Strongly + -cc forcing Generalising MA Our work Mirna D zamonja, - PowerPoint PPT Presentation

κ + -cc and ( < κ ) -closure Strongly κ + -cc forcing Other axioms of the above type were discovered by Mirna Dˇ zamonja, Tutorial 3, Baumgartner (1974), Shelah in several papers and including joint work with J. Cummings Cumming, Dˇ z., Magidor, Morgan and Shelah (2017). and I. Neeman Generalising MA Our work

κ + -cc and ( < κ ) -closure Strongly κ + -cc forcing Other axioms of the above type were discovered by Mirna Dˇ zamonja, Tutorial 3, Baumgartner (1974), Shelah in several papers and including joint work with J. Cummings Cumming, Dˇ z., Magidor, Morgan and Shelah (2017). and I. Neeman Generalising MA No popular analogue of properness, in spite of some Our work known work, notably by Roslanowski and Shelah.

κ + -cc and ( < κ ) -closure Strongly κ + -cc forcing Other axioms of the above type were discovered by Mirna Dˇ zamonja, Tutorial 3, Baumgartner (1974), Shelah in several papers and including joint work with J. Cummings Cumming, Dˇ z., Magidor, Morgan and Shelah (2017). and I. Neeman Generalising MA No popular analogue of properness, in spite of some Our work known work, notably by Roslanowski and Shelah. Recall what I said yesterday about Itay’s new way of seeing iterations, where he has discovered weak analogues of properness for ℵ 2 .

κ + -cc and ( < κ ) -closure Strongly κ + -cc forcing Other axioms of the above type were discovered by Mirna Dˇ zamonja, Tutorial 3, Baumgartner (1974), Shelah in several papers and including joint work with J. Cummings Cumming, Dˇ z., Magidor, Morgan and Shelah (2017). and I. Neeman Generalising MA No popular analogue of properness, in spite of some Our work known work, notably by Roslanowski and Shelah. Recall what I said yesterday about Itay’s new way of seeing iterations, where he has discovered weak analogues of properness for ℵ 2 . Our motivation has been not to generalise properness but to generalise ccc by using non-combinatorial methods, which then give an easily checkable condition.

κ + -cc and ( < κ ) -closure Strongly κ + -cc forcing Other axioms of the above type were discovered by Mirna Dˇ zamonja, Tutorial 3, Baumgartner (1974), Shelah in several papers and including joint work with J. Cummings Cumming, Dˇ z., Magidor, Morgan and Shelah (2017). and I. Neeman Generalising MA No popular analogue of properness, in spite of some Our work known work, notably by Roslanowski and Shelah. Recall what I said yesterday about Itay’s new way of seeing iterations, where he has discovered weak analogues of properness for ℵ 2 . Our motivation has been not to generalise properness but to generalise ccc by using non-combinatorial methods, which then give an easily checkable condition.

κ + -cc and ( < κ ) -closure Strongly κ + -cc forcing Other axioms of the above type were discovered by Mirna Dˇ zamonja, Tutorial 3, Baumgartner (1974), Shelah in several papers and including joint work with J. Cummings Cumming, Dˇ z., Magidor, Morgan and Shelah (2017). and I. Neeman Generalising MA No popular analogue of properness, in spite of some Our work known work, notably by Roslanowski and Shelah. Recall what I said yesterday about Itay’s new way of seeing iterations, where he has discovered weak analogues of properness for ℵ 2 . Our motivation has been not to generalise properness but to generalise ccc by using non-combinatorial methods, which then give an easily checkable condition. I will now report on a preprint with Cummings and Neeman, which is available on the arxiv.

κ + -cc and ( < κ ) -closure Strongly κ + -cc forcing Other axioms of the above type were discovered by Mirna Dˇ zamonja, Tutorial 3, Baumgartner (1974), Shelah in several papers and including joint work with J. Cummings Cumming, Dˇ z., Magidor, Morgan and Shelah (2017). and I. Neeman Generalising MA No popular analogue of properness, in spite of some Our work known work, notably by Roslanowski and Shelah. Recall what I said yesterday about Itay’s new way of seeing iterations, where he has discovered weak analogues of properness for ℵ 2 . Our motivation has been not to generalise properness but to generalise ccc by using non-combinatorial methods, which then give an easily checkable condition. I will now report on a preprint with Cummings and Neeman, which is available on the arxiv. Let κ satisfy κ = κ <κ , say κ ≥ ℵ 1 .

κ + -cc and ( < κ ) -closure Strongly κ + -cc forcing Other axioms of the above type were discovered by Mirna Dˇ zamonja, Tutorial 3, Baumgartner (1974), Shelah in several papers and including joint work with J. Cummings Cumming, Dˇ z., Magidor, Morgan and Shelah (2017). and I. Neeman Generalising MA No popular analogue of properness, in spite of some Our work known work, notably by Roslanowski and Shelah. Recall what I said yesterday about Itay’s new way of seeing iterations, where he has discovered weak analogues of properness for ℵ 2 . Our motivation has been not to generalise properness but to generalise ccc by using non-combinatorial methods, which then give an easily checkable condition. I will now report on a preprint with Cummings and Neeman, which is available on the arxiv. Let κ satisfy κ = κ <κ , say κ ≥ ℵ 1 . Consider elementary submodels M ≺ H ( χ ) = � H ( χ ) , ∈ , < ∗ � with | M | = κ , <κ M ⊆ M , P =the forcing in question ∈ M .

Strongly κ + -cc Strongly κ + -cc forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings and I. Neeman Definition Generalising MA q ∈ P is strongly ( M , P ) -generic if Our work

Strongly κ + -cc Strongly κ + -cc forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings and I. Neeman Definition Generalising MA q ∈ P is strongly ( M , P ) -generic if for every r ≥ q , there is Our work a residue r | M ∈ M , such that any s ≥ r | M with s ∈ M , is compatible with r .

Strongly κ + -cc Strongly κ + -cc forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings and I. Neeman Definition Generalising MA q ∈ P is strongly ( M , P ) -generic if for every r ≥ q , there is Our work a residue r | M ∈ M , such that any s ≥ r | M with s ∈ M , is compatible with r . P is strongly κ + -cc if (there is a stationary set of M ) for which every condition in P is strongly ( M , P ) -generic.

Strongly κ + -cc Strongly κ + -cc forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings and I. Neeman Definition Generalising MA q ∈ P is strongly ( M , P ) -generic if for every r ≥ q , there is Our work a residue r | M ∈ M , such that any s ≥ r | M with s ∈ M , is compatible with r . P is strongly κ + -cc if (there is a stationary set of M ) for which every condition in P is strongly ( M , P ) -generic. Note If r | M is a residue for r and r ≥ t , then r | M is also a residue for t .

Strongly κ + -cc Strongly κ + -cc forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings and I. Neeman Definition Generalising MA q ∈ P is strongly ( M , P ) -generic if for every r ≥ q , there is Our work a residue r | M ∈ M , such that any s ≥ r | M with s ∈ M , is compatible with r . P is strongly κ + -cc if (there is a stationary set of M ) for which every condition in P is strongly ( M , P ) -generic. Note If r | M is a residue for r and r ≥ t , then r | M is also a residue for t . Hence, to prove that a forcing is strongly κ + -cc, it suffices to show that there is a dense set of conditions which are strongly ( M , P ) -generic, for relevant M s.

Strongly κ + -cc forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings Lemma and I. Neeman Strongly κ + -cc implies κ + -cc. Generalising MA Our work

Strongly κ + -cc forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings Lemma and I. Neeman Strongly κ + -cc implies κ + -cc. Generalising MA Our work Proof Let P be strongly κ + -cc and ¯ p = � p i : i < κ + � a sequence of conditions in P .

Strongly κ + -cc forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings Lemma and I. Neeman Strongly κ + -cc implies κ + -cc. Generalising MA Our work Proof Let P be strongly κ + -cc and ¯ p = � p i : i < κ + � a sequence of conditions in P . Choose M ≺ H ( χ ) as above with κ, P , ¯ p ∈ M .

Strongly κ + -cc forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings Lemma and I. Neeman Strongly κ + -cc implies κ + -cc. Generalising MA Our work Proof Let P be strongly κ + -cc and ¯ p = � p i : i < κ + � a sequence of conditions in P . Choose M ≺ H ( χ ) as above p ∈ M . Note that κ + ∩ M = δ is an ordinal. with κ, P , ¯

Strongly κ + -cc forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings Lemma and I. Neeman Strongly κ + -cc implies κ + -cc. Generalising MA Our work Proof Let P be strongly κ + -cc and ¯ p = � p i : i < κ + � a sequence of conditions in P . Choose M ≺ H ( χ ) as above p ∈ M . Note that κ + ∩ M = δ is an ordinal. Let with κ, P , ¯ r = p δ | M .

Strongly κ + -cc forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings Lemma and I. Neeman Strongly κ + -cc implies κ + -cc. Generalising MA Our work Proof Let P be strongly κ + -cc and ¯ p = � p i : i < κ + � a sequence of conditions in P . Choose M ≺ H ( χ ) as above p ∈ M . Note that κ + ∩ M = δ is an ordinal. Let with κ, P , ¯ r = p δ | M . So r ∈ M is compatible with some p j for a j < κ + , namely p δ .

Strongly κ + -cc forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings Lemma and I. Neeman Strongly κ + -cc implies κ + -cc. Generalising MA Our work Proof Let P be strongly κ + -cc and ¯ p = � p i : i < κ + � a sequence of conditions in P . Choose M ≺ H ( χ ) as above p ∈ M . Note that κ + ∩ M = δ is an ordinal. Let with κ, P , ¯ r = p δ | M . So r ∈ M is compatible with some p j for a j < κ + , namely p δ . By elementarity, there is i < δ such that p i and r are compatible.

Strongly κ + -cc forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings Lemma and I. Neeman Strongly κ + -cc implies κ + -cc. Generalising MA Our work Proof Let P be strongly κ + -cc and ¯ p = � p i : i < κ + � a sequence of conditions in P . Choose M ≺ H ( χ ) as above p ∈ M . Note that κ + ∩ M = δ is an ordinal. Let with κ, P , ¯ r = p δ | M . So r ∈ M is compatible with some p j for a j < κ + , namely p δ . By elementarity, there is i < δ such that p i and r are compatible. Then there is s ∈ M such that s ≥ p i , r .

Strongly κ + -cc forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings Lemma and I. Neeman Strongly κ + -cc implies κ + -cc. Generalising MA Our work Proof Let P be strongly κ + -cc and ¯ p = � p i : i < κ + � a sequence of conditions in P . Choose M ≺ H ( χ ) as above p ∈ M . Note that κ + ∩ M = δ is an ordinal. Let with κ, P , ¯ r = p δ | M . So r ∈ M is compatible with some p j for a j < κ + , namely p δ . By elementarity, there is i < δ such that p i and r are compatible. Then there is s ∈ M such that s ≥ p i , r . By the definition of p δ | M , s and p δ are compatible, so p i and p δ are compatible.

Strongly κ + -cc An example forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings and I. Neeman Generalising MA Let Add ( κ, λ ) denote the forcing to add λ Cohen subsets Our work to κ by conditions of size < κ .

Strongly κ + -cc An example forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings and I. Neeman Generalising MA Let Add ( κ, λ ) denote the forcing to add λ Cohen subsets Our work to κ by conditions of size < κ . Lemma Add ( κ, λ ) is strongly κ + -cc.

Strongly κ + -cc An example forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings and I. Neeman Generalising MA Let Add ( κ, λ ) denote the forcing to add λ Cohen subsets Our work to κ by conditions of size < κ . Lemma Add ( κ, λ ) is strongly κ + -cc. Proof Let M ≺ H ( χ ) , | M | = κ , <κ M ⊆ M and κ, λ ∈ M . Then for any condition r in Add ( κ, λ ) , it suffices to let r | M = r ↾ ( dom ( r ) ∩ M ) .

Strongly κ + -cc Closure forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings and I. Neeman A set in a partial order is directed if every two elements in Generalising MA it have a common upper bound. Our work

Strongly κ + -cc Closure forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings and I. Neeman A set in a partial order is directed if every two elements in Generalising MA it have a common upper bound. Our work Definition We say that P is ( < κ ) -strong directed closed if every directed set of length < κ and consisting of conditions in P , has a least upper bound.

Strongly κ + -cc Closure forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings and I. Neeman A set in a partial order is directed if every two elements in Generalising MA it have a common upper bound. Our work Definition We say that P is ( < κ ) -strong directed closed if every directed set of length < κ and consisting of conditions in P , has a least upper bound. Classical methods show that this property is preserved by iteration with ( < κ ) -supports.

Strongly κ + -cc Main result forcing Mirna Dˇ zamonja, Tutorial 3, including joint work Theorem with J. Cummings and I. Neeman An iteration with supports of size ( < κ ) of strongly κ + -cc Generalising MA ( < κ ) -strongly directed closed forcing, is itself strongly Our work κ + -cc.

Strongly κ + -cc Main result forcing Mirna Dˇ zamonja, Tutorial 3, including joint work Theorem with J. Cummings and I. Neeman An iteration with supports of size ( < κ ) of strongly κ + -cc Generalising MA ( < κ ) -strongly directed closed forcing, is itself strongly Our work κ + -cc. I’ll present some elements of the proof.

Strongly κ + -cc Main result forcing Mirna Dˇ zamonja, Tutorial 3, including joint work Theorem with J. Cummings and I. Neeman An iteration with supports of size ( < κ ) of strongly κ + -cc Generalising MA ( < κ ) -strongly directed closed forcing, is itself strongly Our work κ + -cc. I’ll present some elements of the proof. Throughout we use the notation M for appropriate models.

Strongly κ + -cc Main result forcing Mirna Dˇ zamonja, Tutorial 3, including joint work Theorem with J. Cummings and I. Neeman An iteration with supports of size ( < κ ) of strongly κ + -cc Generalising MA ( < κ ) -strongly directed closed forcing, is itself strongly Our work κ + -cc. I’ll present some elements of the proof. Throughout we use the notation M for appropriate models. The support of a condition in an iterated forcing is the set of non-trivial coordinates, denoted by supt.

Strongly κ + -cc Main result forcing Mirna Dˇ zamonja, Tutorial 3, including joint work Theorem with J. Cummings and I. Neeman An iteration with supports of size ( < κ ) of strongly κ + -cc Generalising MA ( < κ ) -strongly directed closed forcing, is itself strongly Our work κ + -cc. I’ll present some elements of the proof. Throughout we use the notation M for appropriate models. The support of a condition in an iterated forcing is the set of non-trivial coordinates, denoted by supt. A set D ⊆ P is open if it is closed upwards, i.e. p ∈ D , q ≥ p = ⇒ q ∈ D .

Strongly κ + -cc Main result forcing Mirna Dˇ zamonja, Tutorial 3, including joint work Theorem with J. Cummings and I. Neeman An iteration with supports of size ( < κ ) of strongly κ + -cc Generalising MA ( < κ ) -strongly directed closed forcing, is itself strongly Our work κ + -cc. I’ll present some elements of the proof. Throughout we use the notation M for appropriate models. The support of a condition in an iterated forcing is the set of non-trivial coordinates, denoted by supt. A set D ⊆ P is open if it is closed upwards, i.e. p ∈ D , q ≥ p = ⇒ q ∈ D . Let γ be the length of the iteration and use G α to denote the P α generic, for α ≤ γ .

Strongly κ + -cc Main result forcing Mirna Dˇ zamonja, Tutorial 3, including joint work Theorem with J. Cummings and I. Neeman An iteration with supports of size ( < κ ) of strongly κ + -cc Generalising MA ( < κ ) -strongly directed closed forcing, is itself strongly Our work κ + -cc. I’ll present some elements of the proof. Throughout we use the notation M for appropriate models. The support of a condition in an iterated forcing is the set of non-trivial coordinates, denoted by supt. A set D ⊆ P is open if it is closed upwards, i.e. p ∈ D , q ≥ p = ⇒ q ∈ D . Let γ be the length of the iteration and use G α to denote the P α generic, for α ≤ γ . Note For a filter G to be M -generic, it suffices that it intersects all open dense sets in M .

Strongly κ + -cc Canonical extensions forcing Mirna Dˇ zamonja, Tutorial 3, Definition including joint work with J. Cummings Given p ∈ P , a canonical extension q ≥ p , if it exists, is and I. Neeman defined by constructing sequences � p i : i < ω � and Generalising MA � H i : i < ω � so that Our work

Strongly κ + -cc Canonical extensions forcing Mirna Dˇ zamonja, Tutorial 3, Definition including joint work with J. Cummings Given p ∈ P , a canonical extension q ≥ p , if it exists, is and I. Neeman defined by constructing sequences � p i : i < ω � and Generalising MA � H i : i < ω � so that Our work for all i , H i ≺ H ( χ ) and | H i | < κ , 1

Strongly κ + -cc Canonical extensions forcing Mirna Dˇ zamonja, Tutorial 3, Definition including joint work with J. Cummings Given p ∈ P , a canonical extension q ≥ p , if it exists, is and I. Neeman defined by constructing sequences � p i : i < ω � and Generalising MA � H i : i < ω � so that Our work for all i , H i ≺ H ( χ ) and | H i | < κ , 1 p , M ∈ H 0 and supt ( p ) ⊆ H 0 , 2

Strongly κ + -cc Canonical extensions forcing Mirna Dˇ zamonja, Tutorial 3, Definition including joint work with J. Cummings Given p ∈ P , a canonical extension q ≥ p , if it exists, is and I. Neeman defined by constructing sequences � p i : i < ω � and Generalising MA � H i : i < ω � so that Our work for all i , H i ≺ H ( χ ) and | H i | < κ , 1 p , M ∈ H 0 and supt ( p ) ⊆ H 0 , 2 p 0 = p , 3

Strongly κ + -cc Canonical extensions forcing Mirna Dˇ zamonja, Tutorial 3, Definition including joint work with J. Cummings Given p ∈ P , a canonical extension q ≥ p , if it exists, is and I. Neeman defined by constructing sequences � p i : i < ω � and Generalising MA � H i : i < ω � so that Our work for all i , H i ≺ H ( χ ) and | H i | < κ , 1 p , M ∈ H 0 and supt ( p ) ⊆ H 0 , 2 p 0 = p , 3 for all i , p i + 1 ≥ p i and p i + 1 ∈ D for every open dense 4 D ∈ H i ,

Strongly κ + -cc Canonical extensions forcing Mirna Dˇ zamonja, Tutorial 3, Definition including joint work with J. Cummings Given p ∈ P , a canonical extension q ≥ p , if it exists, is and I. Neeman defined by constructing sequences � p i : i < ω � and Generalising MA � H i : i < ω � so that Our work for all i , H i ≺ H ( χ ) and | H i | < κ , 1 p , M ∈ H 0 and supt ( p ) ⊆ H 0 , 2 p 0 = p , 3 for all i , p i + 1 ≥ p i and p i + 1 ∈ D for every open dense 4 D ∈ H i , for all i , H i ∪ { p i + 1 , H i } ∪ supt ( p i + 1 ) ⊆ H i + 1 , 5

Strongly κ + -cc Canonical extensions forcing Mirna Dˇ zamonja, Tutorial 3, Definition including joint work with J. Cummings Given p ∈ P , a canonical extension q ≥ p , if it exists, is and I. Neeman defined by constructing sequences � p i : i < ω � and Generalising MA � H i : i < ω � so that Our work for all i , H i ≺ H ( χ ) and | H i | < κ , 1 p , M ∈ H 0 and supt ( p ) ⊆ H 0 , 2 p 0 = p , 3 for all i , p i + 1 ≥ p i and p i + 1 ∈ D for every open dense 4 D ∈ H i , for all i , H i ∪ { p i + 1 , H i } ∪ supt ( p i + 1 ) ⊆ H i + 1 , 5 � p i : i < ω � admits a least upper bound, 6

Strongly κ + -cc Canonical extensions forcing Mirna Dˇ zamonja, Tutorial 3, Definition including joint work with J. Cummings Given p ∈ P , a canonical extension q ≥ p , if it exists, is and I. Neeman defined by constructing sequences � p i : i < ω � and Generalising MA � H i : i < ω � so that Our work for all i , H i ≺ H ( χ ) and | H i | < κ , 1 p , M ∈ H 0 and supt ( p ) ⊆ H 0 , 2 p 0 = p , 3 for all i , p i + 1 ≥ p i and p i + 1 ∈ D for every open dense 4 D ∈ H i , for all i , H i ∪ { p i + 1 , H i } ∪ supt ( p i + 1 ) ⊆ H i + 1 , 5 � p i : i < ω � admits a least upper bound, 6 and then letting q = lub i <ω p i . Let H = � i <ω H i .

Strongly κ + -cc Canonical extensions forcing Mirna Dˇ zamonja, Tutorial 3, Definition including joint work with J. Cummings Given p ∈ P , a canonical extension q ≥ p , if it exists, is and I. Neeman defined by constructing sequences � p i : i < ω � and Generalising MA � H i : i < ω � so that Our work for all i , H i ≺ H ( χ ) and | H i | < κ , 1 p , M ∈ H 0 and supt ( p ) ⊆ H 0 , 2 p 0 = p , 3 for all i , p i + 1 ≥ p i and p i + 1 ∈ D for every open dense 4 D ∈ H i , for all i , H i ∪ { p i + 1 , H i } ∪ supt ( p i + 1 ) ⊆ H i + 1 , 5 � p i : i < ω � admits a least upper bound, 6 and then letting q = lub i <ω p i . Let H = � i <ω H i . Note In our context, every p allows a canonical extension.

Strongly κ + -cc Properties of canonical extensions forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings Lemma and I. Neeman Suppose that q is a canonical extension of p. Then: Generalising MA H ≺ H ( χ ) , | H | < κ and p , M ∈ H, 1 Our work

Strongly κ + -cc Properties of canonical extensions forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings Lemma and I. Neeman Suppose that q is a canonical extension of p. Then: Generalising MA H ≺ H ( χ ) , | H | < κ and p , M ∈ H, 1 Our work if g = { s ∈ P ∩ H : ( ∃ i < ω ) s ≤ p i } then q is the lub 2 of g and g = { s ∈ P ∩ H : s ≤ q } ,

Strongly κ + -cc Properties of canonical extensions forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings Lemma and I. Neeman Suppose that q is a canonical extension of p. Then: Generalising MA H ≺ H ( χ ) , | H | < κ and p , M ∈ H, 1 Our work if g = { s ∈ P ∩ H : ( ∃ i < ω ) s ≤ p i } then q is the lub 2 of g and g = { s ∈ P ∩ H : s ≤ q } , g is a filter on P ∩ H which meets every open set in H 3 that is dense above some p i ,

Strongly κ + -cc Properties of canonical extensions forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings Lemma and I. Neeman Suppose that q is a canonical extension of p. Then: Generalising MA H ≺ H ( χ ) , | H | < κ and p , M ∈ H, 1 Our work if g = { s ∈ P ∩ H : ( ∃ i < ω ) s ≤ p i } then q is the lub 2 of g and g = { s ∈ P ∩ H : s ≤ q } , g is a filter on P ∩ H which meets every open set in H 3 that is dense above some p i , supt ( q ) = H ∩ γ = � i <ω supt ( p i ) , 4

Strongly κ + -cc Properties of canonical extensions forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings Lemma and I. Neeman Suppose that q is a canonical extension of p. Then: Generalising MA H ≺ H ( χ ) , | H | < κ and p , M ∈ H, 1 Our work if g = { s ∈ P ∩ H : ( ∃ i < ω ) s ≤ p i } then q is the lub 2 of g and g = { s ∈ P ∩ H : s ≤ q } , g is a filter on P ∩ H which meets every open set in H 3 that is dense above some p i , supt ( q ) = H ∩ γ = � i <ω supt ( p i ) , 4 H ∩ M ∈ M. 5

Strongly κ + -cc Properties of canonical extensions forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings Lemma and I. Neeman Suppose that q is a canonical extension of p. Then: Generalising MA H ≺ H ( χ ) , | H | < κ and p , M ∈ H, 1 Our work if g = { s ∈ P ∩ H : ( ∃ i < ω ) s ≤ p i } then q is the lub 2 of g and g = { s ∈ P ∩ H : s ≤ q } , g is a filter on P ∩ H which meets every open set in H 3 that is dense above some p i , supt ( q ) = H ∩ γ = � i <ω supt ( p i ) , 4 H ∩ M ∈ M. 5 Given p ∈ P , let q ≥ p be a canonical extension of p , we shall prove that q has a residue over M .

Strongly κ + -cc Canonical extensions have residues forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings and I. Neeman Recall g = { s ∈ P ∩ H : s ≤ q } . Generalising MA Our work

Strongly κ + -cc Canonical extensions have residues forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings and I. Neeman Recall g = { s ∈ P ∩ H : s ≤ q } . Let t = lub ( g ∩ M ) (note Generalising MA g ⊆ H so | g | < κ ). Our work

Strongly κ + -cc Canonical extensions have residues forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings and I. Neeman Recall g = { s ∈ P ∩ H : s ≤ q } . Let t = lub ( g ∩ M ) (note Generalising MA g ⊆ H so | g | < κ ). We claim t = q | M . Our work

Strongly κ + -cc Canonical extensions have residues forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings and I. Neeman Recall g = { s ∈ P ∩ H : s ≤ q } . Let t = lub ( g ∩ M ) (note Generalising MA g ⊆ H so | g | < κ ). We claim t = q | M . t ∈ M by the Our work closure of M .

Strongly κ + -cc Canonical extensions have residues forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings and I. Neeman Recall g = { s ∈ P ∩ H : s ≤ q } . Let t = lub ( g ∩ M ) (note Generalising MA g ⊆ H so | g | < κ ). We claim t = q | M . t ∈ M by the Our work closure of M . q ≥ t since q is a bound for g ⊇ g ∩ M and t is the lub.

Strongly κ + -cc Canonical extensions have residues forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings and I. Neeman Recall g = { s ∈ P ∩ H : s ≤ q } . Let t = lub ( g ∩ M ) (note Generalising MA g ⊆ H so | g | < κ ). We claim t = q | M . t ∈ M by the Our work closure of M . q ≥ t since q is a bound for g ⊇ g ∩ M and t is the lub. Given s ≥ t with s ∈ M , build q ∗ ≥ q , s .

Strongly κ + -cc Canonical extensions have residues forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings and I. Neeman Recall g = { s ∈ P ∩ H : s ≤ q } . Let t = lub ( g ∩ M ) (note Generalising MA g ⊆ H so | g | < κ ). We claim t = q | M . t ∈ M by the Our work closure of M . q ≥ t since q is a bound for g ⊇ g ∩ M and t is the lub. Given s ≥ t with s ∈ M , build q ∗ ≥ q , s . Define q ∗ ↾ α for α ≤ γ .

Strongly κ + -cc Canonical extensions have residues forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings and I. Neeman Recall g = { s ∈ P ∩ H : s ≤ q } . Let t = lub ( g ∩ M ) (note Generalising MA g ⊆ H so | g | < κ ). We claim t = q | M . t ∈ M by the Our work closure of M . q ≥ t since q is a bound for g ⊇ g ∩ M and t is the lub. Given s ≥ t with s ∈ M , build q ∗ ≥ q , s . Define q ∗ ↾ α for α ≤ γ . Suppose q ∗ ↾ α is given, we show how to obtain a P α -name q ∗ ( α ) .

Strongly κ + -cc Canonical extensions have residues forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings and I. Neeman Recall g = { s ∈ P ∩ H : s ≤ q } . Let t = lub ( g ∩ M ) (note Generalising MA g ⊆ H so | g | < κ ). We claim t = q | M . t ∈ M by the Our work closure of M . q ≥ t since q is a bound for g ⊇ g ∩ M and t is the lub. Given s ≥ t with s ∈ M , build q ∗ ≥ q , s . Define q ∗ ↾ α for α ≤ γ . Suppose q ∗ ↾ α is given, we show how to obtain a P α -name q ∗ ( α ) . The interesting case is α ∈ supt ( t ) ∩ supt ( q ) ,

Strongly κ + -cc Canonical extensions have residues forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings and I. Neeman Recall g = { s ∈ P ∩ H : s ≤ q } . Let t = lub ( g ∩ M ) (note Generalising MA g ⊆ H so | g | < κ ). We claim t = q | M . t ∈ M by the Our work closure of M . q ≥ t since q is a bound for g ⊇ g ∩ M and t is the lub. Given s ≥ t with s ∈ M , build q ∗ ≥ q , s . Define q ∗ ↾ α for α ≤ γ . Suppose q ∗ ↾ α is given, we show how to obtain a P α -name q ∗ ( α ) . The interesting case is α ∈ supt ( t ) ∩ supt ( q ) , so α ∈ M since s ∈ M .

Strongly κ + -cc Canonical extensions have residues forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings and I. Neeman Recall g = { s ∈ P ∩ H : s ≤ q } . Let t = lub ( g ∩ M ) (note Generalising MA g ⊆ H so | g | < κ ). We claim t = q | M . t ∈ M by the Our work closure of M . q ≥ t since q is a bound for g ⊇ g ∩ M and t is the lub. Given s ≥ t with s ∈ M , build q ∗ ≥ q , s . Define q ∗ ↾ α for α ≤ γ . Suppose q ∗ ↾ α is given, we show how to obtain a P α -name q ∗ ( α ) . The interesting case is α ∈ supt ( t ) ∩ supt ( q ) , so α ∈ M since s ∈ M . Can assume α ∈ supt ( p i ) for all i .

Strongly κ + -cc Canonical extensions have residues forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings and I. Neeman Recall g = { s ∈ P ∩ H : s ≤ q } . Let t = lub ( g ∩ M ) (note Generalising MA g ⊆ H so | g | < κ ). We claim t = q | M . t ∈ M by the Our work closure of M . q ≥ t since q is a bound for g ⊇ g ∩ M and t is the lub. Given s ≥ t with s ∈ M , build q ∗ ≥ q , s . Define q ∗ ↾ α for α ≤ γ . Suppose q ∗ ↾ α is given, we show how to obtain a P α -name q ∗ ( α ) . The interesting case is α ∈ supt ( t ) ∩ supt ( q ) , so α ∈ M since s ∈ M . Can assume α ∈ supt ( p i ) for all i . Also, α ∈ H since supt ( q ) ⊆ H .

D i Strongly κ + -cc forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings and I. Neeman For each i < ω , let D i collect all u ∈ P such that there is Generalising MA ∈ M ∩ Q r ∗ α with ˜ Our work ˜ u ↾ α � α “ u ( α ) ≥ r ∗ and r ∗ is a residue for p i ( α ) in M [ G ˜ α ]” . ˜ ˜

D i Strongly κ + -cc forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings and I. Neeman For each i < ω , let D i collect all u ∈ P such that there is Generalising MA ∈ M ∩ Q r ∗ α with ˜ Our work ˜ u ↾ α � α “ u ( α ) ≥ r ∗ and r ∗ is a residue for p i ( α ) in M [ G ˜ α ]” . ˜ ˜ D i is in H , since its parameters are in H .

D i Strongly κ + -cc forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings and I. Neeman For each i < ω , let D i collect all u ∈ P such that there is Generalising MA ∈ M ∩ Q r ∗ α with ˜ Our work ˜ u ↾ α � α “ u ( α ) ≥ r ∗ and r ∗ is a residue for p i ( α ) in M [ G ˜ α ]” . ˜ ˜ D i is in H , since its parameters are in H . D i is open.

D i Strongly κ + -cc forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings and I. Neeman For each i < ω , let D i collect all u ∈ P such that there is Generalising MA ∈ M ∩ Q r ∗ α with ˜ Our work ˜ u ↾ α � α “ u ( α ) ≥ r ∗ and r ∗ is a residue for p i ( α ) in M [ G ˜ α ]” . ˜ ˜ D i is in H , since its parameters are in H . D i is open. D i is dense above p i , since Q α is forced to be strongly κ + -cc ˜ and M [ G ˜ α ] to be an appropriate model

D i Strongly κ + -cc forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings and I. Neeman For each i < ω , let D i collect all u ∈ P such that there is Generalising MA ∈ M ∩ Q r ∗ α with ˜ Our work ˜ u ↾ α � α “ u ( α ) ≥ r ∗ and r ∗ is a residue for p i ( α ) in M [ G ˜ α ]” . ˜ ˜ D i is in H , since its parameters are in H . D i is open. D i is dense above p i , since Q α is forced to be strongly κ + -cc ˜ and M [ G ˜ α ] to be an appropriate model (use closure of the forcing to see that M [ G ˜ α ] has to be closed and the chain condition to show it is forced to be ≺ H ( χ )[ G ˜ α ] ).

D i Strongly κ + -cc forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings and I. Neeman For each i < ω , let D i collect all u ∈ P such that there is Generalising MA ∈ M ∩ Q r ∗ α with ˜ Our work ˜ u ↾ α � α “ u ( α ) ≥ r ∗ and r ∗ is a residue for p i ( α ) in M [ G ˜ α ]” . ˜ ˜ D i is in H , since its parameters are in H . D i is open. D i is dense above p i , since Q α is forced to be strongly κ + -cc ˜ and M [ G ˜ α ] to be an appropriate model (use closure of the forcing to see that M [ G ˜ α ] has to be closed and the chain condition to show it is forced to be ≺ H ( χ )[ G ˜ α ] ). So g ∩ D i � = ∅ .

Strongly κ + -cc More induction forcing Mirna Dˇ zamonja, By induction on k < ω we construct � i k : k < ω � , Tutorial 3, including joint work � q k : k < ω � and � r ∗ k : k < ω � such that with J. Cummings and I. Neeman ˜ Generalising MA Our work

Strongly κ + -cc More induction forcing Mirna Dˇ zamonja, By induction on k < ω we construct � i k : k < ω � , Tutorial 3, including joint work � q k : k < ω � and � r ∗ k : k < ω � such that with J. Cummings and I. Neeman ˜ i 0 = 0, Generalising MA Our work

Strongly κ + -cc More induction forcing Mirna Dˇ zamonja, By induction on k < ω we construct � i k : k < ω � , Tutorial 3, including joint work � q k : k < ω � and � r ∗ k : k < ω � such that with J. Cummings and I. Neeman ˜ i 0 = 0, Generalising MA q k ∈ D i k ∩ g as exemplified by r ∗ k and r ∗ k ∈ H , Our work ˜ ˜

Strongly κ + -cc More induction forcing Mirna Dˇ zamonja, By induction on k < ω we construct � i k : k < ω � , Tutorial 3, including joint work � q k : k < ω � and � r ∗ k : k < ω � such that with J. Cummings and I. Neeman ˜ i 0 = 0, Generalising MA q k ∈ D i k ∩ g as exemplified by r ∗ k and r ∗ k ∈ H , Our work ˜ ˜ i k + 1 is such that p i k + 1 ≥ q k ).

Strongly κ + -cc More induction forcing Mirna Dˇ zamonja, By induction on k < ω we construct � i k : k < ω � , Tutorial 3, including joint work � q k : k < ω � and � r ∗ k : k < ω � such that with J. Cummings and I. Neeman ˜ i 0 = 0, Generalising MA q k ∈ D i k ∩ g as exemplified by r ∗ k and r ∗ k ∈ H , Our work ˜ ˜ i k + 1 is such that p i k + 1 ≥ q k ).

Strongly κ + -cc More induction forcing Mirna Dˇ zamonja, By induction on k < ω we construct � i k : k < ω � , Tutorial 3, including joint work � q k : k < ω � and � r ∗ k : k < ω � such that with J. Cummings and I. Neeman ˜ i 0 = 0, Generalising MA q k ∈ D i k ∩ g as exemplified by r ∗ k and r ∗ k ∈ H , Our work ˜ ˜ i k + 1 is such that p i k + 1 ≥ q k ). Use the elementarity of H and the definition of g .

Strongly κ + -cc More induction forcing Mirna Dˇ zamonja, By induction on k < ω we construct � i k : k < ω � , Tutorial 3, including joint work � q k : k < ω � and � r ∗ k : k < ω � such that with J. Cummings and I. Neeman ˜ i 0 = 0, Generalising MA q k ∈ D i k ∩ g as exemplified by r ∗ k and r ∗ k ∈ H , Our work ˜ ˜ i k + 1 is such that p i k + 1 ≥ q k ). Use the elementarity of H and the definition of g . Let r k ∈ P α + 1 have r k ↾ α trivial and r k ( α ) = r ∗ k . ˜

Strongly κ + -cc More induction forcing Mirna Dˇ zamonja, By induction on k < ω we construct � i k : k < ω � , Tutorial 3, including joint work � q k : k < ω � and � r ∗ k : k < ω � such that with J. Cummings and I. Neeman ˜ i 0 = 0, Generalising MA q k ∈ D i k ∩ g as exemplified by r ∗ k and r ∗ k ∈ H , Our work ˜ ˜ i k + 1 is such that p i k + 1 ≥ q k ). Use the elementarity of H and the definition of g . Let r k ∈ P α + 1 have r k ↾ α trivial and r k ( α ) = r ∗ k . Then ˜ r k ∈ M ∩ H (def. of D i ),

Strongly κ + -cc More induction forcing Mirna Dˇ zamonja, By induction on k < ω we construct � i k : k < ω � , Tutorial 3, including joint work � q k : k < ω � and � r ∗ k : k < ω � such that with J. Cummings and I. Neeman ˜ i 0 = 0, Generalising MA q k ∈ D i k ∩ g as exemplified by r ∗ k and r ∗ k ∈ H , Our work ˜ ˜ i k + 1 is such that p i k + 1 ≥ q k ). Use the elementarity of H and the definition of g . Let r k ∈ P α + 1 have r k ↾ α trivial and r k ( α ) = r ∗ k . Then ˜ r k ∈ M ∩ H (def. of D i ), q k ≥ r k .

Strongly κ + -cc More induction forcing Mirna Dˇ zamonja, By induction on k < ω we construct � i k : k < ω � , Tutorial 3, including joint work � q k : k < ω � and � r ∗ k : k < ω � such that with J. Cummings and I. Neeman ˜ i 0 = 0, Generalising MA q k ∈ D i k ∩ g as exemplified by r ∗ k and r ∗ k ∈ H , Our work ˜ ˜ i k + 1 is such that p i k + 1 ≥ q k ). Use the elementarity of H and the definition of g . Let r k ∈ P α + 1 have r k ↾ α trivial and r k ( α ) = r ∗ k . Then ˜ r k ∈ M ∩ H (def. of D i ), q k ≥ r k . Hence q ≥ r k and q ↾ α � α “ r ∗ is a residue for p i k ( α ) in M [ G ˜ α ]” . k ˜

Strongly κ + -cc More induction forcing Mirna Dˇ zamonja, By induction on k < ω we construct � i k : k < ω � , Tutorial 3, including joint work � q k : k < ω � and � r ∗ k : k < ω � such that with J. Cummings and I. Neeman ˜ i 0 = 0, Generalising MA q k ∈ D i k ∩ g as exemplified by r ∗ k and r ∗ k ∈ H , Our work ˜ ˜ i k + 1 is such that p i k + 1 ≥ q k ). Use the elementarity of H and the definition of g . Let r k ∈ P α + 1 have r k ↾ α trivial and r k ( α ) = r ∗ k . Then ˜ r k ∈ M ∩ H (def. of D i ), q k ≥ r k . Hence q ≥ r k and q ↾ α � α “ r ∗ is a residue for p i k ( α ) in M [ G ˜ α ]” . k ˜ Since q ≥ p i k + 1 ≥ q k , and p i s are increasing, we have that q ↾ α � α “ p i ( α ) , s ( α ) are compatible”, for all i .

Strongly κ + -cc More induction forcing Mirna Dˇ zamonja, By induction on k < ω we construct � i k : k < ω � , Tutorial 3, including joint work � q k : k < ω � and � r ∗ k : k < ω � such that with J. Cummings and I. Neeman ˜ i 0 = 0, Generalising MA q k ∈ D i k ∩ g as exemplified by r ∗ k and r ∗ k ∈ H , Our work ˜ ˜ i k + 1 is such that p i k + 1 ≥ q k ). Use the elementarity of H and the definition of g . Let r k ∈ P α + 1 have r k ↾ α trivial and r k ( α ) = r ∗ k . Then ˜ r k ∈ M ∩ H (def. of D i ), q k ≥ r k . Hence q ≥ r k and q ↾ α � α “ r ∗ is a residue for p i k ( α ) in M [ G ˜ α ]” . k ˜ Since q ≥ p i k + 1 ≥ q k , and p i s are increasing, we have that q ↾ α � α “ p i ( α ) , s ( α ) are compatible”, for all i . q ∗ ↾ α forces that there is an upper bound for all p i ( α ) and s ( α ) in Q α , which we then take as q ∗ ( α ) . ˜

Strongly κ + -cc Uses of the theorem forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings and I. Neeman Generalising MA Our work Under exploration.