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SOME CONSEQUENCES OF REFLECTION ON THE APPROACHABILITY IDEAL
, 2010
"... Abstract. We study the approachability ideal I[κ +] in the context of large cardinals and properties of the regular cardinals below a singular κ. As a guiding example consider the approachability ideal I[ℵω+1] assuming that ℵω is a strong limit. In this case we obtain that club many points in ℵω+1 o ..."
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Abstract. We study the approachability ideal I[κ +] in the context of large cardinals and properties of the regular cardinals below a singular κ. As a guiding example consider the approachability ideal I[ℵω+1] assuming that ℵω is a strong limit. In this case we obtain that club many points in ℵω+1 of cofinality ℵn for some n>1 are approachable assuming the joint reflection of countable families of stationary subsets of ℵn. This reflection principle holds under MM for all n>1andforeachn>1 is equiconsistent with ℵn being weakly compact in L. This characterizes the structure of the approachability ideal I[ℵω+1] inmodelsofMM. We also apply our result to show that the Chang conjecture (κ+,κ) ↠ (ℵ2, ℵ1) fails in models of MM for all singular cardinals κ. 1. The approachability ideal In the course of development of the pcftheory of possible cofinalities, Shelah has introduced several interesting stationary sets on the successor of a singular cardinal. 1 Among these are the sets of approachable and weakly approachable points in κ +,whereκis a singular cardinal. Given A = {aα: α<κ +}⊆[κ+]<κ, δ is weakly approachable with respect to A if there is an H unbounded in δ of minimal order type such that {H ∩ γ: γ<δ} is covered2 by {aα: α<δ} and δ is approachable with respect to A if there is an H unbounded in δ of minimal order type such that {H ∩ γ: γ<δ}⊆{aα: α<δ}. Definition 1.1. Let κ be a singular cardinal. S is (weakly) approachable if there is a sequence A = {aα: α<κ +}⊆[κ+]<κ and a club C such that δ is (weakly) approachable with respect to α for all δ ∈ S ∩ C. I[κ +] is the ideal generated by approachable sets; I[κ +,κ] is the ideal generated by weakly approachable sets. It is clear that I[κ +] ⊆I[κ +,κ]. For many of the known applications of approachability, it is irrelevant whether we concentrate on the notion of weak approachability or on the apparently stronger notion of approachability. Moreover in the case that κ is a strong limit and singular, I[κ +]=I[κ+,κ] (section 3.4 and proposition 3.23 of [3]). For this reason we feel free to concentrate our attention Received by the editors April 16, 2008.
On two problems of Erdős and Hechler: New methods in singular madness
, 2004
"... For an infinite cardinal µ, MAD(µ) denotes the set of all cardinalities of nontrivial maximal almost disjoint families over µ. Erdős and Hechler proved in [7] the consistency of µ ∈ MAD(µ) for a singular cardinal µ and asked if it was ever possible for a singular µ that µ / ∈ MAD(µ), and also wheth ..."
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For an infinite cardinal µ, MAD(µ) denotes the set of all cardinalities of nontrivial maximal almost disjoint families over µ. Erdős and Hechler proved in [7] the consistency of µ ∈ MAD(µ) for a singular cardinal µ and asked if it was ever possible for a singular µ that µ / ∈ MAD(µ), and also whether 2 cf µ < µ = ⇒ µ ∈ MAD(µ) for every singular cardinal µ. We introduce a new method for controlling MAD(µ) for a singular µ and, among other new results about the structure of MAD(µ) for singular µ, settle both problems affirmatively.
FALLEN CARDINALS
"... Abstract. We prove that for every singular cardinal µ of cofinality ω, the complete Boolean algebra comp Pµ(µ) contains a complete subalgebra which is isomorphic to the collapse algebra Comp Col(ω1, µ ℵ0). Consequently, adding a generic filter to the quotient algebra Pµ(µ) = P(µ)/[µ] <µ collapse ..."
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Abstract. We prove that for every singular cardinal µ of cofinality ω, the complete Boolean algebra comp Pµ(µ) contains a complete subalgebra which is isomorphic to the collapse algebra Comp Col(ω1, µ ℵ0). Consequently, adding a generic filter to the quotient algebra Pµ(µ) = P(µ)/[µ] <µ collapses µ ℵ0 to ℵ1. Another corollary is that the Baire number of the space U(µ) of all uniform ultrafilters over µ is equal to ω2. The corollaries affirm two conjectures of Balcar and Simon. The proof uses pcf theory. 720 revision:20010627 modified:20020227 1.
The pcf trichotomy theorem does not hold for short sequences
 Archive for Mathematical Logic
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THE CONDITION IN THE TRICHOTOMY THEOREM IS OPTIMAL
"... in the TriAbstract. We show that the assumption λ> κ + chotomy Theorem cannot be relaxed to λ> κ. 673 revision:19971228 modified:20020227 1. ..."
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in the TriAbstract. We show that the assumption λ> κ + chotomy Theorem cannot be relaxed to λ> κ. 673 revision:19971228 modified:20020227 1.