In this paper we shall develop some ideas which lead to a proof of the following Theorem. Suppose there are $n$ Woodin cardinals, where $n<\omega$. Then there is a proper class model $M_n$ such that $M_n$ models ZFC + ``There are $n$ Woodin cardinals'' + ``$\mathbb{R}$ has a $\Delta^1_{n+2}$ wellorder''.

Abstract. An analog of ML-randomness in the effective descriptive set theory setting is studied, where the r.e. objects are replaced by their Π1 1 counterparts. We prove the analogs of the Kraft-Chaitin Theorem and Schnorr’s Theorem. In the new setting, while K-trivial sets exist that are not hyper-arithmetical, each low for random set is. Finally we study a very strong yet effective randomness notion: Z is strongly random if Z is in no null Π1 1 set of reals. We show that there is a greatest Π1 1 null set, that is, a universal test for this notion. 1.

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Received xxx, revised xxx, accepted xxx Published online xxx Key words increasing chain, order modulo an ideal MSC (2000) 03E05 03E10 We prove that, e.g., in (ω3) (ω3) there is no sequence of length ω4 increasing modulo the ideal of countable sets. Copyright line will be provided by the publisher This note is concerned with the depth of the partial order of the functions in κ γ modulo the ideal of the form I = [κ] <µ. Let us recall the following definitions. Definition 1 For a partial order (P, ⊏) we define • Depth(P, ⊏) = sup{|F | : F ⊆ P is well–ordered by ⊏ } [the depth] • cf(P, ⊏) = min{|F | : F ⊆ P is ⊏–cofinal which mean that for every p ∈ P there is q ∈ F such that p ⊑ p} [the cofinality]. 908 revision:2009-08-23 modified:2009-08-24 Our result (Theorem 4) states that under suitable assumptions the depth of the partial order ( κ γ, < [κ] <µ) is at most |γ|. In particular, letting µ = ℵ1, κ = |γ | = ℵ3 we obtain that in (ω3) (ω3) there is no sequence of length ω4 increasing modulo the ideal of countable sets. Let κ = cf(κ)> ℵ0. If µ = κ, then Depth ( κ κ, < J bd κ) can be (forced to be) large. But for µ> Depth ( κ µ, < J bd κ) this implies pcf results (see [Sh 410], [Sh 589]). However, e.g., for the ideal I = [ω3] ≤ℵ0 it is harder to get long increasing sequence, as above for “high µ”, this leads to pcf results e.g. if we assume that ¯ λ = 〈λi: i < ω3 〉 ∈ ω3 Reg, and in ( ∏ ¯ λ, <I) there is an increasing sequence moduo I of length say> 2 ℵ3 +sup{λi: i < ω3} are much stronger than known consistency results. Even for I = [ω1] ≤ℵ0 we do not know, for I = [ℶω] ≤ℵ0 we know ([Sh 460]), so even [ℵω] ≤ℵ0 would be interesting good news. We hope sometime to prove, e.g., Conjecture 2 For every µ> θ, in (θ+3) µ there is no increasing sequence of length µ modulo [θ

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An analog of ML-randomness in the effective descriptive set theory setting is studied, where the r.e. objects are replaced by their Π1 1 counterparts. We prove the analogs of the Kraft-Chaitin Theorem and Schnorr’s Theorem. In the new setting, while K-trivial sets exist that are not hyperarithmetical, each low for random set is. Finally, we begin to study a very strong yet effective randomness notion: Z is Π1 1 random if Z is in no null Π1 1 class. There is a greatest Π1 1 null class, that is, a universal test for this notion.

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Abstract. We investigate complete Boolean algebras that carry a continuous submeasure.

Contributions to the theory of weakly distributive complete Boolean algebras Bohuslav BALCAR and Thomas JECH 1