ⇒ { α < κ | f ( α ) ≤ g ( α ) } ∈ Dual ( I ) Sean Cox Bounding by canonical functions

The partial order ≤ I on κ ORD Let κ be regular, uncountable and I ⊂ ℘ ( κ ) a normal ideal. e.g. I := NS κ ; or I := NS ↾ S for some stationary S ⊂ κ . Define ≤ I on κ ORD by: f ≤ I g ⇐ ⇒ { α < κ | f ( α ) ≤ g ( α ) } ∈ Dual ( I ) ≤ I is wellfounded Sean Cox Bounding by canonical functions

Canonical functions on κ Definition (Canonical functions on κ ) By recursion: h ν : ≃ the ≤ NS κ -least upper bound of � h µ | µ < ν � ( if such a l.u.b. exists) View each h ν as an equivalence class in κ ORD / = NS κ . Sean Cox Bounding by canonical functions

Canonical functions on κ Definition (Canonical functions on κ ) By recursion: h ν : ≃ the ≤ NS κ -least upper bound of � h µ | µ < ν � ( if such a l.u.b. exists) View each h ν as an equivalence class in κ ORD / = NS κ . The “first few” (i.e. for ν < κ + ); these all map into κ : h 0 : α �→ 0 h ν +1 : α �→ h ν ( α ) + 1 For limit ν < κ + : h ν can be defined from earlier ones using sups or diagonal sups Sean Cox Bounding by canonical functions

Canonical functions on κ Definition (Canonical functions on κ ) By recursion: h ν : ≃ the ≤ NS κ -least upper bound of � h µ | µ < ν � ( if such a l.u.b. exists) View each h ν as an equivalence class in κ ORD / = NS κ . The “first few” (i.e. for ν < κ + ); these all map into κ : h 0 : α �→ 0 h ν +1 : α �→ h ν ( α ) + 1 For limit ν < κ + : h ν can be defined from earlier ones using sups or diagonal sups Theorem (Jech-Shelah; Hajnal) Existence of h κ + is independent of ZFC. Sean Cox Bounding by canonical functions

Canonical functions and ultrapowers κ + V κ V Sean Cox Bounding by canonical functions

Canonical functions and ultrapowers Let U ⊂ P ( κ ) be normal w.r.t. V κ + V κ V Sean Cox Bounding by canonical functions

Canonical functions and ultrapowers Let U ⊂ P ( κ ) be normal w.r.t. V Possibly U / ∈ V : e.g. κ = ω 1 and U is any V -generic for ( ℘ ( ω 1 ) / NS ω 1 , ⊂ NS ω κ + V κ V Sean Cox Bounding by canonical functions

Canonical functions and ultrapowers Let U ⊂ P ( κ ) be normal w.r.t. V Possibly U / ∈ V : e.g. κ = ω 1 and U is any V -generic for ( ℘ ( ω 1 ) / NS ω 1 , ⊂ NS ω Or possibly U ∈ V ; e.g. if κ is a measurable cardinal in V κ + V κ V Sean Cox Bounding by canonical functions

Canonical functions and ultrapowers Let U ⊂ P ( κ ) be normal w.r.t. V Possibly U / ∈ V : e.g. κ = ω 1 and U is any V -generic for ( ℘ ( ω 1 ) / NS ω 1 , ⊂ NS ω Or possibly U ∈ V ; e.g. if κ is a measurable cardinal in V ? κ + V ? κ ult ( V , U ) V Sean Cox Bounding by canonical functions

Canonical functions and ultrapowers Let U ⊂ P ( κ ) be normal w.r.t. V Possibly U / ∈ V : e.g. κ = ω 1 and U is any V -generic for ( ℘ ( ω 1 ) / NS ω 1 , ⊂ NS ω Or possibly U ∈ V ; e.g. if κ is a measurable cardinal in V ? κ + V ? ν [ h ν ] U κ ult ( V , U ) V Sean Cox Bounding by canonical functions

Other characterizations of the first κ + canonical functions Could have equivalently used ≤ I for any normal ideal I ⊂ ℘ ( κ ) Sean Cox Bounding by canonical functions

Other characterizations of the first κ + canonical functions Could have equivalently used ≤ I for any normal ideal I ⊂ ℘ ( κ ) Non-recursive characterizations of h ν (for ν < κ + ): “the” function which represents ν in any generic ultrapower by a normal ideal on κ Fix any surjection g ν : κ → ν and set h ν ( α ) := otp ( g ′′ ν α ) Fix any wellorder ∆ of H κ + and set h ν ( α ) : ≃ otp ( M ∩ ν ) for any M ≺ ( H κ + , ∈ , ∆ , { ν } ) such that α = M ∩ κ Sean Cox Bounding by canonical functions

Bounding by canonical functions Definition For a normal ideal I ⊂ ℘ ( κ ), Bound ( I ) means that { h ν | ν < κ + } is cofinal in ( κ κ, ≤ I ). Sean Cox Bounding by canonical functions

Bounding by canonical functions Definition For a normal ideal I ⊂ ℘ ( κ ), Bound ( I ) means that { h ν | ν < κ + } is cofinal in ( κ κ, ≤ I ). Lemma Suppose κ is a successor cardinal. Bound ( I ) implies that if U is an ultrafilter on V ∩ ℘ ( κ ) such that: U is normal w.r.t. sequences from V U extends the dual of I and j : V → U ult ( V , U ) is the ultrapower embedding, then j ( κ ) = κ + V . Sean Cox Bounding by canonical functions

Bounding by canonical functions Definition For a normal ideal I ⊂ ℘ ( κ ), Bound ( I ) means that { h ν | ν < κ + } is cofinal in ( κ κ, ≤ I ). Lemma Suppose κ is a successor cardinal. Bound ( I ) implies that if U is an ultrafilter on V ∩ ℘ ( κ ) such that: U is normal w.r.t. sequences from V U extends the dual of I and j : V → U ult ( V , U ) is the ultrapower embedding, then j ( κ ) = κ + V . One can always obtain such a U (even if κ is a successor cardinal) by forcing with P I := ( P ( κ ) / I , ⊆ I ). Sean Cox Bounding by canonical functions

Assuming κ is successor, Bound ( I ), and U ⊃ Dual ( I ): κ + V κ + V κ ult ( V , U ) V Sean Cox Bounding by canonical functions

Saturation implies bounding Definition Let I be a normal ideal on κ . I is saturated iff P I := ( ℘ ( κ ) / I , ⊆ I ) has the κ + -cc. Lemma (folklore) If I is saturated then Bound ( I ) holds. Sean Cox Bounding by canonical functions

Saturation implies bounding κ + -cc of P I (and that κ is a successor cardinal) implies G ( κ ) = κ + V � P I j ˙ Then for every f : κ → κ : D f := { S ∈ I + | ∃ ν < κ + f < h ν on S } is dense in P I For each S ∈ D f pick a ν S < κ + such that f < h ν S on S Let A f ⊂ D f be a maximal antichain. Set µ := sup { ν S | S ∈ A f } ; µ < κ + by κ + -cc of P I . Maximality of A f implies that f ≤ I h µ . Sean Cox Bounding by canonical functions

♦ implies failure of Bounding Lemma (folklore?) ♦ κ = ⇒ ¬ Bound ( NS κ ) Sean Cox Bounding by canonical functions

♦ implies failure of Bounding Lemma (folklore?) ♦ κ = ⇒ ¬ Bound ( NS κ ) Suppose � A α | α < κ � is a ♦ κ sequence, p : κ × κ ↔ bij κ , and � otp ( A α ) if A α codes a wellorder (via p ↾ ( α × α ) ) f ( α ) := 0 otherwise Fix ν < κ + . Fix b ⊂ κ coding ν . b ∩ α = A α for stationarily many α otp ( b ∩ α ) = h ν ( α ) for club-many α So f ( α ) = h ν ( α ) for stationarily many α . So f ≮ NS h ν Sean Cox Bounding by canonical functions

Chang’s Conjecture and bounding Lemma ( κ + , κ ) ։ ( κ, < κ ) implies a weak variation of Bound ( NS κ ) . Sean Cox Bounding by canonical functions

Bound ( NS ω 1 ) is well-understood Theorem (Larson-Shelah; Deiser-Donder) The following are equiconsistent: ZFC + Bound ( NS ω 1 ) ZFC + there is an inaccessible limit of measurable cardinals Moreover, saturation of NS ω 1 (which implies Bound ( NS ω 1 )) is known to be consistent relative to a Woodin cardinal (Shelah). Sean Cox Bounding by canonical functions

What about Bound ( NS ω 2 )? NOTATION: S m n := ω m ∩ cof ( ω n ) Theorem (Shelah) Suppose I is a normal ideal on ω 2 such that S 2 0 ∈ I + . Then I is not saturated. In particular, NS ω 2 is never saturated. Theorem (Woodin; building on work of Kunen and Magidor) It is consistent relative to an almost huge cardinal that there is some stationary S ⊆ S 2 1 such that NS ω 2 ↾ S is saturated. (Recall this implies Bound ( NS ω 2 ↾ S ) ) Question (Well-known open problems) 1 Can NS ω 2 ↾ S 2 1 be saturated? 2 Can Bound ( NS ω 2 ) hold? What about Bound ( NS ω 2 ↾ S 2 1 ) ? Sean Cox Bounding by canonical functions

Big gap in known consistency bounds Question What is the consistency strength of: “Bound ( I ) holds for some normal ideal I ⊂ ℘ ( ω 2 ) ”? Best known upper bound: almost huge cardinal (Kunen, Magidor, Woodin) Best known lower bound (even assuming that F = NS ω 2 ): inaccessible limit of measurables ! (Deiser-Donder) Sean Cox Bounding by canonical functions

Big gap in known consistency bounds Question What is the consistency strength of: “Bound ( I ) holds for some normal ideal I ⊂ ℘ ( ω 2 ) ”? Best known upper bound: almost huge cardinal (Kunen, Magidor, Woodin) Best known lower bound (even assuming that F = NS ω 2 ): inaccessible limit of measurables ! (Deiser-Donder) Lower bound for Bound ( ω 2 ) hasn’t even escaped “easy” inner model theory. Sean Cox Bounding by canonical functions