F o r ing, Combinato ri s and De�nabilit y Summa ry: T o da y: De�nable W ello rders La rge a rdinals F o r ing axioms Ca rdinal ha ra teristi s (new!) Other ontexts T uesda y: Ca rdinal Cha ra teristi s on κ Rather new topi : Many op en questions Continuum fun tion 2 κ Dominating, b ounding numb ers Co�nalit y of the symmetri group Almost disjointness, splitt i ng numb ers W ednesda y: Mo dels of PF A , BPF A
De�nable W ello rders In ZF , A C is equivalent to: H ( κ + ) an b e w ello rdered fo r every κ When an w e obtain a de�nab le w ello rder of H ( κ + ) ? de�nab le w ello rder of H ( κ + ) : W ello rder of H ( κ + ) whi h is Σ n n de�nab le over H ( κ + ) with κ as a pa rameter Rema rks: 1. If n is at least 3, then κ an b e elimi nated, as { κ } is Π de�nab le 2 2. If λ is a limi t a rdinal and H ( κ + ) has a de�nab le w ello rder fo r o�nally many κ < λ , then H ( λ ) has a de�nab le w ello rder de�nab le w ello rder of H ( κ + ) with pa rameters : W ello rder of n Σ H ( κ + ) whi h is Σ de�nab le over H ( κ + ) with a rbitra ry elements of n H ( κ + ) as pa rameters Σ
De�nable W ello rders: La rge a rdinals and H ( ω 1 ) The b est situation: V = L → Ea h H ( κ + ) = L κ + has a Σ de�nab le w ello rder 1 De�nable w ello rders and La rge Ca rdinals H ( ω 1 ) de�nab le w ello rder of H ( ω (with pa rameters) ∼ n 1 ) 1 de�nab le w ello rder of the reals (with real pa rameters) n + 1 Theo rem 1 (Mans�eld) Σ w ello rder of the reals → every real b elongs to L. 2 1 (Ma rtin-Steel) A Σ w ello rder of the reals is onsistent with n n + 2 W o o din a rdinals but in onsistent with n W o o din a rdinals and a measurable a rdinal ab ove them. Σ Σ
De�nable W ello rders: La rge Ca rdinals and H ( ω 2 ) H ( ω 2 ) A fo r ing is small i� it has size less than the least ina essible. Small fo r ings p reserve all la rge a rdinals. Theo rem (Asp er�-F) There is a small fo r ing whi h fo r es CH and a de�nab le w ello rder of H ( ω . 2 ) The ab ove w ello rder is not Σ . In fa t: 1 Theo rem (W o o din) Measurable W o o din a rdinal + CH → there is no w ello rder of the reals whi h is Σ over H ( ω . 1 2 ) Ho w ever:
De�nable W ello rders: La rge a rdinals and H ( ω 2 ) Theo rem (A vraham-Shelah) There is a small fo r ing whi h fo r es ∼ CH and a w ello rder of the reals whi h is Σ over H ( ω . 1 2 ) Question 1. Is there a small fo r ing whi h fo r es a Σ w ello rder of 2 H ( ω ? 2 )
De�nable W ello rders: La rge a rdinals and H ( ω 2 ) Ab out the p ro of of: Theo rem (Asp er�-F) There is a small fo r ing whi h fo r es CH and a de�nab le w ello rder of H ( ω . 2 ) T w o ingredients: Canoni al fun tion o ding Strongly t yp e-guessing o ding (Asp er�)
De�nable W ello rders: La rge a rdinals and H ( ω 2 ) Canoni al funtion o ding F o r ea h α < ω ho ose f α : ω onto and de�ne g α : ω 2 1 → α 1 → ω 1 b y: g α ( γ ) = o rdert yp e f α [ γ ] . g α is a � anoni al fun tion� fo r α . No w o de A ⊆ ω b y B ⊆ ω as follo ws: 2 1 A i� g α ( γ ) ∈ B fo r a lub of γ Assuming GCH , the fo r ing to do this is ω -strategi all y losed and - . 2 α ∈ ω
De�nable W ello rders: La rge a rdinals and H ( ω 2 ) Asp er� o ding A lub-sequen e in ω of height τ is a sequen e � C = ( C δ | δ ∈ S ) 1 where S ⊆ ω is stationa ry and ea h C δ is lub in δ of o rdert yp e τ . 1 C is strongly t yp e-guessing i� fo r every lub C ⊆ ω there is a lub 1 D ⊆ ω su h that fo r all δ in D ∩ S , o rdert yp e ( C ∩ C + , where 1 C + denotes the set of su esso r elements of C δ . An o rdinal γ is p erfe t i� ω γ = γ. Lemma (Asp er�) Assume GCH . Let B ⊆ ω . Then there is an 1 -strategi all y losed, ω - fo r ing that fo r es: γ ∈ B i� the γ -th 2 p erfe t o rdinal is the height of a strongly t yp e-guessing lub � sequen e. δ ) = τ δ ω
De�nable W ello rders: La rge a rdinals and H ( ω 2 ) T o p rove: Theo rem (Asp er�-F) There is a small fo r ing whi h fo r es CH and a de�nab le w ello rder of H ( ω . 2 ) Assume GCH . W rite H ( ω as L ω A ] , A ⊆ ω . 2 ) 2 2 [ Use Canoni al fun tion o ding to o de A b y B ⊆ ω . 1 Use Asp er� o ding to o de B de�nab ly over H ( ω . 2 ) Problem: B only o des H ( ω of the ground mo del, not H ( ω of 2 ) 2 ) the extension! Solution: P erfo rm b oth o dings �simult aneously�. The fo r ing is a hyb rid fo r ing: halfw a y b et w een iteration and p ro du t.
De�nable W ello rders: La rge a rdinals and H ( κ ) H ( κ ) Theo rem (Asp er�-F) There is a lass fo r ing whi h fo r es GCH , p reserves all sup er ompa t a rdinals (as w ell as a p rop er lass of n -huge a rdinals fo r ea h n ) and adds a de�nab le w ello rder of H ( κ + ) fo r all regula r κ ≥ ω . 1 Co rolla ry There is a lass fo r ing whi h fo r es GCH , p reserves all sup er ompa t a rdinals (as w ell as a p rop er lass of n -huge a rdinals fo r ea h n ) and adds a pa rameter-free de�nab le w ello rder of H ( δ ) fo r all a rdinals δ ≥ ω whi h a re not su esso rs of 2 singula rs. Su esso rs of singula rs? Σ de�nab le w ello rders? 1
De�nable W ello rders: La rge a rdinals and H ( κ ) Su esso rs of singula rs: Theo rem (Asp er�-F) Supp ose that there is a j : L ( H ( λ + )) → L ( H ( λ + )) �xing λ , with riti al p oint < λ . Then there is no de�nab le w ello rder of H ( λ + ) with pa rameters. Question 2. Is there a small fo r ing that adds a de�nab le w ello rder of H ( ℵ ω + 1 ) with pa rameters? de�nab le w ello rders: 1 Theo rem There is a lass fo r ing whi h fo r es GCH , p reserves all sup er ompa t a rdinals (as w ell as a p rop er lass of n -huge a rdinals fo r ea h n ) and adds a Σ de�nab le w ello rder of H ( κ + ) 1 with pa rameters fo r all regula r κ ≥ ω . 1 Σ
De�nable W ello rders and F o r ing Axioms Question 3. Is there a small fo r ing that adds a Σ de�nab le 1 w ello rder of H ( ω ? 3 ) De�nable w ello rders and F o r ing Axioms H ( ω 1 ) Theo rem 1 MA is onsistent with a Σ w ello rder of the reals. 3 L (Cai edo-F) BPF A + ω (whi h is onsistent relative to a 1 = ω 1 1 re�e ting a rdinal) implies that there is a Σ w ello rder of the reals. 3 Theo rem (Hjo rth) Assume ∼ CH and every real has a # . Then there is no 1 w ello rder of the reals. 3 Σ
De�nable W ello rders and F o r ing Axioms Question 4. Do es BPF A + 0 # do es not exist imply that there is a 1 w ello rder of the reals? 3 Question 5. Is BMM onsistent with a p roje tive w ello rder of the reals? PF A is not. Question 6. Is MA onsistent with the nonexisten e of a p roje tive w ello rder of the reals? F o r H ( ω : 2 ) Theo rem L (Cai edo-V Σ eli k ovi ) BPF A + ω implies that there is a Σ 1 = ω 1 1 de�nab le w ello rder of H ( ω . 2 ) Theo rem (La rson) Relative to enough sup er ompa ts, there is a mo del of MM with a de�nab le w ello rder of H ( ω . 2 )
De�nable W ello rders and F o r ing Axioms F o r la rger H ( κ ) : Theo rem MA is onsistent with a de�nab le w ello rder of H ( κ + ) fo r all κ . (Re�e ting a rdinal) BSPF A is onsistent with a de�nab le w ello rder of H ( κ + ) fo r all κ . (Enough sup er ompa ts) MM is onsistent with a de�nab le w ello rder of H ( κ + ) fo r all regula r κ ≥ ω . 1
De�nable W ello rders and Ca rdinal Cha ra teris ti s New ontext fo r de�nab le w ello rders: Ca rdinal Cha ra teristi s T emplate iteration T : A ountable supp o rt, ω - iteration whi h 2 1 adds a Σ w ello rder of the reals (and a Σ w ello rder of H ( ω ). It 3 1 2 ) is not p rop er, but is S -p rop er fo r ertain stationa ry S ⊆ ω . 1 Broad p roje t: Mix the template iteration with a va riet y of iterations fo r ontrolling a rdinal ha ra teristi s. Theo rem 1 (V.Fis her - F) Ea h of the follo wing is onsistent with a Σ 3 w ello rder of the reals: d < c , b < a = s , b < g . the b ounding numb er, a = the almost disjointness numb er, the splitt i ng numb er, g = the group wise densit y numb er b = s =
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