The Experts below are selected from a list of 303 Experts worldwide ranked by ideXlab platform
Saharon Shelah - One of the best experts on this subject based on the ideXlab platform.
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on the Cofinality of the splitting number
Indagationes Mathematicae, 2018Co-Authors: Saharon ShelahAbstract:Abstract The splitting number s can be singular. The key method is to construct a forcing poset with finite support matrix iterations of ccc posets introduced by Blass and Shelah (1989).
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on the Cofinality of the splitting number
2018Co-Authors: Saharon ShelahAbstract:The splitting number can be singular. The key method is to construct a forcing poset with finite support matrix iterations of ccc posets introduced by Blass and the second author "Ultrafilters with small generating sets", Israel J. Math., 65, (1989)
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model theoretic applications of Cofinality spectrum problems
Israel Journal of Mathematics, 2017Co-Authors: M Malliaris, Saharon ShelahAbstract:We apply the recently developed technology of Cofinality spectrum problems to prove a range of theorems in model theory. First, we prove that any model of Peano arithmetic is λ-saturated iff it has Cofinality ≥ λ and the underlying order has no (κ, κ)-gaps for regular κ < λ. We also answer a question about balanced pairs of models of PA. Second, assuming instances of GCH, we prove that SOP 2 characterizes maximality in the interpretability order ∇*, settling a prior conjecture and proving that SOP 2 is a real dividing line. Third, we establish the beginnings of a structure theory for NSOP 2, proving that NSOP 2 can be characterized by the existence of few so-called higher formulas. In the course of the paper, we show that ps = ts in any weak Cofinality spectrum problem closed under exponentiation (naturally defined). We also prove that the local versions of these cardinals need not coincide, even in Cofinality spectrum problems arising from Peano arithmetic.
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Cofinality spectrum problems the axiomatic approach
Topology and its Applications, 2016Co-Authors: M Malliaris, Saharon ShelahAbstract:Abstract Our investigations are framed by two overlapping problems: finding the right axiomatic framework for so-called Cofinality spectrum problems, and a 1985 question of Dow on the conjecturally nonempty (in ZFC) region of OK but not good ultrafilters. We define the lower-Cofinality spectrum for a regular ultrafilter D on λ and show that this spectrum may consist of a strict initial segment of cardinals below λ and also that it may finitely alternate. We define so-called ‘automorphic ultrafilters’ and prove that the ultrafilters which are automorphic for some, equivalently every, unstable theory are precisely the good ultrafilters. We axiomatize a bare-bones framework called “lower Cofinality spectrum problems”, consisting essentially of a single tree projecting onto two linear orders. We prove existence of a lower Cofinality function in this context and show by example that it holds of certain theories whose model theoretic complexity is bounded.
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Cofinality of normal ideals on $$[\lambda ]^{
Archive for Mathematical Logic, 2016Co-Authors: Pierre Matet, Cedric Pean, Saharon ShelahAbstract:An ideal J on $$[\lambda ]^{
Marion Scheepers - One of the best experts on this subject based on the ideXlab platform.
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Cardinals of countable Cofinality and eventual domination
Order, 1994Co-Authors: Marion ScheepersAbstract:We prove embedding theorems for ω-sequences of ordinals below certain ordinals of countable Cofinality.
Pierre Matet - One of the best experts on this subject based on the ideXlab platform.
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Cofinality of normal ideals on $$[\lambda ]^{
Archive for Mathematical Logic, 2016Co-Authors: Pierre Matet, Cedric Pean, Saharon ShelahAbstract:An ideal J on $$[\lambda ]^{
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non saturation of the non stationary ideal on pκ λ with λ of countable Cofinality
Mathematical Logic Quarterly, 2012Co-Authors: Pierre MatetAbstract:Given a regular uncountable cardinal κ and a cardinal λ > κ of Cofinality ω, we show that the restriction of the non-stationary ideal on Pκ(λ) to the set of all a with is not λ++-saturated (and even not -saturated in case 2λ = λ+). We actually prove the stronger result that there is with |Q| = λ++ such that A∩B is a non-cofinal subset of Pκ(λ) for any two distinct members A, B of Q, where NGκ, λ denotes the game ideal on Pκ(λ). We also remark that for κ > ω1, adding λ+3 Cohen subsets of ω1 to makes NGκ, λ λ+3-saturated.
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Non‐saturation of the non‐stationary ideal on Pκ (λ) with λ of countable Cofinality
Mathematical Logic Quarterly, 2012Co-Authors: Pierre MatetAbstract:Given a regular uncountable cardinal κ and a cardinal λ > κ of Cofinality ω, we show that the restriction of the non-stationary ideal on Pκ(λ) to the set of all a with is not λ++-saturated (and even not -saturated in case 2λ = λ+). We actually prove the stronger result that there is with |Q| = λ++ such that A∩B is a non-cofinal subset of Pκ(λ) for any two distinct members A, B of Q, where NGκ, λ denotes the game ideal on Pκ(λ). We also remark that for κ > ω1, adding λ+3 Cohen subsets of ω1 to makes NGκ, λ λ+3-saturated.
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Cofinality of normal ideals onpκ λ ii
Israel Journal of Mathematics, 2005Co-Authors: Pierre Matet, Saharon Shelah, Cedric PeanAbstract:For an idealJ on an infinite setX with add(J)=κ, let\(\overline {cof} (J)\) be the smallest size of any subfamilyY ofJ with the property that any member ofJ can be covered by less than κ members ofY. We study the value of\(\overline {cof} (NS_{\kappa ,\lambda }^{[\delta ]^{< \theta } } |A)\) forA in\((NS_{\kappa ,\lambda }^{[\delta ]^{< \theta } } )^ + \), where\((NS_{\kappa ,\lambda }^{[\delta ]^{< \theta } } \) denotes the smallest [δ]<θ ideal onPκ(λ). We also discuss the problem of whether there exists a setA such that\((NS_{\kappa ,\lambda }^{[\delta ]^{< \theta } } = I_{\kappa ,\lambda } |A\), or even\((NS_{\kappa ,\lambda }^{[\delta ]^{< \theta } } |A = I_{\kappa ,\lambda } |A\).
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Cofinality OF THE NONSTATIONARY IDEAL
Transactions of the American Mathematical Society, 2005Co-Authors: Pierre Matet, Andrzej Roslanowski, Saharon ShelahAbstract:We show that the reduced Cofinality of the nonstationary ideal AS κ on a regular uncountable cardinal κ may be less than its Cofinality, where the reduced Cofinality of NS κ is the least cardinality of any family F of nonstationary subsets of κ such that every nonstationary subset of κ can be covered by less than κ many members of F. For this we investigate connections of the various cofinalities of NS κ with other cardinal characteristics of κ κ and we also give a property of forcing notions (called manageability) which is preserved in
Oleg Pikhurko - One of the best experts on this subject based on the ideXlab platform.
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On the Cofinality of Infinite Partially Ordered Sets: Factoring a Poset into Lean Essential Subsets
Order, 2003Co-Authors: Reinhard Diestel, Oleg PikhurkoAbstract:We study which infinite posets have simple cofinal subsets such as chains, or decompose canonically into such subsets. The posets of countable Cofinality admitting such a decomposition are characterized by a forbidden substructure; the corresponding problem for uncountable Cofinality remains open.
Simon Thomas - One of the best experts on this subject based on the ideXlab platform.
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groupwise density and the Cofinality of the infinite symmetric group
Archive for Mathematical Logic, 1998Co-Authors: Simon ThomasAbstract:We study the relationship between the Cofinality \(c(Sym(\omega))\) of the infinite symmetric group and the cardinal invariants \(\frak{u}\) and \(\frak{g}\). In particular, we prove the following two results. Theorem 0.1It is consistent with ZFC that there exists a simple $P_{\omega_{1}}$-point and that $c(Sym(\omega)) = \omega_{2} = 2^{\omega}$. Theorem 0.2If there exist both a simple $P_{\omega_{1}}$-point and a $P_{\omega_{2}}$-point, then $c(Sym(\omega)) = \omega_{1}$.
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the Cofinality spectrum of the infinite symmetric group
Journal of Symbolic Logic, 1997Co-Authors: Saharon Shelah, Simon ThomasAbstract:Let S be the group of all permutations of the set of natural numbers. The Cofinality spectrum CF(S) of S is the set of all regular cardinals A such that S can be expressed as the union of a chain of i proper subgroups. This paper investigates which sets C of regular uncountable cardinals can be the Cofinality spectrum of S. The following theorem is the main result of this paper. THEOREM. Suppose that V t GCH. Let C be a set of regular uncountable cardinals which satisfies the jollowing coalitions. (a) C contains a maximum element. (b) Iju is an inaccessible cardinal such that ui = sup(C n iu), then ,u E C. (c) I'li is a singular cardinal such that pi = sup(C n iu), then i + E C. Then there exists a ce..c. notion offorcing P such that VP t CF(S) = C. We shall also investigate the connections between the Cofinality spectrum and pef theory; and show that CF(S) cannot be an arbitrarily prescribed set of regular uncountable cardinals. ?
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Unbounded families and the Cofinality of the infinite symmetric group
Archive for Mathematical Logic, 1995Co-Authors: James D. Sharp, Simon ThomasAbstract:In this paper, we study the relationship between the Cofinality c (Sym(ω)) of the infinite symmetric group and the minimal cardinality $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{b} $$ of an unbounded family F of^ ω ω .
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the Cofinality spectrum of the infinite symmetric group
arXiv: Logic, 1994Co-Authors: Saharon Shelah, Simon ThomasAbstract:A group G that is not finitely generated can be written as the union of a chain of proper subgroups. The Cofinality spectrum of G, written CF(S), is the set of regular cardinals lambda such that G can be expressed as the union of a chain of lambda proper subgroups. The Cofinality of G, written c(G), is the least element of CF(G). We show that it is consistent that CF(S) is quite a bizarre set of cardinals. For example, we prove Theorem (A): Let T be any subset of omega setminus {0}. Then it is consistent that aleph_n in CF(S) if and only if n in T . One might suspect that it is consistent that CF(S) is an arbitrarily prescribed set of regular uncountable cardinals, subject only to the above mentioned constraint. This is not the case. Theorem (B): If aleph_n in CF(S) for all n in omega setminus {0}, then aleph_{omega +1} in CF(S) .
