A correction to "A non-implication between fragments of Martin's Axiom related to a property which comes from Aronszajn trees"

In the paper A non-implication between fragments of Martin’s Axiom related to a property which comes from Aronszajn trees [1], Proposition 2.7 is not true. To avoid this error and correct Proposition 2.7, the definition of the property R1,א1 is changed. In [1], all proofs of lemmas and theorems but Lemma 6.9 are valid about this definition without changing the proofs. We give a new statement and a new proof of Lemma 6.9. In the paper A non-implication between fragments of Martin’s Axiom related to a property which comes from Aronszajn trees [1], Proposition 2.7 is not true. For example, T is an Aronszajn tree, t1 and t3 are incomparable node of T in a model N , t2 is a node of T such that t2 6∈ N and t1 <T t2, σ := {t2, t3} (which is in a(T )) and I be an uncountable subset of a(T ) which forms a ∆-system with root {t1, t3}. Then σ ∩ N = {t1} ⊆ {t1, t3}, but every element of I is incompatible with σ in a(T ). To avoid this error and correct Proposition 2.7, the definition of the property R1,א1 is changed as follows. Theorem 2.6. A forcing notion Q in FSCO has the property R1,א1 if for any regular cardinal κ larger than א1, countable elementary submodel N of H(κ) which has the set {Q}, I ∈ [Q]א1 ∩N and σ ∈ Q\N , if I forms a ∆-system with root (exactly) σ ∩N , then there exists I ′ ∈ [I]א1 ∩N such that every member of I ′ is compatible with σ in Q. Similarly, we should also change Proposition 2.8 and Proposition 2.10.2 as follows. Proposition 2.8. The property R1,א1 is closed under finite support products in the following sense. If {Qξ; ξ ∈ Σ} is a set of forcing notions in FSCO with the property R1,א1 , κ is a large enough regular cardinal, N is a countable elementary submodel of Email addresses: [email protected] (Teruyuki Yorioka) Preprint submitted to Annals of Pure and Applied Logic February 10, 2011 H(κ) which has the set {{Qξ; ξ ∈ Σ}}, I is an uncountable subset of the finite support product ∏ ξ∈Σ Qξ in N , ~σ ∈ ∏ ξ∈Σ Qξ \N , I forms a ∆-system with root (exactly) ~σ ∩N , that is, • the set {supp(~τ); τ ∈ I} forms a ∆-system with root (exactly) supp(~σ)∩N , where supp(~τ) := {ξ ∈ Σ;~τ(ξ) 6= ∅}, • for each ξ ∈ supp(~σ)∩N , the set {~τ(ξ); τ ∈ I} forms a ∆-system with root (exactly) ~σ(ξ) ∩N , then there exists I ′ ∈ [I]א1 ∩ N such that every element of I ′ is compatible with ~σ in ∏ ξ∈Σ Qξ. Proposition 2.10.2. Let Q be a forcing notion in FSCO with the property R1,א1 . Suppose that κ is a regular cardinal larger than א1, N is a countable elementary submodel of H(κ) which has the set {Q}, 〈Ii; i ∈ n〉 is a finite sequence of members of the set [Q]א1 ∩ N , and σ ∈ Q \ N such that the union ∪ i∈n Ii forms a ∆-system with root (exactly) σ ∩N . Then there exists 〈τi; i ∈ n〉 ∈ ∏ i∈n Ii such that there exists a common extension of σ and the τi in Q. The new definition of the property R1,א1 is less restrictive. All examples in the paper [1] has this property. In [1], all proofs of lemmas and theorems but Lemma 6.9 are valid about this definition without changing the proofs. For example, in the proof of Proposition 2.7, we have only to check for an uncountable subset I of a(P) in a countable elementary submodel N of H(κ) and σ ∈ a(P) \N such that I forms a ∆-system with root σ ∩N . The proof of this proposition is completely same to the one in [1]. The proofs of Theorems 5.3 and 5.4 are adopted fot this new definition. Because the property R1,א1 are applied for uncountable sets which form ∆-systems with root exact “ τ ∩N ” in the proofs of Theorems 5.3 and 5.4 in [1]. We apply the new Proposition 2.10.2 to these ∆-systems. We have to change only the statement and the proof of Lemma 6.9 as follows. Lemma 6.9. Suppose that Q is a forcing notion in FSCO with the property R1,א1 , I is an uncountable subset of Q such that • I forms a ∆-system with root , and • for every σ and τ in I, either max(σ \ ) < min(τ \ ) or max(τ \ ) < min(σ \ ), ~ M = 〈Mα;α ∈ ω1〉 is a sequence of countable elementary submodels of H(א2) such that {Q, I} ∈ M0, and for every α ∈ ω1, 〈Mβ ;β ∈ α〉 ∈ Mα, and S ⊆ ω1 \ {0} is stationary. Then Q(Q, I, ~ M, S) is (T, S)-preserving.