On the existence of large subsets of [λ] <κ
نویسنده
چکیده
On the existence of large subsets of [λ] <κ which contain no unbounded non-stationary subsets * Here we deal with some problems posed by Matet. The first section deals with the existence of stationary subsets of [λ] <κ with no unbounded subsets which are not stationary, where, of course, κ is regular uncountable ≤ λ. In the second section we deal with the existence of such clubs. The proofs are easy but the result seems to be very surprising. Theorem 1.2 was proved some time ago by Baumgartner (see Theorem 2.3 of [Jo88]) and is presented here for the sake of completeness. Section 1: On stationary sets with no unbounded stationary subsets The following quite answers the question of Matet indicated in the title of this section. Theorems 1.2 and 1.4 yield existence under reasonable cardinal-arithmetical assumptions. Theorem 1.5 yields consistency. Remark 1.3 recalls that under the assumption λ = λ <κ there is no such set. 1.0 NOTATION: H(χ) is the family of sets x such that TC(x), the transitive closure of x, has cardinality < χ, and < * χ is any well ordering of H(χ). 1.1 DEFINITION: (1) For κ regular uncountable and κ ≤ λ, let E λ,κ , the club filter of [λ] <κ , be the filter generated by the clubs of [λ] <κ , where a club of [λ] <κ (or an E λ,κ-club) is a set of the form C M = def {a ∈ [λ] <κ : a = cl M (a) ∧ a ∩ κ ∈ κ} where M is a model with universe λ and countable vocabulary. Note that we could have required in the definition of C M that M restricted to a be an elementary submodel of M, but it does not matter as we can expand M by Skolem functions. Note that, as a consequence, only the functions of M matter. (2) We call S ⊆ [λ] <κ stationary if it's complement [λ] <κ \ S does not belongs to E λ,κ. (3) We call S ⊆ [λ] <κ unbounded if every member of [λ] <κ is contained in some member of S, so cf [λ] <κ , ⊆ is exactly M in{|U | : U ⊆ [λ] <κ is unbounded}. (4) In part (1) we can replace λ by any set A which includes κ. We deal first with a special case. 1.2 THEOREM: if …
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