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.

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.

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.

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.

On the ordering of certain large cardinals

1979 ◽  
Vol 44 (4) ◽  
pp. 563-565
Author(s):  
Carl F. Morgenstern
Keyword(s):  
Measurable Cardinal ◽  
Large Cardinals ◽  
Large Cardinal ◽  
Inaccessible Cardinal ◽  
Elementary Embedding ◽  
Strongly Compact Cardinal ◽  
Weakly Compact ◽  
Compact Cardinal ◽  
The Universe ◽  
Weakly Compact Cardinal

It is well known that the first strongly inaccessible cardinal is strictly less than the first weakly compact cardinal which in turn is strictly less than the first Ramsey cardinal, etc. However, once one passes the first measurable cardinal the inequalities are no longer strict. Magidor [3] has shown that the first strongly compact cardinal may be equal to the first measurable cardinal or equal to the first super-compact cardinal (the first supercompact cardinal is strictly larger than the first measurable cardinal). In this note we will indicate how Magidor's methods can be used to show that it is undecidable whether one cardinal (the first strongly compact) is greater than or less than another large cardinal (the first huge cardinal). We assume that the reader is familiar with the ultrapower construction of Scott, as presented in Drake [1] or Kanamori, Reinhardt and Solovay [2].Definition. A cardinal κ is huge (or 1-huge) if there is an elementary embedding j of the universe V into a transitive class M such that M contains the ordinals, is closed under j(κ) sequences, j(κ) > κ and j ↾ Rκ = id. Let κ denote the first huge cardinal, and let λ = j(κ).One can see from easy reflection arguments that κ and λ are inaccessible in V and, in fact, that κ is measurable in V.

On the consistency of the Definable Tree Property on ℵ1

2000 ◽  
Vol 65 (3) ◽  
pp. 1204-1214 ◽  
Author(s):  
Amir Leshem
Keyword(s):  
Tree Property ◽  
Consistency Strength ◽  
Weakly Compact ◽  
Compact Cardinal ◽  
First Order ◽  
Weakly Compact Cardinal

AbstractIn this paper we prove the equiconsistency of “Every ω1 –tree which is first order definable over (, ε) has a cofinal branch” with the existence of a reflecting cardinal. We also prove that the addition of MA to the definable tree property increases the consistency strength to that of a weakly compact cardinal. Finally we comment on the generalization to higher cardinals.

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