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Saharon Shelah - One of the best experts on this subject based on the ideXlab platform.
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Generalizing random real forcing for inaccessible Cardinals
arXiv: Logic, 2016Co-Authors: Shani Cohen, Saharon ShelahAbstract:The two parallel concepts of "small" sets of the real line are meagre sets and null sets. Those are equivalent to Cohen forcing and Random real forcing for aleph_0^aleph_0; in spite of this similarity, the Cohen forcing and Random Real Forcing have very different shapes. One of these differences is in the fact that the Cohen forcing has an easy natural generalization for lambda 2 while lambda greater than aleph 0, corresponding to an extension for the meagre sets, while the Random real forcing didn't seem to have a natural generalization, as Lebesgue measure doesn't have a generalization for space 2 lambda while lambda greater than aleph 0. In work [1], Shelah found a forcing resembling the properties of Random Real Forcing for 2 lambda while lambda is a Weakly Compact Cardinal. Here we describe, with additional assumptions, such a forcing for 2 lambda while lambda is an Inaccessible Cardinal; this forcing is less than lambda-complete and satisfies the lambda^+-c.c hence preserves Cardinals and cofinalities, however unlike Cohen forcing, does not add an undominated real.
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Model-theoretic properties of ultrafilters built by independent families of functions.” math.LO/1208.2579
2015Co-Authors: Mary Malliaris, Saharon ShelahAbstract:Abstract. Via two short proofs and three constructions, we show how to increase the model-theoretic precision of a widely used method for building ultrafilters. We begin by showing that any flexible regular ultrafilter makes the product of an unbounded sequence of finite Cardinals large, thus saturating any stable theory. We then prove directly that a “bottleneck ” in the inductive construction of a regular ultrafilter on λ (i.e. a point after which all antichains of P(λ)/D have Cardinality less than λ) essentially prevents any subsequent ultrafilter from being flexible, thus from saturating any non-low theory. The constructions are as follows. First, we construct a regular filter D on λ so that any ultrafilter extending D fails to λ+-saturate ultrapowers of the random graph, thus of any unstable theory. The proof constructs the omitted random graph type directly. Second, assuming existence of a measurable Cardinal κ, we construct a regular ultrafilter on λ> κ which is λ-flexible but not κ++-good, improving our previous answer to a question raised in Dow 1975. Third, assuming a Weakly Compact Cardinal κ, we construct an ultrafilter to show that lcf(ℵ0) may be small while all symmetric cuts of cofinality κ are realized. Thus certain families of pre-cuts may be realized while still failing to saturate any unstable theory. Our methods advance the general problem of constructing ultrafilters whose ultrapowers have a precise degree of saturation. 1
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model theoretic properties of ultrafilters built by independent families of functions
Journal of Symbolic Logic, 2014Co-Authors: Mary Malliaris, Saharon ShelahAbstract:Our results in this paper increase the model-theoretic precision of a widely used method for building ultrafilters, and so advance the general problem of constructing ultrafilters whose ultrapowers have a precise degree of saturation. We begin by showing that any flexible regular ultrafilter makes the product of an unbounded sequence of finite Cardinals large, thus saturating any stable theory. We then prove directly that a “bottleneck” in the inductive construction of a regular ultrafilter on λ (i.e., a point after which all antichains of have Cardinality less than λ) essentially prevents any subsequent ultrafilter from being flexible, thus from saturating any nonlow theory. The constructions are as follows. First, we construct a regular filter on λ so that any ultrafilter extending fails to -saturate ultrapowers of the random graph, thus of any unstable theory. The proof constructs the omitted random graph type directly. Second, assuming existence of a measurable Cardinal κ, we construct a regular ultrafilter on which is λ-flexible but not -good, improving our previous answer to a question raised in Dow (1985). Third, assuming a Weakly Compact Cardinal κ, we construct an ultrafilter to show that may be small while all symmetric cuts of cofinality κ are realized. Thus certain families of precuts may be realized while still failing to saturate any unstable theory.
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Clones on regular Cardinals
2012Co-Authors: Martin Goldstern, Saharon ShelahAbstract:Abstract. We investigate the structure of the lattice of clones on an infinite set X. We first observe that ultrafilters naturally induce clones; this yields a simple proof of Rosenberg’s theorem: there are 22λ many maximal ( = “precomplete”) clones on a set of size λ. The clones we construct do not contain all unary functions. We then investigate clones that do contain all unary functions. Using a strong negative partition theorem we show that for many Cardinals λ (in particular, for all successors of regulars) there are 22λ many such clones on a set of size λ. Finally, we show that on a Weakly Compact Cardinal there are exactly 2 maximal clones which contain all unary functions. 1
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Clones on regular Cardinals
2012Co-Authors: Martin Goldstern, Saharon ShelahAbstract:Abstract. We investigate the structure of the lattice of clones on an infinite set X. We first observe that ultrafilters naturally induce clones; this yields a simple proof of Rosenberg’s theorem: there are 22λ many maximal ( = “precomplete”) clones on a set of size λ. The clones we construct do not contain all unary functions. We then investigate clones that do contain all unary functions. Using a strong negative partition theorem from pcf theory we show that for many Cardinals λ (in particular, for all successors of regulars) there are 22λ many such clones on a set of size λ. Finally, we show that on a Weakly Compact Cardinal there are exactly 2 maximal clones which contain all unary functions. 1
Brent Cody - One of the best experts on this subject based on the ideXlab platform.
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forcing a κ like principle to hold at a Weakly Compact Cardinal
Annals of Pure and Applied Logic, 2021Co-Authors: Brent Cody, Victoria Gitman, Chris LambiehansonAbstract:Abstract Hellsten [Hel03a] proved that when κ is Π n 1 -indescribable, the n-club subsets of κ provide a filter base for the Π n 1 -indescribability ideal, and hence can also be used to give a characterization of Π n 1 -indescribable sets which resembles the definition of stationarity: a set S ⊆ κ is Π n 1 -indescribable if and only if S ∩ C ≠ ∅ for every n-club C ⊆ κ . By replacing clubs with n-clubs in the definition of □ ( κ ) , one obtains a □ ( κ ) -like principle □ n ( κ ) , a version of which was first considered by Brickhill and Welch [BW] . The principle □ n ( κ ) is consistent with the Π n 1 -indescribability of κ but inconsistent with the Π n + 1 1 -indescribability of κ. By generalizing the standard forcing to add a □ ( κ ) -sequence, we show that if κ is κ + -Weakly Compact and GCH holds then there is a cofinality-preserving forcing extension in which κ remains κ + -Weakly Compact and □ 1 ( κ ) holds. If κ is Π 2 1 -indescribable and GCH holds then there is a cofinality-preserving forcing extension in which κ is κ + -Weakly Compact, □ 1 ( κ ) holds and every Weakly Compact subset of κ has a Weakly Compact proper initial segment. As an application, we prove that, relative to a Π 2 1 -indescribable Cardinal, it is consistent that κ is κ + -Weakly Compact, every Weakly Compact subset of κ has a Weakly Compact proper initial segment, and there exist two Weakly Compact subsets S 0 and S 1 of κ such that there is no β κ for which both S 0 ∩ β and S 1 ∩ β are Weakly Compact.
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The Weakly Compact reflection principle need not imply a high order of weak Compactness
Archive for Mathematical Logic, 2019Co-Authors: Brent Cody, Hiroshi SakaiAbstract:The Weakly Compact reflection principle\({\mathrm{Refl}}_{{\mathrm{wc}}}(\kappa )\) states that \(\kappa \) is a Weakly Compact Cardinal and every Weakly Compact subset of \(\kappa \) has a Weakly Compact proper initial segment. The Weakly Compact reflection principle at \(\kappa \) implies that \(\kappa \) is an \(\omega \)-Weakly Compact Cardinal. In this article we show that the Weakly Compact reflection principle does not imply that \(\kappa \) is \((\omega +1)\)-Weakly Compact. Moreover, we show that if the Weakly Compact reflection principle holds at \(\kappa \) then there is a forcing extension preserving this in which \(\kappa \) is the least \(\omega \)-Weakly Compact Cardinal. Along the way we generalize the well-known result which states that if \(\kappa \) is a regular Cardinal then in any forcing extension by \(\kappa \)-c.c. forcing the nonstationary ideal equals the ideal generated by the ground model nonstationary ideal; our generalization states that if \(\kappa \) is a Weakly Compact Cardinal then after forcing with a ‘typical’ Easton-support iteration of length \(\kappa \) the Weakly Compact ideal equals the ideal generated by the ground model Weakly Compact ideal.
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Forcing a $\square(\kappa)$-like principle to hold at a Weakly Compact Cardinal
arXiv: Logic, 2019Co-Authors: Brent Cody, Victoria Gitman, Chris Lambie-hansonAbstract:Hellsten \cite{MR2026390} proved that when $\kappa$ is $\Pi^1_n$-indescribable, the \emph{$n$-club} subsets of $\kappa$ provide a filter base for the $\Pi^1_n$-indescribability ideal, and hence can also be used to give a characterization of $\Pi^1_n$-indescribable sets which resembles the definition of stationarity: a set $S\subseteq\kappa$ is $\Pi^1_n$-indescribable if and only if $S\cap C\neq\emptyset$ for every $n$-club $C\subseteq\kappa$. By replacing clubs with $n$-clubs in the definition of $\Box(\kappa)$, one obtains a $\Box(\kappa)$-like principle $\Box_n(\kappa)$, a version of which was first considered by Brickhill and Welch \cite{BrickhillWelch}. The principle $\Box_n(\kappa)$ is consistent with the $\Pi^1_n$-indescribability of $\kappa$ but inconsistent with the $\Pi^1_{n+1}$-indescribability of $\kappa$. By generalizing the standard forcing to add a $\Box(\kappa)$-sequence, we show that if $\kappa$ is $\kappa^+$-Weakly Compact and $\mathrm{GCH}$ holds then there is a cofinality-preserving forcing extension in which $\kappa$ remains $\kappa^+$-Weakly Compact and $\Box_1(\kappa)$ holds. If $\kappa$ is $\Pi^1_2$-indescribable and $\mathrm{GCH}$ holds then there is a cofinality-preserving forcing extension in which $\kappa$ is $\kappa^+$-Weakly Compact, $\Box_1(\kappa)$ holds and every Weakly Compact subset of $\kappa$ has a Weakly Compact proper initial segment. As an application, we prove that, relative to a $\Pi^1_2$-indescribable Cardinal, it is consistent that $\kappa$ is $\kappa^+$-Weakly Compact, every Weakly Compact subset of $\kappa$ has a Weakly Compact proper initial segment, and there exist two Weakly Compact subsets $S^0$ and $S^1$ of $\kappa$ such that there is no $\beta
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the Weakly Compact reflection principle need not imply a high order of weak Compactness
arXiv e-prints, 2017Co-Authors: Brent Cody, Hiroshi SakaiAbstract:The Weakly Compact reflection principle $\text{Refl}_{\text{wc}}(\kappa)$ states that $\kappa$ is a Weakly Compact Cardinal and every Weakly Compact subset of $\kappa$ has a Weakly Compact proper initial segment. The Weakly Compact reflection principle at $\kappa$ implies that $\kappa$ is an $\omega$-Weakly Compact Cardinal. In this article we show that the Weakly Compact reflection principle does not imply that $\kappa$ is $(\omega+1)$-Weakly Compact. Moreover, we show that if the Weakly Compact reflection principle holds at $\kappa$ then there is a forcing extension preserving this in which $\kappa$ is the least $\omega$-Weakly Compact Cardinal. Along the way we generalize the well-known result which states that if $\kappa$ is a regular Cardinal then in any forcing extension by $\kappa$-c.c. forcing the nonstationary ideal equals the ideal generated by the ground model nonstationary ideal; our generalization states that if $\kappa$ is a Weakly Compact Cardinal then after forcing with a `typical' Easton-support iteration of length $\kappa$ the Weakly Compact ideal equals the ideal generated by the ground model Weakly Compact ideal.
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the least Weakly Compact Cardinal can be unfoldable Weakly measurable and nearly θ superCompact
Archive for Mathematical Logic, 2015Co-Authors: Brent Cody, Joel David Hamkins, Moti Gitik, Jason SchankerAbstract:We prove from suitable large Cardinal hypotheses that the least Weakly Compact Cardinal can be unfoldable, Weakly measurable and even nearly $${\theta}$$?-superCompact, for any desired $${\theta}$$?. In addition, we prove several global results showing how the entire class of Weakly CompactCardinals, a proper class, can be made to coincide with the class of unfoldable Cardinals, with the class of Weakly measurable Cardinals or with the class of nearly $${\theta_\kappa}$$??-superCompact Cardinals $${\kappa}$$?, for nearly any desired function $${\kappa\mapsto\theta_\kappa}$$????. These results answer several questions that had been open in the literature and extend to these large Cardinals the identity-crises phenomenon, first identified by Magidor with the strongly Compact Cardinals.
Hiroshi Sakai - One of the best experts on this subject based on the ideXlab platform.
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The Weakly Compact reflection principle need not imply a high order of weak Compactness
Archive for Mathematical Logic, 2019Co-Authors: Brent Cody, Hiroshi SakaiAbstract:The Weakly Compact reflection principle\({\mathrm{Refl}}_{{\mathrm{wc}}}(\kappa )\) states that \(\kappa \) is a Weakly Compact Cardinal and every Weakly Compact subset of \(\kappa \) has a Weakly Compact proper initial segment. The Weakly Compact reflection principle at \(\kappa \) implies that \(\kappa \) is an \(\omega \)-Weakly Compact Cardinal. In this article we show that the Weakly Compact reflection principle does not imply that \(\kappa \) is \((\omega +1)\)-Weakly Compact. Moreover, we show that if the Weakly Compact reflection principle holds at \(\kappa \) then there is a forcing extension preserving this in which \(\kappa \) is the least \(\omega \)-Weakly Compact Cardinal. Along the way we generalize the well-known result which states that if \(\kappa \) is a regular Cardinal then in any forcing extension by \(\kappa \)-c.c. forcing the nonstationary ideal equals the ideal generated by the ground model nonstationary ideal; our generalization states that if \(\kappa \) is a Weakly Compact Cardinal then after forcing with a ‘typical’ Easton-support iteration of length \(\kappa \) the Weakly Compact ideal equals the ideal generated by the ground model Weakly Compact ideal.
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the Weakly Compact reflection principle need not imply a high order of weak Compactness
arXiv e-prints, 2017Co-Authors: Brent Cody, Hiroshi SakaiAbstract:The Weakly Compact reflection principle $\text{Refl}_{\text{wc}}(\kappa)$ states that $\kappa$ is a Weakly Compact Cardinal and every Weakly Compact subset of $\kappa$ has a Weakly Compact proper initial segment. The Weakly Compact reflection principle at $\kappa$ implies that $\kappa$ is an $\omega$-Weakly Compact Cardinal. In this article we show that the Weakly Compact reflection principle does not imply that $\kappa$ is $(\omega+1)$-Weakly Compact. Moreover, we show that if the Weakly Compact reflection principle holds at $\kappa$ then there is a forcing extension preserving this in which $\kappa$ is the least $\omega$-Weakly Compact Cardinal. Along the way we generalize the well-known result which states that if $\kappa$ is a regular Cardinal then in any forcing extension by $\kappa$-c.c. forcing the nonstationary ideal equals the ideal generated by the ground model nonstationary ideal; our generalization states that if $\kappa$ is a Weakly Compact Cardinal then after forcing with a `typical' Easton-support iteration of length $\kappa$ the Weakly Compact ideal equals the ideal generated by the ground model Weakly Compact ideal.
Chris Lambie-hanson - One of the best experts on this subject based on the ideXlab platform.
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Simultaneously vanishing higher derived limits
Cambridge University Press (CUP), 2020Co-Authors: Jeffrey Bergfalk, Chris Lambie-hansonAbstract:Abstract In 1988, Sibe Mardešić and Andrei Prasolov isolated an inverse system $\textbf {A}$ with the property that the additivity of strong homology on any class of spaces which includes the closed subsets of Euclidean space would entail that $\lim ^n\textbf {A}$ (the nth derived limit of $\textbf {A}$ ) vanishes for every $n>0$ . Since that time, the question of whether it is consistent with the $\mathsf {ZFC}$ axioms that $\lim ^n \textbf {A}=0$ for every $n>0$ has remained open. It remains possible as well that this condition in fact implies that strong homology is additive on the category of metric spaces. We show that assuming the existence of a Weakly Compact Cardinal, it is indeed consistent with the $\mathsf {ZFC}$ axioms that $\lim ^n \textbf {A}=0$ for all $n>0$ . We show this via a finite-support iteration of Hechler forcings which is of Weakly Compact length. More precisely, we show that in any forcing extension by this iteration, a condition equivalent to $\lim ^n\textbf {A}=0$ will hold for each $n>0$ . This condition is of interest in its own right; namely, it is the triviality of every coherent n-dimensional family of certain specified sorts of partial functions $\mathbb {N}^2\to \mathbb {Z}$ which are indexed in turn by n-tuples of functions $f:\mathbb {N}\to \mathbb {N}$ . The triviality and coherence in question here generalise the classical and well-studied case of $n=1$ .
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Forcing a $\square(\kappa)$-like principle to hold at a Weakly Compact Cardinal
arXiv: Logic, 2019Co-Authors: Brent Cody, Victoria Gitman, Chris Lambie-hansonAbstract:Hellsten \cite{MR2026390} proved that when $\kappa$ is $\Pi^1_n$-indescribable, the \emph{$n$-club} subsets of $\kappa$ provide a filter base for the $\Pi^1_n$-indescribability ideal, and hence can also be used to give a characterization of $\Pi^1_n$-indescribable sets which resembles the definition of stationarity: a set $S\subseteq\kappa$ is $\Pi^1_n$-indescribable if and only if $S\cap C\neq\emptyset$ for every $n$-club $C\subseteq\kappa$. By replacing clubs with $n$-clubs in the definition of $\Box(\kappa)$, one obtains a $\Box(\kappa)$-like principle $\Box_n(\kappa)$, a version of which was first considered by Brickhill and Welch \cite{BrickhillWelch}. The principle $\Box_n(\kappa)$ is consistent with the $\Pi^1_n$-indescribability of $\kappa$ but inconsistent with the $\Pi^1_{n+1}$-indescribability of $\kappa$. By generalizing the standard forcing to add a $\Box(\kappa)$-sequence, we show that if $\kappa$ is $\kappa^+$-Weakly Compact and $\mathrm{GCH}$ holds then there is a cofinality-preserving forcing extension in which $\kappa$ remains $\kappa^+$-Weakly Compact and $\Box_1(\kappa)$ holds. If $\kappa$ is $\Pi^1_2$-indescribable and $\mathrm{GCH}$ holds then there is a cofinality-preserving forcing extension in which $\kappa$ is $\kappa^+$-Weakly Compact, $\Box_1(\kappa)$ holds and every Weakly Compact subset of $\kappa$ has a Weakly Compact proper initial segment. As an application, we prove that, relative to a $\Pi^1_2$-indescribable Cardinal, it is consistent that $\kappa$ is $\kappa^+$-Weakly Compact, every Weakly Compact subset of $\kappa$ has a Weakly Compact proper initial segment, and there exist two Weakly Compact subsets $S^0$ and $S^1$ of $\kappa$ such that there is no $\beta
Luca Motto Ros - One of the best experts on this subject based on the ideXlab platform.
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The descriptive set-theoretical complexity of the embeddability relations on models of large size
2015Co-Authors: Luca Motto RosAbstract:Abstract. We show that if κ is a Weakly Compact Cardinal then the embed-dability relation on (generalized) trees of size κ is invariantly universal. This means that for every analytic quasi-order R on the generalized Cantor space κ2 there is an Lκ+κ-sentence ϕ such that the embeddability relation on its models of size κ, which are all trees, is Borel bi-reducible (and, in fact, class-wise Borel isomorphic) to R. In particular, this implies that the relation of embeddability on trees of size κ is complete for analytic quasi-orders on κ2. These facts generalize analogous results for κ = ω obtained in [LR05, FMR11], and it also partially extends a result from [Bau76] concerning the structure of the embeddability relation on linear orders of size κ. 1
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the descriptive set theoretical complexity of the embeddability relation on models of large size
Annals of Pure and Applied Logic, 2013Co-Authors: Luca Motto RosAbstract:Abstract We show that if κ is a Weakly Compact Cardinal then the embeddability relation on (generalized) trees of size κ is invariantly universal. This means that for every analytic quasi-order R on the generalized Cantor space 2 κ there is an L κ + κ -sentence φ such that the embeddability relation on its models of size κ , which are all trees, is Borel bi-reducible (and, in fact, classwise Borel isomorphic) to R . In particular, this implies that the relation of embeddability on trees of size κ is complete for analytic quasi-orders on 2 κ . These facts generalize analogous results for κ = ω obtained in Louveau and Rosendal (2005) [17] and Friedman and Motto Ros (2011) [6] , and it also partially extends a result from Baumgartner (1976) [3] concerning the structure of the embeddability relation on linear orders of size κ .
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the descriptive set theoretical complexity of the embeddability relation on models of large size
arXiv: Logic, 2011Co-Authors: Luca Motto RosAbstract:We show that if \kappa\ is a Weakly Compact Cardinal then the embeddability relation on (generalized) trees of size \kappa\ is invariantly universal. This means that for every analytic quasi-order R on the generalized Cantor space 2^\kappa\ there is an L_{\kappa^+ \kappa}-sentence \phi\ such that the embeddability relation on its models of size \kappa, which are all trees, is Borel bireducible (and, in fact, classwise Borel isomorphic) to R. In particular, this implies that the relation of embeddability on trees of size \kappa\ is complete for analytic quasi-orders. These facts generalize analogous results for \kappa=\omega\ obtained in [LR05, FMR11], and it also partially extends a result from [Bau76] concerning the structure of the embeddability relation on linear orders of size \kappa.
