scholarly journals THE HALPERN–LÄUCHLI THEOREM AT A MEASURABLE CARDINAL

2017 ◽  
Vol 82 (4) ◽  
pp. 1560-1575 ◽  
Author(s):  
NATASHA DOBRINEN ◽  
DAN HATHAWAY
Keyword(s):  
Measurable Cardinal ◽  
Weakly Compact ◽  
Compact Cardinal ◽  
Strong Cardinal ◽  
Partition Relations ◽  
Weakly Compact Cardinal

AbstractSeveral variants of the Halpern–Läuchli Theorem for trees of uncountable height are investigated. Forκweakly compact, we prove that the various statements are all equivalent, and hence, the strong tree version holds for one tree on any weakly compact cardinal. For any finited≥ 2, we prove the consistency of the Halpern–Läuchli Theorem ondmany normalκ-trees at a measurable cardinalκ, given the consistency of aκ+d-strong cardinal. This follows from a more general consistency result at measurableκ, which includes the possibility of infinitely many trees, assuming partition relations which hold in models of AD.

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.

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 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.

On large cardinals and partition relations

1971 ◽  
Vol 36 (2) ◽  
pp. 305-308 ◽  
Author(s):  
E. M. Kleinberg ◽  
R. A. Shore
Keyword(s):  
Set Theory ◽  
Measurable Cardinal ◽  
Order Type ◽  
Large Cardinals ◽  
Infinite Cardinal ◽  
Partition Relation ◽  
Weakly Compact ◽  
Original Result ◽  
Partition Relations ◽  
Uncountable Cardinal

A significant portion of the study of large cardinals in set theory centers around the concept of “partition relation”. To best capture the basic idea here, we introduce the following notation: for x and y sets, κ an infinite cardinal, and γ an ordinal less than κ, we let [x]γ denote the collection of subsets of x of order-type γ and abbreviate with the partition relation for each function F frominto y there exists a subset C of κ of cardinality κ such that (such that for each α < γ) the range of F on [С]γ ([С]α) has cardinality 1. Now although each infinite cardinal κ satisfies the relation for each n and m in ω (F. P. Ramsey [8]), a connection with large cardinals arises when one asks, “For which uncountable κ do we have κ → (κ)2?” Indeed, any uncountable cardinal κ which satisfies κ → (κ)2 is strongly inaccessible and weakly compact (see [9]). As another example one can look at the improvements of Scott's original result to the effect that if there exists a measurable cardinal then there exists a nonconstructible set. Indeed, if κ is a measurable cardinal then κ → (κ)< ω, and as Solovay [11] has shown, if there exists a cardinal κ such that κ → (κ)< ω3 (κ → (ℵ1)< ω, even) then there exists a nonconstructible set of integers.

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