scholarly journals WEAK DISTRIBUTIVITY IMPLYING DISTRIBUTIVITY

2016 ◽  
Vol 81 (2) ◽  
pp. 711-717
Author(s):  
DAN HATHAWAY
Keyword(s):  
Boolean Algebra ◽  
Complete Boolean Algebra ◽  
Infinite Cardinal ◽  
Similar Argument ◽  
Weakly Compact ◽  
Compact Cardinal ◽  
Image Position ◽  
Weakly Compact Cardinal

AbstractLet $B$ be a complete Boolean algebra. We show that if λ is an infinite cardinal and $B$ is weakly (λω, ω)-distributive, then $B$ is (λ, 2)-distributive. Using a similar argument, we show that if κ is a weakly compact cardinal such that $B$ is weakly (2κ, κ)-distributive and $B$ is (α, 2)-distributive for each α < κ, then $B$ is (κ, 2)-distributive.

A polarized partition relation for weakly compact cardinals using elementary substructures

2006 ◽  
Vol 71 (4) ◽  
pp. 1342-1352 ◽  
Author(s):  
Albin L. Jones
Keyword(s):  
Partition Relation ◽  
Weakly Compact ◽  
Compact Cardinal ◽  
Image Position ◽  
Weakly Compact Cardinal

AbstractWe show that if κ is a weakly compact cardinal, thenfor any ordinals α < κ+ and μ < κ, and any finite ordinals m and n. This polarized partition relation represents the statement that for any partitionof κ × κ+ into m + μ pieces either there are A ∈ [κ]κ, B ∈ [κ]+]α and i < m with A × B ⊆ Ki or there are C ∈ [κ]κ, , and j < μ with C × D ⊆ Lj. Related results for measurable and almost measurable κ are also investigated. Our proofs of these relations involve the use of elementary substructures of set models of large fragments of ZFC.

THE EIGHTFOLD WAY

2018 ◽  
Vol 83 (1) ◽  
pp. 349-371
Author(s):  
JAMES CUMMINGS ◽  
SY-DAVID FRIEDMAN ◽  
MENACHEM MAGIDOR ◽  
ASSAF RINOT ◽  
DIMA SINAPOVA
Keyword(s):  
Set Theory ◽  
Large Cardinal ◽  
Tree Property ◽  
Weakly Compact ◽  
Compact Cardinal ◽  
Combinatorial Properties ◽  
Mutual Independence ◽  
Stationary Set ◽  
Image Position ◽  
Weakly Compact Cardinal

AbstractThree central combinatorial properties in set theory are the tree property, the approachability property and stationary reflection. We prove the mutual independence of these properties by showing that any of their eight Boolean combinations can be forced to hold at${\kappa ^{ + + }}$, assuming that$\kappa = {\kappa ^{ < \kappa }}$and there is a weakly compact cardinal aboveκ.If in additionκis supercompact then we can forceκto be${\aleph _\omega }$in the extension. The proofs combine the techniques of adding and then destroying a nonreflecting stationary set or a${\kappa ^{ + + }}$-Souslin tree, variants of Mitchell’s forcing to obtain the tree property, together with the Prikry-collapse poset for turning a large cardinal into${\aleph _\omega }$.

CHAIN CONDITIONS OF PRODUCTS, AND WEAKLY COMPACT CARDINALS

2014 ◽  
Vol 20 (3) ◽  
pp. 293-314 ◽  
Author(s):  
ASSAF RINOT
Keyword(s):  
Regular Cardinal ◽  
Chain Condition ◽  
Topological Spaces ◽  
Weakly Compact ◽  
Compact Cardinal ◽  
Inner Model ◽  
Aronszajn Tree ◽  
History Of ◽  
Image Position ◽  
Weakly Compact Cardinal

AbstractThe history of productivity of theκ-chain condition in partial orders, topological spaces, or Boolean algebras is surveyed, and its connection to the set-theoretic notion of a weakly compact cardinal is highlighted. Then, it is proved that for every regular cardinal$\kappa > \aleph _1 {\rm{,}}$the principle □(k) is equivalent to the existence of a certain strong coloring$c\,:\,[k]^2 \, \to $kfor which the family of fibers${\cal T}\left( c \right)$is a nonspecialκ-Aronszajn tree.The theorem follows from an analysis of a new characteristic function for walks on ordinals, and implies in particular that if theκ-chain condition is productive for a given regular cardinal$\kappa > \aleph _1 {\rm{,}}$thenκis weakly compact in some inner model of ZFC. This provides a partial converse to the fact that ifκis a weakly compact cardinal, then theκ-chain condition is productive.

scholarly journals The least weakly compact cardinal can be unfoldable, weakly measurable and nearly $${\theta}$$ θ -supercompact

2015 ◽  
Vol 54 (5-6) ◽  
pp. 491-510 ◽  
Author(s):  
Brent Cody ◽  
Moti Gitik ◽  
Joel David Hamkins ◽  
Jason A. Schanker
Keyword(s):  
Weakly Compact ◽  
Compact Cardinal ◽  
Weakly Compact Cardinal

scholarly journals Simultaneously vanishing higher derived limits

2021 ◽  
Vol 9 ◽  
Author(s):  
Jeffrey Bergfalk ◽  
Chris Lambie-Hanson
Keyword(s):  
Euclidean Space ◽  
Metric Spaces ◽  
Inverse System ◽  
Finite Support ◽  
Partial Functions ◽  
Weakly Compact ◽  
Compact Cardinal ◽  
Strong Homology ◽  
Finite Support Iteration ◽  
Weakly Compact Cardinal

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 $n=1$ .

scholarly journals PROOF THEORY OF WEAK COMPACTNESS

2013 ◽  
Vol 13 (01) ◽  
pp. 1350003 ◽  
Author(s):  
TOSHIYASU ARAI
Keyword(s):  
Set Theory ◽  
Proof Theory ◽  
Weak Compactness ◽  
Weakly Compact ◽  
Compact Cardinal ◽  
Weakly Compact Cardinal

We show that the existence of a weakly compact cardinal over the Zermelo–Fraenkel's set theory ZF is proof-theoretically reducible to iterations of Mostowski collapsings and Mahlo operations.

Power-collapsing games

2008 ◽  
Vol 73 (4) ◽  
pp. 1433-1457 ◽  
Author(s):  
Miloš S. Kurilić ◽  
Boris Šobot
Keyword(s):  
Boolean Algebra ◽  
Dense Subset ◽  
Winning Strategy ◽  
Zero Element ◽  
The Other ◽  
Complete Boolean Algebra ◽  
Infinite Cardinal ◽  
Other Hand ◽  
Game Theoretic

AbstractThe game is played on a complete Boolean algebra , by two players. White and Black, in κ-many moves (where κ is an infinite cardinal). At the beginning White chooses a non-zero element p ∈ . In the α-th move White chooses pα ∈ (0, p) and Black responds choosing iα ∈{0, 1}. White winsthe play iff . where and .The corresponding game theoretic properties of c.B.a.'s are investigated. So, Black has a winning strategy (w.s.) if κ ≥ π() or if contains a κ-closed dense subset. On the other hand, if White has a w.s., then κ ∈ . The existence of w.s. is characterized in a combinatorial way and in terms of forcing. In particular, if 2<κ = κ ∈ Reg and forcing by preserves the regularity of κ, then White has a w.s. iff the power 2κ is collapsed to κ in some extension. It is shown that, under the GCH, for each set S ⊆ Reg there is a c.B.a. such that White (respectively. Black) has a w.s. for each infinite cardinal κ ∈ S (resp. κ ∉ S). Also it is shown consistent that for each κ ∈ Reg there is a c.B.a. on which the game is undetermined.

Complete Boolean ultraproducts

1987 ◽  
Vol 52 (2) ◽  
pp. 530-542
Author(s):  
R. Michael Canjar
Keyword(s):  
Boolean Algebra ◽  
Regular Cardinal ◽  
Complete Boolean Algebra ◽  
Full Structure ◽  
Truth Value ◽  
Image Position

Throughout this paper, B will always be a Boolean algebra and Γ an ultrafilter on B. We use + and Σ for the Boolean join operation and · and Π for the Boolean meet.κ is always a regular cardinal. C(κ) is the full structure of κ, the structure with universe κ and whose functions and relations consist of all unitary functions and relations on κ. κB is the collection of all B-valued names for elements of κ. We use symbols f, g, h for members of κB. Formally an element f ∈ κB is a mapping κ → B with the properties that Σα∈κf(α) = 1B and that f(α) · f(β) = 0B whenever α ≠ β. We view f(α) as the Boolean-truth value indicating the extent to which the name f is equal to α, and we will hereafter write ∥f = α∥ for f(α). For every α ∈ κ there is a canonical name fα ∈ κB which has the property that ∥fα = α∥ = 1. Hereafter we identify α and fα.If B is a κ+-complete Boolean algebra and Γ is an ultrafilter on B, then we may define the Boolean ultraproduct C(κ)B/Γ in the following manner. If ϕ(x0, x1, …, xn) is a formula of Lκ, the language for C(κ) (which has symbols for all finitary functions and relations on κ), and f0, f1, …, fn−1 are elements of κB then we define the Boolean-truth value of ϕ(f0, f1, …, fn−1) as

The monadic theory of ω2

1983 ◽  
Vol 48 (2) ◽  
pp. 387-398 ◽  
Author(s):  
Yuri Gurevich ◽  
Menachem Magidor ◽  
Saharon Shelah
Keyword(s):  
Order Theory ◽  
Second Order ◽  
The Other ◽  
Weakly Compact ◽  
Compact Cardinal ◽  
Monadic Theory ◽  
Weakly Compact Cardinal

AbstractAssume ZFC + “There is a weakly compact cardinal” is consistent. Then:(i) For every S ⊆ ω, ZFC + “S and the monadic theory of ω2 are recursive each in the other” is consistent; and(ii) ZFC + “The full second-order theory of ω2 is interpretable in the monadic theory of ω2” is consistent.

Two weak consequences of 0#

1985 ◽  
Vol 50 (3) ◽  
pp. 597-603
Author(s):  
M. Gitik ◽  
M. Magidor ◽  
H. Woodin
Keyword(s):  
Limit Point ◽  
Inaccessible Cardinal ◽  
Weakly Compact ◽  
Compact Cardinal ◽  
Weakly Compact Cardinal

AbstractIt is proven that the following statement:“there exists a club C ⊆ κ such that every α ∈ C is an inaccessible cardinal in L and, for every δ a limit point of C, C ∩ δ is almost contained in every club of δ of L”is equiconsistent with a weakly compact cardinal if δ = ℵ1, and with a weakly compact cardinal of order 1 if δ = ℵ2.

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