Forcing with adequate sets of models as side conditions

Mota

2015

J. Math. Log.

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Abstract. We develop a general framework for forcing with coherent adequate sets on H(λ) as side conditions, where λ ≥ ω 2 is a cardinal of uncountable cofinality. We describe a class of forcing posets which we call coherent adequate type forcings. The main theorem of the paper is that any coherent adequate type forcing preserves CH. We show that there exists a forcing poset for adding a club subset of ω 2 with finite conditions while preserving CH, solving a problem of Friedman [3].

Section: Adding a Club Preserving Chsupporting

confidence: 56%

“…Since (z, C) is closed, if M and M ′ are isomorphic sets in C, then for any a ∈ M ∩ z, σ M,M ′ (a) ∈ z. Thus s satisfies requirement (5) in the definition of P. Since x r ∪ x w ⊆ z and A r ∪ A w ⊆ C, it follows that if s is a condition, then s ≤ r, w.…”

Section: Adding a Club Preserving Chmentioning

confidence: 95%

Coherent adequate forcing and preserving CH

Mota

2015

J. Math. Log.

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“…The next lemma provides some useful technical facts about comparison points. Statement (4) is not very intuitive; however it turns out that this observation simplifies some of the material in the original development of adequate sets in [6]. Lemma 1.16.…”

Section: Introductionmentioning

confidence: 94%

“…An important distinction between Neeman's side conditions and those of Friedman and Mitchell is that the two-type side conditions include both countable and uncountable models. A couple of years later, Krueger [6] developed an alternative framework of side conditions called adequate sets. This approach bases the analysis of side conditions on the ideas of the comparison point and remainder points of two countable models.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Mitchell's theorem revisited

Gilton

Annals of Pure and Applied Logic

2017

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The approachability ideal without a maximal set

Annals of Pure and Applied Logic

2019

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We develop a forcing poset with finite conditions which adds a partial square sequence on a given stationary set, with adequate sets of models as side conditions. We then develop a kind of side condition product forcing for simultaneously adding partial square sequences on multiple stationary sets. We show that certain quotients of such forcings have the ω 1 -approximation property. We apply these ideas to prove, assuming the consistency of a greatly Mahlo cardinal, that it is consistent that the approachability ideal I[ω 2 ] does not have a maximal set modulo clubs.Definition 2.14. A finite set A ⊆ X is said to be adequate if for all M and N in A, either M < N , M ∼ N , or N < M .Note that A is adequate iff for all M and N in A, {M, N } is adequate. If A is adequate and B ⊆ A, then B is adequate. If M and N are in an adequate set A, then either M ≤ N or N ≤ M .This proves the first equality. For the second equality, the reverse inclusion is trivial, and the forward inclusion follows from Proposition 2.11.It turns out that if {M, N } is adequate, then which relation holds between M and N is determined by comparing the ordinals M ∩ ω 1 and N ∩ ω 1 .Lemma 2.17. Let {M, N } be adequate. Then:(Proof. Suppose that M < N , and we will show that M ∩ ω 1 < N ∩ ω 1 . Since β M,N has uncountable cofinality, ω 1 ≤ β M,N . Therefore, M ∩ ω 1 is an initial segment of M ∩ β M,N , and hence is in N . So M ∩ ω 1 < N ∩ ω 1 . Suppose that M ∼ N , and we will show that M ∩ω 1 = N ∩ω 1 .Conversely, if M ∩ ω 1 < N ∩ ω 1 , then the implications which we just proved rule out the possibilities that N < M and N ∼ M . Therefore, M < N . This completes the proof of (1) and (2), and (3) follows immediately.Lemma 2.18. Let A be an adequate set. Then the relation < is irreflexive and transitive on A, ∼ is an equivalence relation on A, ≤ is transitive on A, and the relations < and ≤ respect ∼.Proof. Immediate from Lemma 2.17.We state a closure property of X as an assumption. Assumption 2.19. Suppose that M and N are in X and {M, N } is adequate. Then M ∩ N ∈ X .Proposition 2.28 (Amalgamation over uncountable models). Let A be adequate, P ∈ Y, and suppose that for all M ∈ A, M ∩ P ∈ A. Assume that P is simple. Suppose that B is adequate and A ∩ P ⊆ B ⊆ P. Then A ∪ B is adequate. Proof. See [5, Proposition 1.35]. Lemma 2.29. Suppose that M ∈ X and P ∈ Y. Then M ∼ M ∩ P .Note that by Assumption 2.5(2), M ∩ P ∈ X .Proof. Applying Proposition 2.27 to the adequate set {M }, we get that {M, M ∩P } is adequate. Since ω 1 ≤ P ∩ κ, we have that M ∩ ω 1 = M ∩ P ∩ ω 1 . Hence, by Lemma 2.17(2), M ∼ M ∩ P .We will need one more result about simple models.Lemma 2.30. Suppose that N ∈ X is simple and P ∈ Y ∩ N is simple. Then N ∩ P is simple.Note that by Assumption 2.5(2), N ∩ P ∈ X .Proof. Let M ∈ X be such that M < N ∩ P , and we will show that M ∩ (N ∩ P ) ∈ N ∩ P . It suffices to show that M ∩ N ∩ P < N . For then, since N is simple, M ∩ (N ∩ P ) = (M ∩ N ∩ P ) ∩ N ∈ N, and since P is simple, M ∩ (N ∩ P ) = (M ∩ N ∩ P ) ∩ P ∈ P.