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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, 2020Co-Authors: Brent Cody, Hiroshi SakaiAbstract:The Weakly Compact reflection principle $${\mathrm{Refl}}_{{\mathrm{wc}}}(\kappa )$$ Refl wc ( κ ) states that $$\kappa $$ κ is a Weakly Compact cardinal and every Weakly Compact subset of $$\kappa $$ κ has a Weakly Compact proper initial segment. The Weakly Compact reflection principle at $$\kappa $$ κ implies that $$\kappa $$ κ is an $$\omega $$ ω -Weakly Compact cardinal. In this article we show that the Weakly Compact reflection principle does not imply that $$\kappa $$ κ is $$(\omega +1)$$ ( ω + 1 ) -Weakly Compact. Moreover, we show that if the Weakly Compact reflection principle holds at $$\kappa $$ κ then there is a forcing extension preserving this in which $$\kappa $$ κ is the least $$\omega $$ ω -Weakly Compact cardinal. Along the way we generalize the well-known result which states that if $$\kappa $$ κ is a regular cardinal then in any forcing extension by $$\kappa $$ κ -c.c. forcing the nonstationary ideal equals the ideal generated by the ground model nonstationary ideal; our generalization states that if $$\kappa $$ κ is a Weakly Compact cardinal then after forcing with a ‘typical’ Easton-support iteration of length $$\kappa $$ κ 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
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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How much weak Compactness does the Weakly Compact reflection principle imply
arXiv: Logic, 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.
Yu Zhou - One of the best experts on this subject based on the ideXlab platform.
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on super Weakly Compact convex sets and representation of the dual of the normed semigroup they generate
Canadian Mathematical Bulletin, 2013Co-Authors: Li Xin Cheng, Zhenghua Luo, Yu ZhouAbstract:In this note, we first give a characterization of super Weakly Compact convex sets of a Banach space X: a closed bounded convex set K X is super Weakly Compact if and only if there exists a w lower semicontinuous seminorm p with p K sup x2K h ; xi such that p 2 is uniformly Fr´ differentiable on each bounded set of X. Then we present a representation theorem for the dual of the semigroup swcc(X) consisting of all the nonempty super Weakly Compact convex sets of the space X.
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, 2020Co-Authors: Brent Cody, Hiroshi SakaiAbstract:The Weakly Compact reflection principle $${\mathrm{Refl}}_{{\mathrm{wc}}}(\kappa )$$ Refl wc ( κ ) states that $$\kappa $$ κ is a Weakly Compact cardinal and every Weakly Compact subset of $$\kappa $$ κ has a Weakly Compact proper initial segment. The Weakly Compact reflection principle at $$\kappa $$ κ implies that $$\kappa $$ κ is an $$\omega $$ ω -Weakly Compact cardinal. In this article we show that the Weakly Compact reflection principle does not imply that $$\kappa $$ κ is $$(\omega +1)$$ ( ω + 1 ) -Weakly Compact. Moreover, we show that if the Weakly Compact reflection principle holds at $$\kappa $$ κ then there is a forcing extension preserving this in which $$\kappa $$ κ is the least $$\omega $$ ω -Weakly Compact cardinal. Along the way we generalize the well-known result which states that if $$\kappa $$ κ is a regular cardinal then in any forcing extension by $$\kappa $$ κ -c.c. forcing the nonstationary ideal equals the ideal generated by the ground model nonstationary ideal; our generalization states that if $$\kappa $$ κ is a Weakly Compact cardinal then after forcing with a ‘typical’ Easton-support iteration of length $$\kappa $$ κ 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
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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How much weak Compactness does the Weakly Compact reflection principle imply
arXiv: Logic, 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 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.
Li Xin Cheng - One of the best experts on this subject based on the ideXlab platform.
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on super Weakly Compact convex sets and representation of the dual of the normed semigroup they generate
Canadian Mathematical Bulletin, 2013Co-Authors: Li Xin Cheng, Zhenghua Luo, Yu ZhouAbstract:In this note, we first give a characterization of super Weakly Compact convex sets of a Banach space X: a closed bounded convex set K X is super Weakly Compact if and only if there exists a w lower semicontinuous seminorm p with p K sup x2K h ; xi such that p 2 is uniformly Fr´ differentiable on each bounded set of X. Then we present a representation theorem for the dual of the semigroup swcc(X) consisting of all the nonempty super Weakly Compact convex sets of the space X.
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every Weakly Compact set can be uniformly embedded into a reflexive banach space
Acta Mathematica Sinica, 2009Co-Authors: Li Xin Cheng, Qingjin Cheng, Wen ZhangAbstract:Based on an application of the Davis-Figiel-Johnson-Pelzyśki procedure, this note shows that every Weakly Compact subset of a Banach space can be uniformly embedded into a reflexive Banach space. As its application, we present the recent renorming theorems for reflexive spaces of Odell-Schlumprecht and Hajek-Johanis can be extended and localized to Weakly Compact convex subsets of an arbitrary Banach space.
Surjit Singh Khurana - One of the best experts on this subject based on the ideXlab platform.
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Weakly Compact sets in L∞(μ,E)
Journal of Functional Analysis, 2012Co-Authors: Surjit Singh KhuranaAbstract:Abstract Using the theory of liftings, we give simple new proofs of the characterizations of the relatively Weakly Compact subsets and weak Cauchy sequences of L ∞ ( E ) . Also a different proof, of a deep result of J. Diestel, W. Ruess, W. Schachermayer about weak Compactness in L 1 ( E ) , is given.
