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

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

Strongly κ+-cc forcing

Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings and I. Neeman 28 June, 2019

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

κ+-cc and (< κ)-closure

We shall consider κ such that κ = κ<κ.

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

κ+-cc and (< κ)-closure

We shall consider κ such that κ = κ<κ.

Definition

A forcing notion P is κ+-cc iff every antichain in P has size ≤ κ.

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

κ+-cc and (< κ)-closure

We shall consider κ such that κ = κ<κ.

Definition

A forcing notion P is κ+-cc iff every antichain in P has size ≤ κ. Facts (1) If a forcing notion P has an antichain of every length λ < κ and κ is singular, then P has an antichain of length κ (Erd¨

• s-Tarski 1943).

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

κ+-cc and (< κ)-closure

We shall consider κ such that κ = κ<κ.

Definition

A forcing notion P is κ+-cc iff every antichain in P has size ≤ κ. Facts (1) If a forcing notion P has an antichain of every length λ < κ and κ is singular, then P has an antichain of length κ (Erd¨

• s-Tarski 1943).

(2) A κ+-cc forcing preserves cardinals ≥ κ+.

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

κ+-cc and (< κ)-closure

We shall consider κ such that κ = κ<κ.

Definition

A forcing notion P is κ+-cc iff every antichain in P has size ≤ κ. Facts (1) If a forcing notion P has an antichain of every length λ < κ and κ is singular, then P has an antichain of length κ (Erd¨

• s-Tarski 1943).

(2) A κ+-cc forcing preserves cardinals ≥ κ+. Remark (2) uses the fact that for any function f : Ord → Ord in the extension by a κ+-cc forcing, there is a function F in the ground model such that f(α) ∈ F(α) and |F(α)| ≤ κ.

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

Definition

A forcing notion P is (< κ)-closed iff every increasing sequence of length < κ in P has an upper bound.

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

Definition

A forcing notion P is (< κ)-closed iff every increasing sequence of length < κ in P has an upper bound.

Lemma

Countably closed forcing preserves ℵ1, (< κ)-cc closed forcing preserves cardinals ≤ κ.

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

Definition

A forcing notion P is (< κ)-closed iff every increasing sequence of length < κ in P has an upper bound.

Lemma

Countably closed forcing preserves ℵ1, (< κ)-cc closed forcing preserves cardinals ≤ κ.

• Proof. Suppose that f

˜ is a P-name for a surjection from ω to ωV

1 .

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

Definition

A forcing notion P is (< κ)-closed iff every increasing sequence of length < κ in P has an upper bound.

Lemma

Countably closed forcing preserves ℵ1, (< κ)-cc closed forcing preserves cardinals ≤ κ.

• Proof. Suppose that f

˜ is a P-name for a surjection from ω to ωV

1 . Consider the set

D = {p ∈ P : p f ˜ = g for some function g ∈ V}, it is in V, show it is dense.

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

Definition

A forcing notion P is (< κ)-closed iff every increasing sequence of length < κ in P has an upper bound.

Lemma

Countably closed forcing preserves ℵ1, (< κ)-cc closed forcing preserves cardinals ≤ κ.

• Proof. Suppose that f

˜ is a P-name for a surjection from ω to ωV

1 . Consider the set

D = {p ∈ P : p f ˜ = g for some function g ∈ V}, it is in V, show it is dense. Start with p0 and build inductively an increasing sequence p0 ≤ p1 ≤ . . . pn ≤ of conditions such that pn+1 forces a value f ˜ (n)

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

Definition

A forcing notion P is (< κ)-closed iff every increasing sequence of length < κ in P has an upper bound.

Lemma

Countably closed forcing preserves ℵ1, (< κ)-cc closed forcing preserves cardinals ≤ κ.

• Proof. Suppose that f

˜ is a P-name for a surjection from ω to ωV

1 . Consider the set

D = {p ∈ P : p f ˜ = g for some function g ∈ V}, it is in V, show it is dense. Start with p0 and build inductively an increasing sequence p0 ≤ p1 ≤ . . . pn ≤ of conditions such that pn+1 forces a value f ˜ (n) (note that we can do this ! general theory of forcing).

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

Definition

A forcing notion P is (< κ)-closed iff every increasing sequence of length < κ in P has an upper bound.

Lemma

Countably closed forcing preserves ℵ1, (< κ)-cc closed forcing preserves cardinals ≤ κ.

• Proof. Suppose that f

˜ is a P-name for a surjection from ω to ωV

1 . Consider the set

D = {p ∈ P : p f ˜ = g for some function g ∈ V}, it is in V, show it is dense. Start with p0 and build inductively an increasing sequence p0 ≤ p1 ≤ . . . pn ≤ of conditions such that pn+1 forces a value f ˜ (n) (note that we can do this ! general theory of forcing). Let q ≥ pn for every n, then p0 ≤ q and g ∈ D

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

Definition

A forcing notion P is (< κ)-closed iff every increasing sequence of length < κ in P has an upper bound.

Lemma

Countably closed forcing preserves ℵ1, (< κ)-cc closed forcing preserves cardinals ≤ κ.

• Proof. Suppose that f

˜ is a P-name for a surjection from ω to ωV

1 . Consider the set

D = {p ∈ P : p f ˜ = g for some function g ∈ V}, it is in V, show it is dense. Start with p0 and build inductively an increasing sequence p0 ≤ p1 ≤ . . . pn ≤ of conditions such that pn+1 forces a value f ˜ (n) (note that we can do this ! general theory of forcing). Let q ≥ pn for every n, then p0 ≤ q and g ∈ D as exemplified by g(n) being the value that q forces to f ˜ (n).

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

Conclusion

κ+-cc (< κ)-closed forcing preserves cardinals.

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

Conclusion

κ+-cc (< κ)-closed forcing preserves cardinals. Problem Not iterable (examples: see Shelah ”Generalized MA” 1978).

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

Conclusion

κ+-cc (< κ)-closed forcing preserves cardinals. Problem Not iterable (examples: see Shelah ”Generalized MA” 1978). Iterable conditions in this line exist, obtained by strengthening both κ+-cc and (< κ)-closed.

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

Conclusion

κ+-cc (< κ)-closed forcing preserves cardinals. Problem Not iterable (examples: see Shelah ”Generalized MA” 1978). Iterable conditions in this line exist, obtained by strengthening both κ+-cc and (< κ)-closed. For example,

Definition

P is κ+-stationary-cc if for every pi : i < κ+ in P there is a club C in κ+ and a regressive function f, such that i, j ∈ Sκ+

κ

∩ C and f(i) = f(j) = ⇒ pi, pj countable .

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

Conclusion

κ+-cc (< κ)-closed forcing preserves cardinals. Problem Not iterable (examples: see Shelah ”Generalized MA” 1978). Iterable conditions in this line exist, obtained by strengthening both κ+-cc and (< κ)-closed. For example,

Definition

P is κ+-stationary-cc if for every pi : i < κ+ in P there is a club C in κ+ and a regressive function f, such that i, j ∈ Sκ+

κ

∩ C and f(i) = f(j) = ⇒ pi, pj countable .

Theorem (Shelah 1978)

An iteration with (< κ)-supports of κ+-stationary-cc (< κ)-closed forcing in which every two conditions have lub, is κ+-cc.

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

κ+-cc and (< κ)-closure

Other axioms of the above type were discovered by Baumgartner (1974), Shelah in several papers and Cumming, Dˇ z., Magidor, Morgan and Shelah (2017).

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

κ+-cc and (< κ)-closure

Other axioms of the above type were discovered by Baumgartner (1974), Shelah in several papers and Cumming, Dˇ z., Magidor, Morgan and Shelah (2017). No popular analogue of properness, in spite of some known work, notably by Roslanowski and Shelah.

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

κ+-cc and (< κ)-closure

Other axioms of the above type were discovered by Baumgartner (1974), Shelah in several papers and Cumming, Dˇ z., Magidor, Morgan and Shelah (2017). No popular analogue of properness, in spite of some 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 ℵ2.

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

κ+-cc and (< κ)-closure

Other axioms of the above type were discovered by Baumgartner (1974), Shelah in several papers and Cumming, Dˇ z., Magidor, Morgan and Shelah (2017). No popular analogue of properness, in spite of some 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 ℵ2. Our motivation has been not to generalise properness but to generalise ccc by using non-combinatorial methods, which then give an easily checkable condition.

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

κ+-cc and (< κ)-closure

Other axioms of the above type were discovered by Baumgartner (1974), Shelah in several papers and Cumming, Dˇ z., Magidor, Morgan and Shelah (2017). No popular analogue of properness, in spite of some 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 ℵ2. Our motivation has been not to generalise properness but to generalise ccc by using non-combinatorial methods, which then give an easily checkable condition.

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

κ+-cc and (< κ)-closure

Other axioms of the above type were discovered by Baumgartner (1974), Shelah in several papers and Cumming, Dˇ z., Magidor, Morgan and Shelah (2017). No popular analogue of properness, in spite of some 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 ℵ2. 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.

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

κ+-cc and (< κ)-closure

Other axioms of the above type were discovered by Baumgartner (1974), Shelah in several papers and Cumming, Dˇ z., Magidor, Morgan and Shelah (2017). No popular analogue of properness, in spite of some 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 ℵ2. 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 ≥ ℵ1.

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

κ+-cc and (< κ)-closure

Other axioms of the above type were discovered by Baumgartner (1974), Shelah in several papers and Cumming, Dˇ z., Magidor, Morgan and Shelah (2017). No popular analogue of properness, in spite of some 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 ℵ2. 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 ≥ ℵ1. Consider elementary submodels M ≺ H(χ) = H(χ), ∈, <∗ with |M| = κ,

<κM ⊆ M, P=the forcing in question ∈ M.

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

Strongly κ+-cc

Definition

q ∈ P is strongly (M, P)-generic if

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

Strongly κ+-cc

Definition

q ∈ P is strongly (M, P)-generic if for every r ≥ q, there is a residue r|M ∈ M, such that any s ≥ r|M with s ∈ M, is compatible with r.

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

Strongly κ+-cc

Definition

q ∈ P is strongly (M, P)-generic if for every r ≥ q, there is 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 forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings and I. Neeman Generalising MA Our work

Strongly κ+-cc

Definition

q ∈ P is strongly (M, P)-generic if for every r ≥ q, there is 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 forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings and I. Neeman Generalising MA Our work

Strongly κ+-cc

Definition

q ∈ P is strongly (M, P)-generic if for every r ≥ q, there is 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 Ms.

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

Lemma

Strongly κ+-cc implies κ+-cc.

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

Lemma

Strongly κ+-cc implies κ+-cc. Proof Let P be strongly κ+-cc and ¯ p = pi : i < κ+ a sequence of conditions in P.

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

Lemma

Strongly κ+-cc implies κ+-cc. Proof Let P be strongly κ+-cc and ¯ p = pi : 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 and I. Neeman Generalising MA Our work

Lemma

Strongly κ+-cc implies κ+-cc. Proof Let P be strongly κ+-cc and ¯ p = pi : i < κ+ a sequence of conditions in P. Choose M ≺ H(χ) as above with κ, P, ¯ p ∈ M . Note that κ+ ∩ M = δ is an ordinal.

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

Lemma

Strongly κ+-cc implies κ+-cc. Proof Let P be strongly κ+-cc and ¯ p = pi : i < κ+ a sequence of conditions in P. Choose M ≺ H(χ) as above with κ, P, ¯ p ∈ M . Note that κ+ ∩ M = δ is an ordinal. Let r = pδ|M.

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

Lemma

Strongly κ+-cc implies κ+-cc. Proof Let P be strongly κ+-cc and ¯ p = pi : i < κ+ a sequence of conditions in P. Choose M ≺ H(χ) as above with κ, P, ¯ p ∈ M . Note that κ+ ∩ M = δ is an ordinal. Let r = pδ|M. So r ∈ M is compatible with some pj for a j < κ+, namely pδ.

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

Lemma

Strongly κ+-cc implies κ+-cc. Proof Let P be strongly κ+-cc and ¯ p = pi : i < κ+ a sequence of conditions in P. Choose M ≺ H(χ) as above with κ, P, ¯ p ∈ M . Note that κ+ ∩ M = δ is an ordinal. Let r = pδ|M. So r ∈ M is compatible with some pj for a j < κ+, namely pδ. By elementarity, there is i < δ such that pi and r are compatible.

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

Lemma

Strongly κ+-cc implies κ+-cc. Proof Let P be strongly κ+-cc and ¯ p = pi : i < κ+ a sequence of conditions in P. Choose M ≺ H(χ) as above with κ, P, ¯ p ∈ M . Note that κ+ ∩ M = δ is an ordinal. Let r = pδ|M. So r ∈ M is compatible with some pj for a j < κ+, namely pδ. By elementarity, there is i < δ such that pi and r are compatible. Then there is s ∈ M such that s ≥ pi, r.

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

Lemma

Strongly κ+-cc implies κ+-cc. Proof Let P be strongly κ+-cc and ¯ p = pi : i < κ+ a sequence of conditions in P. Choose M ≺ H(χ) as above with κ, P, ¯ p ∈ M . Note that κ+ ∩ M = δ is an ordinal. Let r = pδ|M. So r ∈ M is compatible with some pj for a j < κ+, namely pδ. By elementarity, there is i < δ such that pi and r are compatible. Then there is s ∈ M such that s ≥ pi, r. By the definition of pδ|M, s and pδ are compatible, so pi and pδ are compatible.

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

An example

Let Add(κ, λ) denote the forcing to add λ Cohen subsets to κ by conditions of size < κ.

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

An example

Let Add(κ, λ) denote the forcing to add λ Cohen subsets to κ by conditions of size < κ.

Lemma

Add(κ, λ) is strongly κ+-cc.

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

An example

Let Add(κ, λ) denote the forcing to add λ Cohen subsets to κ by conditions of size < κ.

Lemma

Add(κ, λ) is strongly κ+-cc. Proof Let M ≺ H(χ), |M| = κ, <κM ⊆ 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 forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings and I. Neeman Generalising MA Our work

Closure

A set in a partial order is directed if every two elements in it have a common upper bound.

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

Closure

A set in a partial order is directed if every two elements in it have a common upper bound.

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 forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings and I. Neeman Generalising MA Our work

Closure

A set in a partial order is directed if every two elements in it have a common upper bound.

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 forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings and I. Neeman Generalising MA Our work

Main result

Theorem

An iteration with supports of size (< κ) of strongly κ+-cc (< κ)-strongly directed closed forcing, is itself strongly κ+-cc.

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

Main result

Theorem

An iteration with supports of size (< κ) of strongly κ+-cc (< κ)-strongly directed closed forcing, is itself strongly κ+-cc. I’ll present some elements of the proof.

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

Main result

Theorem

An iteration with supports of size (< κ) of strongly κ+-cc (< κ)-strongly directed closed forcing, is itself strongly κ+-cc. I’ll present some elements of the proof. Throughout we use the notation M for appropriate models.

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

Main result

Theorem

An iteration with supports of size (< κ) of strongly κ+-cc (< κ)-strongly directed closed forcing, is itself strongly κ+-cc. I’ll present some elements of the proof. Throughout we use the notation M for appropriate models. The support

• f a condition in an iterated forcing is the set of non-trivial

coordinates, denoted by supt.

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

Main result

Theorem

An iteration with supports of size (< κ) of strongly κ+-cc (< κ)-strongly directed closed forcing, is itself strongly κ+-cc. I’ll present some elements of the proof. Throughout we use the notation M for appropriate models. The support

• f 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 forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings and I. Neeman Generalising MA Our work

Main result

Theorem

An iteration with supports of size (< κ) of strongly κ+-cc (< κ)-strongly directed closed forcing, is itself strongly κ+-cc. I’ll present some elements of the proof. Throughout we use the notation M for appropriate models. The support

• f 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 Gα to denote the Pα generic, for α ≤ γ.

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

Main result

Theorem

An iteration with supports of size (< κ) of strongly κ+-cc (< κ)-strongly directed closed forcing, is itself strongly κ+-cc. I’ll present some elements of the proof. Throughout we use the notation M for appropriate models. The support

• f 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 Gα to denote the Pα 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 forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings and I. Neeman Generalising MA Our work

Canonical extensions

Definition

Given p ∈ P, a canonical extension q ≥ p, if it exists, is defined by constructing sequences pi : i < ω and Hi : i < ω so that

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

Canonical extensions

Definition

Given p ∈ P, a canonical extension q ≥ p, if it exists, is defined by constructing sequences pi : i < ω and Hi : i < ω so that

1

for all i, Hi ≺ H(χ) and |Hi| < κ,

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

Canonical extensions

Definition

Given p ∈ P, a canonical extension q ≥ p, if it exists, is defined by constructing sequences pi : i < ω and Hi : i < ω so that

1

for all i, Hi ≺ H(χ) and |Hi| < κ,

2

p, M ∈ H0 and supt(p) ⊆ H0,

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

Canonical extensions

Definition

Given p ∈ P, a canonical extension q ≥ p, if it exists, is defined by constructing sequences pi : i < ω and Hi : i < ω so that

1

for all i, Hi ≺ H(χ) and |Hi| < κ,

2

p, M ∈ H0 and supt(p) ⊆ H0,

3

p0 = p,

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

Canonical extensions

Definition

Given p ∈ P, a canonical extension q ≥ p, if it exists, is defined by constructing sequences pi : i < ω and Hi : i < ω so that

1

for all i, Hi ≺ H(χ) and |Hi| < κ,

2

p, M ∈ H0 and supt(p) ⊆ H0,

3

p0 = p,

4

for all i, pi+1 ≥ pi and pi+1 ∈ D for every open dense D ∈ Hi,

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

Canonical extensions

Definition

Given p ∈ P, a canonical extension q ≥ p, if it exists, is defined by constructing sequences pi : i < ω and Hi : i < ω so that

1

for all i, Hi ≺ H(χ) and |Hi| < κ,

2

p, M ∈ H0 and supt(p) ⊆ H0,

3

p0 = p,

4

for all i, pi+1 ≥ pi and pi+1 ∈ D for every open dense D ∈ Hi,

5

for all i, Hi ∪ {pi+1, Hi} ∪ supt(pi+1) ⊆ Hi+1,

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

Canonical extensions

Definition

Given p ∈ P, a canonical extension q ≥ p, if it exists, is defined by constructing sequences pi : i < ω and Hi : i < ω so that

1

for all i, Hi ≺ H(χ) and |Hi| < κ,

2

p, M ∈ H0 and supt(p) ⊆ H0,

3

p0 = p,

4

for all i, pi+1 ≥ pi and pi+1 ∈ D for every open dense D ∈ Hi,

5

for all i, Hi ∪ {pi+1, Hi} ∪ supt(pi+1) ⊆ Hi+1,

6

pi : i < ω admits a least upper bound,

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

Canonical extensions

Definition

Given p ∈ P, a canonical extension q ≥ p, if it exists, is defined by constructing sequences pi : i < ω and Hi : i < ω so that

1

for all i, Hi ≺ H(χ) and |Hi| < κ,

2

p, M ∈ H0 and supt(p) ⊆ H0,

3

p0 = p,

4

for all i, pi+1 ≥ pi and pi+1 ∈ D for every open dense D ∈ Hi,

5

for all i, Hi ∪ {pi+1, Hi} ∪ supt(pi+1) ⊆ Hi+1,

6

pi : i < ω admits a least upper bound, and then letting q = lubi<ωpi. Let H =

i<ω Hi.

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

Canonical extensions

Definition

Given p ∈ P, a canonical extension q ≥ p, if it exists, is defined by constructing sequences pi : i < ω and Hi : i < ω so that

1

for all i, Hi ≺ H(χ) and |Hi| < κ,

2

p, M ∈ H0 and supt(p) ⊆ H0,

3

p0 = p,

4

for all i, pi+1 ≥ pi and pi+1 ∈ D for every open dense D ∈ Hi,

5

for all i, Hi ∪ {pi+1, Hi} ∪ supt(pi+1) ⊆ Hi+1,

6

pi : i < ω admits a least upper bound, and then letting q = lubi<ωpi. Let H =

i<ω Hi.

Note In our context, every p allows a canonical extension.

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

Properties of canonical extensions

Lemma

Suppose that q is a canonical extension of p. Then:

1

H ≺ H(χ), |H| < κ and p, M ∈ H,

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

Properties of canonical extensions

Lemma

Suppose that q is a canonical extension of p. Then:

1

H ≺ H(χ), |H| < κ and p, M ∈ H,

2

if g = {s ∈ P ∩ H : (∃i < ω)s ≤ pi} then q is the lub

• f g and g = {s ∈ P ∩ H : s ≤ q},

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

Properties of canonical extensions

Lemma

Suppose that q is a canonical extension of p. Then:

1

H ≺ H(χ), |H| < κ and p, M ∈ H,

2

if g = {s ∈ P ∩ H : (∃i < ω)s ≤ pi} then q is the lub

• f g and g = {s ∈ P ∩ H : s ≤ q},

3

g is a filter on P ∩ H which meets every open set in H that is dense above some pi,

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

Properties of canonical extensions

Lemma

Suppose that q is a canonical extension of p. Then:

1

H ≺ H(χ), |H| < κ and p, M ∈ H,

2

if g = {s ∈ P ∩ H : (∃i < ω)s ≤ pi} then q is the lub

• f g and g = {s ∈ P ∩ H : s ≤ q},

3

g is a filter on P ∩ H which meets every open set in H that is dense above some pi,

4

supt(q) = H ∩ γ =

i<ω supt(pi),

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

Properties of canonical extensions

Lemma

Suppose that q is a canonical extension of p. Then:

1

H ≺ H(χ), |H| < κ and p, M ∈ H,

2

if g = {s ∈ P ∩ H : (∃i < ω)s ≤ pi} then q is the lub

• f g and g = {s ∈ P ∩ H : s ≤ q},

3

g is a filter on P ∩ H which meets every open set in H that is dense above some pi,

4

supt(q) = H ∩ γ =

i<ω supt(pi),

5

H ∩ M ∈ M.

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

Properties of canonical extensions

Lemma

Suppose that q is a canonical extension of p. Then:

1

H ≺ H(χ), |H| < κ and p, M ∈ H,

2

if g = {s ∈ P ∩ H : (∃i < ω)s ≤ pi} then q is the lub

• f g and g = {s ∈ P ∩ H : s ≤ q},

3

g is a filter on P ∩ H which meets every open set in H that is dense above some pi,

4

supt(q) = H ∩ γ =

i<ω supt(pi),

5

H ∩ M ∈ M. 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 forcing Mirna Dˇ zamonja, Tutorial 3, including joint work with J. Cummings and I. Neeman Generalising MA Our work

Canonical extensions have residues

Recall g = {s ∈ P ∩ H : s ≤ q}.

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

Canonical extensions have residues

Recall g = {s ∈ P ∩ H : s ≤ q}. Let t = lub(g ∩ M) (note g ⊆ H so |g| < κ).

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

Canonical extensions have residues

Recall g = {s ∈ P ∩ H : s ≤ q}. Let t = lub(g ∩ M) (note g ⊆ H so |g| < κ). We claim t = q|M.

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

Canonical extensions have residues

Recall g = {s ∈ P ∩ H : s ≤ q}. Let t = lub(g ∩ M) (note g ⊆ H so |g| < κ). We claim t = q|M. t ∈ M by the closure of M.

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

Canonical extensions have residues

Recall g = {s ∈ P ∩ H : s ≤ q}. Let t = lub(g ∩ M) (note g ⊆ H so |g| < κ). We claim t = q|M. t ∈ M by the closure of M. q ≥ t since q is a bound for g ⊇ g ∩ M and t is the lub.

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

Canonical extensions have residues

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

Canonical extensions have residues

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

Canonical extensions have residues

Recall g = {s ∈ P ∩ H : s ≤ q}. Let t = lub(g ∩ M) (note g ⊆ H so |g| < κ). We claim t = q|M. t ∈ M by the 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 Pα-name q∗(α).

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

Canonical extensions have residues

Recall g = {s ∈ P ∩ H : s ≤ q}. Let t = lub(g ∩ M) (note g ⊆ H so |g| < κ). We claim t = q|M. t ∈ M by the 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 Pα-name q∗(α). The interesting case is α ∈ supt(t) ∩ supt(q),

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

Canonical extensions have residues

Recall g = {s ∈ P ∩ H : s ≤ q}. Let t = lub(g ∩ M) (note g ⊆ H so |g| < κ). We claim t = q|M. t ∈ M by the 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 Pα-name q∗(α). The interesting case is α ∈ supt(t) ∩ supt(q), so α ∈ M since s ∈ M.

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

Canonical extensions have residues

Recall g = {s ∈ P ∩ H : s ≤ q}. Let t = lub(g ∩ M) (note g ⊆ H so |g| < κ). We claim t = q|M. t ∈ M by the 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 Pα-name q∗(α). The interesting case is α ∈ supt(t) ∩ supt(q), so α ∈ M since s ∈ M. Can assume α ∈ supt(pi) for all i.

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

Canonical extensions have residues

Recall g = {s ∈ P ∩ H : s ≤ q}. Let t = lub(g ∩ M) (note g ⊆ H so |g| < κ). We claim t = q|M. t ∈ M by the 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 Pα-name q∗(α). The interesting case is α ∈ supt(t) ∩ supt(q), so α ∈ M since s ∈ M. Can assume α ∈ supt(pi) for all i. Also, α ∈ H since supt(q) ⊆ H.

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

Di

For each i < ω, let Di collect all u ∈ P such that there is r ∗ ˜ ∈ M ∩ Q ˜

α with

u ↾ α α “u(α) ≥ r ∗ ˜ and r ∗ ˜ is a residue for pi(α) in M[G ˜ α]”.

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

Di

For each i < ω, let Di collect all u ∈ P such that there is r ∗ ˜ ∈ M ∩ Q ˜

α with

u ↾ α α “u(α) ≥ r ∗ ˜ and r ∗ ˜ is a residue for pi(α) in M[G ˜ α]”. Di is in H, since its parameters are in H.

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

Di

For each i < ω, let Di collect all u ∈ P such that there is r ∗ ˜ ∈ M ∩ Q ˜

α with

u ↾ α α “u(α) ≥ r ∗ ˜ and r ∗ ˜ is a residue for pi(α) in M[G ˜ α]”. Di is in H, since its parameters are in H. Di is open.

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

Di

For each i < ω, let Di collect all u ∈ P such that there is r ∗ ˜ ∈ M ∩ Q ˜

α with

u ↾ α α “u(α) ≥ r ∗ ˜ and r ∗ ˜ is a residue for pi(α) in M[G ˜ α]”. Di is in H, since its parameters are in H. Di is open. Di is dense above pi, since Q ˜

α is forced to be strongly κ+-cc

and M[G ˜ α] to be an appropriate model

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

Di

For each i < ω, let Di collect all u ∈ P such that there is r ∗ ˜ ∈ M ∩ Q ˜

α with

u ↾ α α “u(α) ≥ r ∗ ˜ and r ∗ ˜ is a residue for pi(α) in M[G ˜ α]”. Di is in H, since its parameters are in H. Di is open. Di is dense above pi, 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]).

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

Di

For each i < ω, let Di collect all u ∈ P such that there is r ∗ ˜ ∈ M ∩ Q ˜

α with

u ↾ α α “u(α) ≥ r ∗ ˜ and r ∗ ˜ is a residue for pi(α) in M[G ˜ α]”. Di is in H, since its parameters are in H. Di is open. Di is dense above pi, 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 ∩ Di = ∅.

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

More induction

By induction on k < ω we construct ik : k < ω, qk : k < ω and r ∗ ˜

k : k < ω such that

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

More induction

By induction on k < ω we construct ik : k < ω, qk : k < ω and r ∗ ˜

k : k < ω such that

i0 = 0,

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

More induction

By induction on k < ω we construct ik : k < ω, qk : k < ω and r ∗ ˜

k : k < ω such that

i0 = 0, qk ∈ Dik ∩ g as exemplified by r ∗ ˜

k and r ∗

˜

k ∈ H,

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

More induction

By induction on k < ω we construct ik : k < ω, qk : k < ω and r ∗ ˜

k : k < ω such that

i0 = 0, qk ∈ Dik ∩ g as exemplified by r ∗ ˜

k and r ∗

˜

k ∈ H,

ik+1 is such that pik+1 ≥ qk).

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

More induction

By induction on k < ω we construct ik : k < ω, qk : k < ω and r ∗ ˜

k : k < ω such that

i0 = 0, qk ∈ Dik ∩ g as exemplified by r ∗ ˜

k and r ∗

˜

k ∈ H,

ik+1 is such that pik+1 ≥ qk).

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

More induction

By induction on k < ω we construct ik : k < ω, qk : k < ω and r ∗ ˜

k : k < ω such that

i0 = 0, qk ∈ Dik ∩ g as exemplified by r ∗ ˜

k and r ∗

˜

k ∈ H,

ik+1 is such that pik+1 ≥ qk). Use the elementarity of H and the definition of g.

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

More induction

By induction on k < ω we construct ik : k < ω, qk : k < ω and r ∗ ˜

k : k < ω such that

i0 = 0, qk ∈ Dik ∩ g as exemplified by r ∗ ˜

k and r ∗

˜

k ∈ H,

ik+1 is such that pik+1 ≥ qk). Use the elementarity of H and the definition of g. Let rk ∈ Pα+1 have rk ↾ α trivial and rk(α) = r ∗ ˜

k.

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

More induction

By induction on k < ω we construct ik : k < ω, qk : k < ω and r ∗ ˜

k : k < ω such that

i0 = 0, qk ∈ Dik ∩ g as exemplified by r ∗ ˜

k and r ∗

˜

k ∈ H,

ik+1 is such that pik+1 ≥ qk). Use the elementarity of H and the definition of g. Let rk ∈ Pα+1 have rk ↾ α trivial and rk(α) = r ∗ ˜

• k. Then

rk ∈ M ∩ H (def. of Di),

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

More induction

By induction on k < ω we construct ik : k < ω, qk : k < ω and r ∗ ˜

k : k < ω such that

i0 = 0, qk ∈ Dik ∩ g as exemplified by r ∗ ˜

k and r ∗

˜

k ∈ H,

ik+1 is such that pik+1 ≥ qk). Use the elementarity of H and the definition of g. Let rk ∈ Pα+1 have rk ↾ α trivial and rk(α) = r ∗ ˜

• k. Then

rk ∈ M ∩ H (def. of Di), qk ≥ rk.

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

More induction

By induction on k < ω we construct ik : k < ω, qk : k < ω and r ∗ ˜

k : k < ω such that

i0 = 0, qk ∈ Dik ∩ g as exemplified by r ∗ ˜

k and r ∗

˜

k ∈ H,

ik+1 is such that pik+1 ≥ qk). Use the elementarity of H and the definition of g. Let rk ∈ Pα+1 have rk ↾ α trivial and rk(α) = r ∗ ˜

• k. Then

rk ∈ M ∩ H (def. of Di), qk ≥ rk. Hence q ≥ rk and q ↾ α α “r ∗

k

˜ is a residue for pik(α) in M[G ˜ α]”.

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

More induction

By induction on k < ω we construct ik : k < ω, qk : k < ω and r ∗ ˜

k : k < ω such that

i0 = 0, qk ∈ Dik ∩ g as exemplified by r ∗ ˜

k and r ∗

˜

k ∈ H,

ik+1 is such that pik+1 ≥ qk). Use the elementarity of H and the definition of g. Let rk ∈ Pα+1 have rk ↾ α trivial and rk(α) = r ∗ ˜

• k. Then

rk ∈ M ∩ H (def. of Di), qk ≥ rk. Hence q ≥ rk and q ↾ α α “r ∗

k

˜ is a residue for pik(α) in M[G ˜ α]”. Since q ≥ pik+1 ≥ qk, and pis are increasing, we have that q ↾ α α “pi(α), s(α) are compatible”, for all i.

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

More induction

By induction on k < ω we construct ik : k < ω, qk : k < ω and r ∗ ˜

k : k < ω such that

i0 = 0, qk ∈ Dik ∩ g as exemplified by r ∗ ˜

k and r ∗

˜

k ∈ H,

ik+1 is such that pik+1 ≥ qk). Use the elementarity of H and the definition of g. Let rk ∈ Pα+1 have rk ↾ α trivial and rk(α) = r ∗ ˜

• k. Then

rk ∈ M ∩ H (def. of Di), qk ≥ rk. Hence q ≥ rk and q ↾ α α “r ∗

k

˜ is a residue for pik(α) in M[G ˜ α]”. Since q ≥ pik+1 ≥ qk, and pis are increasing, we have that q ↾ α α “pi(α), s(α) are compatible”, for all i. q∗ ↾ α forces that there is an upper bound for all pi(α) and s(α) in Q ˜

α, which we then take as q∗(α).

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

Uses of the theorem

Under exploration.

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

Uses of the theorem

Under exploration. An alternative proof of a result of Shelah on the consistent existence of a universal graph

• n κ+ with 2κ > κ+.

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

Uses of the theorem

Under exploration. An alternative proof of a result of Shelah on the consistent existence of a universal graph

• n κ+ with 2κ > κ+. A whole plethora of universality

results.