Proper and Improper Forcing

Shelah considered a certain version of Strong Chang's Conjecture which we denote SCC cof , and proved that it is equivalent to several statements, including the assertion that Namba forcing is semiproper. We introduce an apparently weaker version, denoted SCC split , and prove an analogous characterization of it. In particular, SCC split is equivalent to the assertion that the the Friedman‐Krueger poset is semiproper. This strengthens and sharpens results by Cox and sheds some light on problems posed by Usuba, Torres‐Perez and Wu.

Section: Claim 34 Suppose M ∈ S Then Whenever Y Is a Chang -Extensimentioning

confidence: 99%

A variant of Shelah's characterization of Strong Chang's Conjecture

Cox

Sakai

2019

“…Definition 1. 6 We write CC * for the assertion that, for every large regular ϑ, well-ordering on H ϑ , and every countable M ≺ H ϑ , ∈, , there is an ω 1…”

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confidence: 99%

Bounded dagger principles

Usuba

2014

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confidence: 99%

“…The reason is that there are two versions of the Proper Iteration Lemma in the literature. The version which is better known, which we may call the "Weak Proper Iteration Lemma", is given in [1, Lemma 2.6], [2, Lemma 6.1.3], [3, Lemma 3.18], [8,9, Theorem III.3.2], and [9, Lemma III.3.3H].The stronger version of the Proper Iteration Lemma uses a weaker hypothesis. A proof of this Lemma is given in [7, Lemma 25].…”

mentioning

confidence: 99%

“…There are two versions of the Proper Iteration Lemma. The stronger (but less well-known) version can be used to give simpler proofs of iteration theorems (e.g., [7, Lemma 24] versus [9, Theorem IX.4.7]). In this paper we give another demonstration of the fecundity of the stronger version by giving a short proof of Shelah's theorem on the preservation of the ω ω -bounding property.…”

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confidence: 99%

See 1 more Smart Citation

A short proof of the preservation of the ω^ω‐bounding property

Schlindwein

2003

There are two versions of the Proper Iteration Lemma. The stronger (but less well-known) version can be used to give simpler proofs of iteration theorems (e.g., [7, Lemma 24] versus [9, Theorem IX.4.7]). In this paper we give another demonstration of the fecundity of the stronger version by giving a short proof of Shelah's theorem on the preservation of the ω ω -bounding property.In this paper we give an account of the following theorem of Shelah:Theorem A Suppose P η : η ≤ κ is a countable support forcing iteration of ω ω -bounding proper forcing notions. Then P κ is ω ω -bounding.The proof we give is a simplified version of [9, Conclusion VI.2.8D] building on the main idea of [9, Theorem VI.1.12]. The proof of Theorem A given in [9] is also given in [1]. We also note [3], which gives a slightly weaker result. The argument of [5] is not correct ([6]).There is an interesting reason why we are able to give a simpler argument. The reason is that there are two versions of the Proper Iteration Lemma in the literature. The version which is better known, which we may call the "Weak Proper Iteration Lemma", is given in [1, Lemma 2.6], [2, Lemma 6.1.3], [3, Lemma 3.18], [8, 9, Theorem III.3.2], and [9, Lemma III.3.3H]. The stronger version of the Proper Iteration Lemma uses a weaker hypothesis. A proof of this Lemma is given in [7, Lemma 25]. Here we merely give the statement of the Lemma and refer the reader to [7] for a proof.Proper Iteration Lemma Suppose P η : η ≤ κ is a countable support forcing iteration based on Q η : η < κ and for every η < κ we have that 1 Pη "Q η is proper". Suppose also that α < κ, λ is a sufficiently large regular cardinal, N is a countable elementary submodel of H λ , {P κ , α} ∈ N , p ∈ P α is N -generic, and p "q ∈Ṗ α,κ ∩ N [G Pα ]". Then there is r ∈ P κ such that r is N -generic, r α = p, p "r [α, κ) ≤q", and supp(r) ⊆ α ∪ N .We now turn to the ω ω -bounding property. Definition Suppose f and g are elements of ω ω.P r o o f o f T h e o r e m A . We show that P κ is ω ω -bounding by induction on κ. We may assume that κ is a limit ordinal. Suppose p ∈ P κ and p "f ∈ ω ω". We show that there is an h ∈ ω ω and r ≤ p such that r "f ≤ * ȟ ".Let λ be a sufficiently large regular cardinal (in particular, the set H(λ) contains the power set of P κ , the power set of ω 1 , and the name f ) and let N be a countable elementary substructure of H λ containing {P