Perfect-Set Properties inL(R)[U]

Prisco, Carlos Augusto Di; Todorčević, Stevo

doi:10.1006/aima.1998.1752

Cited by 15 publications

(19 citation statements)

References 20 publications

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“…It is consistent with ZF + DC that (i) there exists a nonprincipal ultrafilter U over ω; and [7] have shown that many of the regularity properties of…”

Section: Theorem 16 (Lc)supporting

confidence: 59%

On the Steinhaus and Bergman Properties for Infinite Products of Finite Groups

Thomas¹,

Zapletal²

2012

Confluentes Math.

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“…It is consistent with ZF + DC that (i) there exists a nonprincipal ultrafilter U over ω; and [7] have shown that many of the regularity properties of…”

Section: Theorem 16 (Lc)supporting

confidence: 59%

On the Steinhaus and Bergman Properties for Infinite Products of Finite Groups

Thomas¹,

Zapletal²

2012

Confluentes Math.

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“…In [3] there is a proof of the following general result, that we are going to use, about Solovay models. 1…”

Section: Solovay Models and The Abstract Mathias Forcingmentioning

confidence: 99%

“…. , b r be a list of all the b's in AR(B n ) with depth Bn (b) = n. P r o o f. We adapt the argument from [3]. Take (a, A) ∈ M. Let p be the real parameter in the definition of S. Let α < κ be such that p, [a, A] ∈ L[G α0 ] (where G α0 = G ∩ Coll(ω, ≤ α 0 )).…”

Section: Solovay Models and The Abstract Mathias Forcingmentioning

confidence: 99%

A notion of selective ultrafilter corresponding to topological Ramsey spaces

Mijares¹

2007

Mathematical Logic Qtrly

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We introduce the relation of almost-reduction in an arbitrary topological Ramsey space R as a generalization of the relation of almost-inclusion on N [∞] . This leads us to a type of ultrafilter U ⊆ R which corresponds to the well-known notion of selective ultrafilter on N. The relationship turns out to be rather exact in the sense that it permits us to lift several well-known facts about selective ultrafilters on N and the Ellentuck space N [∞] to the ultrafilter U and the Ramsey space R. For example, we prove that the open coloring axiom holds on L(R) [U], extending therefore the result from [3] which gives the same conclusion for the Ramsey space N [∞] . PreliminariesWe follow [13] in describing what a topological Ramsey space is, rather than the earlier reference [2], where a slightly different definition is given. Consider triplets of the form (R, ≤, (r n ) n∈N ), where R is a set, ≤ is a pre-order on R, and for every n ∈ N, r n : R −→ AR n is a function with range AR n . If A ≤ B, we say that A is a reduction of B; and for each A ∈ R, we say that r n (A) is the nth approximation of A. We will assume that the following are satisfied: (A1) For any A ∈ R, r 0 (A) = ∅. (A2) For any A, B ∈ R, if A = B, then there exists n such that r n (A) = r n (B). (A3) If r n (A) = r m (B), then n = m and for all i < n, r i (A) = r i (B).These three assumptions allow us to identify each A ∈ R with the sequence (r n (A)) n of its approximations. In this way, if we consider the space AR := n AR n with the discrete topology, we can identify R with a subspace of the (metric) space AR N (with the product topology) of all the sequences of elements of AR. Via this identification, we will regard R as a subspace of AR N , and we will say that R is metrically closed if it is a closed subspace of AR N .Moreover, for a ∈ AR we define the length of a, |a|, as the unique n such that a = r n (A) for some A ∈ R. We will further identify a with the sequence {r i (A)} i≤n . So if a = r n (A) and a = r i (A) for i ≤ n, then we write a = r i (a) (that is, we extend the domain of the function r i to the set of a ∈ AR with i ≤ |a|). In this case, we also write a a and say that a is an initial segment of a. We shall also consider on R the Ellentuck type neighborhoods [a, A] = {B ∈ R : ∃n(a = r n (B)) and B ≤ A}, where a ∈ AR and A ∈ R. If [a, A] = ∅, we will say that a is compatible with A (or A is compatible with a). Let AR(A) = {a ∈ AR : a is compatible with A}. *

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“…On one hand, this model contains an ultrafilter on ω, which is a non-measurable set without the Baire property. On the other hand, many consequences of determinacy survive, such as the perfect set property [5]. In this section, we will study the reducibility of Borel equivalence relations in this model and show how the work of the previous chapters can be applied to this end.…”

Section: Ultrafilter Modelsmentioning

confidence: 99%

Canonical Ramsey Theory on Polish Spaces

Kanovei

Sabok

Zapletal

2013

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This book lays the foundations for an exciting new area of research in descriptive set theory. It develops a robust connection between two active topics: forcing and analytic equivalence relations. This in turn allows the authors to develop a generalization of classical Ramsey theory. Given an analytic equivalence relation on a Polish space, can one find a large subset of the space on which it has a simple form? The book provides many positive and negative general answers to this question. The proofs feature proper forcing and Gandy–Harrington forcing, as well as partition arguments. The results include strong canonization theorems for many classes of equivalence relations and sigma-ideals, as well as ergodicity results in cases where canonization theorems are impossible to achieve. Ideal for graduate students and researchers in set theory, the book provides a useful springboard for further research.

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Perfect-Set Properties inL(R)[U]

Abstract: We study perfect-set properties in the model L(R)[U] when L(R) is (elementarily equivalent to) a Solovay model and U is a selective ultrafilter on the integers, generic over L(R).

Cited by 15 publications

References 20 publications

On the Steinhaus and Bergman Properties for Infinite Products of Finite Groups

On the Steinhaus and Bergman Properties for Infinite Products of Finite Groups

A notion of selective ultrafilter corresponding to topological Ramsey spaces

Canonical Ramsey Theory on Polish Spaces

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