scholarly journals MODEL-THEORETIC PROPERTIES OF ULTRAFILTERS BUILT BY INDEPENDENT FAMILIES OF FUNCTIONS

2014 ◽  
Vol 79 (01) ◽  
pp. 103-134 ◽  
Author(s):  
M. MALLIARIS ◽  
S. SHELAH
Keyword(s):  
Random Graph ◽  
Measurable Cardinal ◽  
Degree Of Saturation ◽  
Stable Theory ◽  
Weakly Compact ◽  
Compact Cardinal ◽  
Graph Type ◽  
Inductive Construction ◽  
Image Position ◽  
Precise Degree

Abstract 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 ${\cal P}\left( \lambda \right)/{\cal D}$ 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 ${\cal D}$ on λ so that any ultrafilter extending ${\cal D}$ fails to ${\lambda ^ + }$ -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 $\lambda > \kappa$ which is λ-flexible but not ${\kappa ^{ + + }}$ -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 ${\rm{lcf}}\left( {{\aleph _0}} \right)$ 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.

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 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${\aleph _\omega }$.

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.

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.

Higher order reflection principles

1989 ◽  
Vol 54 (2) ◽  
pp. 474-489 ◽  
Author(s):  
M. Victoria Marshall R.
Keyword(s):  
Measurable Cardinal ◽  
Reflection Principle ◽  
Higher Order ◽  
Weakly Compact ◽  
Class Theory ◽  
First Order ◽  
Transitive Set ◽  
Reflection Principles ◽  
Image Position ◽  
Order Reflection

In [1] and [2] there is a development of a class theory, whose axioms were formulated by Bernays and based on a reflection principle. See [3]. These axioms are formulated in first order logic with ∈:(A1)Extensionality.(A2)Class specification. Ifϕis a formula andAis not free inϕ, thenNote that “xis a set“ can be written as “∃u(x∈u)”.(A3)Subsets.Note also that “B⊆A” can be written as “∀x(x∈B→x∈A)”.(A4)Reflection principle. Ifϕ(x)is a formula, thenwhere “uis a transitive set” is the formula “∃v(u∈v) ∧ ∀x∀y(x∈y∧y∈u→x∈u)” andϕPuis the formulaϕrelativized to subsets ofu.(A5)Foundation.(A6)Choice for sets.We denote byB1the theory with axioms (A1) to (A6).The existence of weakly compact and-indescribable cardinals for everynis established inB1by the method of defining all metamathematical concepts forB1in a weaker theory of classes where the natural numbers can be defined and using the reflection principle to reflect the satisfaction relation; see [1]. There is a proof of the consistency ofB1assuming the existence of a measurable cardinal; see [4] and [5]. In [6] several set and class theories with reflection principles are developed. In them, the existence of inaccessible cardinals and some kinds of indescribable cardinals can be proved; and also there is a generalization of indescribability for higher-order languages using only class parameters.The purpose of this work is to develop higher order reflection principles, including higher order parameters, in order to obtain other large cardinals.

ON ${\omega _1}$-STRONGLY COMPACT CARDINALS

2014 ◽  
Vol 79 (01) ◽  
pp. 266-278 ◽  
Author(s):  
JOAN BAGARIA ◽  
MENACHEM MAGIDOR
Keyword(s):  
Measurable Cardinal ◽  
Second Order ◽  
Singular Cardinal ◽  
Strongly Compact Cardinal ◽  
Compact Cardinal ◽  
Relational Structures ◽  
Uncountable Cardinal ◽  
Image Position ◽  
Strongly Compact Cardinals ◽  
Order Reflection

Abstract An uncountable cardinal κ is called ${\omega _1}$ -strongly compact if every κ-complete ultrafilter on any set I can be extended to an ${\omega _1}$ -complete ultrafilter on I. We show that the first ${\omega _1}$ -strongly compact cardinal, ${\kappa _0}$ , cannot be a successor cardinal, and that its cofinality is at least the first measurable cardinal. We prove that the Singular Cardinal Hypothesis holds above ${\kappa _0}$ . We show that the product of Lindelöf spaces is κ-Lindelöf if and only if $\kappa \ge {\kappa _0}$ . Finally, we characterize ${\kappa _0}$ in terms of second order reflection for relational structures and we give some applications. For instance, we show that every first-countable nonmetrizable space has a nonmetrizable subspace of size less than ${\kappa _0}$ .

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 $n=1$ .

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