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43
Iteration trees
 J. OF AMER. MATH. SOCIETY
, 1994
"... 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 $\Delta^1_{n+2}$ wello ..."
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Cited by 20 (2 self)
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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''.
Randomness in effective descriptive set theory
 London. Math. Soc
"... Abstract. An analog of MLrandomness 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 KraftChaitin Theorem and Schnorr’s Theorem. In the new setting, while Ktrivial sets exist that are not hyper ..."
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Cited by 9 (3 self)
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Abstract. An analog of MLrandomness 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 KraftChaitin Theorem and Schnorr’s Theorem. In the new setting, while Ktrivial sets exist that are not hyperarithmetical, 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.
On long increasing chains modulo flat ideals ∗ 1,2
, 2009
"... 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 Th ..."
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Cited by 3 (1 self)
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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:20090823 modified:20090824 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
The effective theory of Borel equivalence relations
 Annals of Pure and Applied Logic
"... The study of Borel equivalence relations under Borel reducibility has developed into an important area of descriptive set theory. The dichotomies of Silver ([19]) and HarringtonKechrisLouveau ([5]) show that with respect to Borel reducibility, any Borel equivalence relation strictly above equality ..."
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Cited by 2 (2 self)
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The study of Borel equivalence relations under Borel reducibility has developed into an important area of descriptive set theory. The dichotomies of Silver ([19]) and HarringtonKechrisLouveau ([5]) show that with respect to Borel reducibility, any Borel equivalence relation strictly above equality on ω is above equality on P(ω), the power set of ω, and any Borel equivalence relation strictly above equality on the reals is above equality modulo finite on P(ω). In this article we examine the effective content of these and related results by studying effectively Borel equivalence relations under effectively Borel reducibility. The resulting structure is complex, even for equivalence relations with finitely many equivalence classes. However use of Kleene’s O as a parameter is sufficient to restore the picture from the noneffective setting. A key lemma is the existence of two effectively Borel sets of reals, neither of which contains the range of the other under any effectively Borel function; the proof of this result applies Barwise compactness to a deep theorem of Harrington (see [6]) establishing for any recursive ordinal α the existence of Π0 1 singletons whose αjumps are Turing incomparable. 1
RANDOMNESS VIA EFFECTIVE DESCRIPTIVE SET THEORY
"... An analog of MLrandomness 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 KraftChaitin Theorem and Schnorr’s Theorem. In the new setting, while Ktrivial sets exist that are not hyperarithmetica ..."
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Cited by 1 (1 self)
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An analog of MLrandomness 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 KraftChaitin Theorem and Schnorr’s Theorem. In the new setting, while Ktrivial 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.
www.stacsconf.org IS RAMSEY’S THEOREM ωAUTOMATIC?
"... Centre national de la recherche scientifique (CNRS) and ..."
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Centre national de la recherche scientifique (CNRS) and
Strong limits and Inaccessibility with nonwellorderable powersets
 ILLC PUBLICATIONS
, 2006
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