Stationary subsets of [ℵω]<ωn

1993 ◽  
Vol 58 (4) ◽  
pp. 1201-1218 ◽  
Author(s):  
Kecheng Liu
Keyword(s):  
Large Cardinals ◽  
Image Position ◽  
Stationary Subsets

AbstractIn this paper, assuming large cardinals, we prove the consistency of the following:Let n ∈ ω and k1, k2 ≤ n. Let f: ω → {k1, k2} be such that for all n1 < n2 ∈ f−1{k1},n2 − n1 ≥ 4. Then the setis stationary in The above is equivalent to the statement that for any structure on on ℵω, there is ≺ A such that ∣∣ = ωn and for all m > n, cf( ∩ ωm) = ωf(m).

Ultrafilters on spaces of partitions

1982 ◽  
Vol 47 (1) ◽  
pp. 137-146 ◽  
Author(s):  
James M. Henle ◽  
William S. Zwicker
Keyword(s):  
Large Cardinals ◽  
Loss Of Information ◽  
Gainful Employment ◽  
Image Position

Qκλ. Pκλ the space of all < κ-sized subsets of λ, has provided numerous opportunities for the gainful employment of set theorists in recent years, thanks to its combinatorial richness and to its relationships with various large cardinals. In the spirit of Pκλ we offer the following definition:For κ ≤ λ both cardinals, Qκλ is the set of all partitions of λ into < κ-many pieces (an element of q ∈ Qκλ is called a piece of q). EquivalentlyAn element of Pκλ may be viewed as an injection from a < κ-sized set into λ, with some information thrown away. An element of Qκλ is a surjection from λ onto a < κ-sized set, with analogous loss of information.For p, q ∈ Qκλ, we say p ≤ q iff q is a refinement of p (every piece of q is contained in a piece of p).

scholarly journals IDEAL PROJECTIONS AND FORCING PROJECTIONS

2014 ◽  
Vol 79 (4) ◽  
pp. 1247-1285 ◽  
Author(s):  
SEAN COX ◽  
MARTIN ZEMAN
Keyword(s):  
Set Theory ◽  
Negative Answer ◽  
Large Cardinals ◽  
Normal Ideal ◽  
List Type ◽  
Type Number ◽  
Inner Model ◽  
Image Position ◽  
Open Question ◽  
Woodin Cardinal

AbstractIt is well known that saturation of ideals is closely related to the “antichain-catching” phenomenon from Foreman–Magidor–Shelah [10]. We consider several antichain-catching properties that are weaker than saturation, and prove:(1)If${\cal I}$is a normal ideal on$\omega _2 $which satisfiesstationary antichain catching, then there is an inner model with a Woodin cardinal;(2)For any$n \in \omega $, it is consistent relative to large cardinals that there is a normal ideal${\cal I}$on$\omega _n $which satisfiesprojective antichain catching, yet${\cal I}$is not saturated (or even strong). This provides a negative answer to Open Question number 13 from Foreman’s chapter in the Handbook of Set Theory ([7]).

Σ1(κ)-DEFINABLE SUBSETS OF H(κ+)

2017 ◽  
Vol 82 (3) ◽  
pp. 1106-1131 ◽  
Author(s):  
PHILIPP LÜCKE ◽  
RALF SCHINDLER ◽  
PHILIPP SCHLICHT
Keyword(s):  
Measurable Cardinal ◽  
Large Cardinal ◽  
Woodin Cardinals ◽  
Perfect Set ◽  
Nonstationary Ideal ◽  
Definable Sets ◽  
Image Position ◽  
Stationary Subsets ◽  
Woodin Cardinal ◽  
Generic Ultrapowers

AbstractWe study Σ1(ω1)-definable sets (i.e., sets that are equal to the collection of all sets satisfying a certain Σ1-formula with parameter ω1 ) in the presence of large cardinals. Our results show that the existence of a Woodin cardinal and a measurable cardinal above it imply that no well-ordering of the reals is Σ1(ω1)-definable, the set of all stationary subsets of ω1 is not Σ1(ω1)-definable and the complement of every Σ1(ω1)-definable Bernstein subset of ${}_{}^{{\omega _1}}\omega _1^{}$ is not Σ1(ω1)-definable. In contrast, we show that the existence of a Woodin cardinal is compatible with the existence of a Σ1(ω1)-definable well-ordering of H(ω2) and the existence of a Δ1(ω1)-definable Bernstein subset of ${}_{}^{{\omega _1}}\omega _1^{}$. We also show that, if there are infinitely many Woodin cardinals and a measurable cardinal above them, then there is no Σ1(ω1)-definable uniformization of the club filter on ω1. Moreover, we prove a perfect set theorem for Σ1(ω1)-definable subsets of ${}_{}^{{\omega _1}}\omega _1^{}$, assuming that there is a measurable cardinal and the nonstationary ideal on ω1 is saturated. The proofs of these results use iterated generic ultrapowers and Woodin’s ℙmax-forcing. Finally, we also prove variants of some of these results for Σ1(κ)-definable subsets of κκ, in the case where κ itself has certain large cardinal properties.

scholarly journals Possible behaviours of the reflection ordering of stationary sets

1995 ◽  
Vol 60 (2) ◽  
pp. 534-547 ◽  
Author(s):  
Jiří Witzany
Keyword(s):  
Partial Ordering ◽  
Generic Extension ◽  
Maximal Antichain ◽  
Uncountable Cardinal ◽  
Image Position ◽  
Stationary Subsets ◽  
Almost All

AbstractIf S, T are stationary subsets of a regular uncountable cardinal κ, we say that S reflects fully in T, S < T, if for almost all α ∈ T (except a nonstationary set) S ∩ α stationary in α. This relation is known to be a well-founded partial ordering. We say that a given poset P is realized by the reflection ordering if there is a maximal antichain 〈Xp: p ∈ P〉 of stationary subsets of Reg(κ) so thatWe prove that if , and P is an arbitrary well-founded poset of cardinality ≤ κ+ then there is a generic extension where P is realized by the reflection ordering on κ.

THE TREE PROPERTY AT AND

2018 ◽  
Vol 83 (2) ◽  
pp. 669-682 ◽  
Author(s):  
DIMA SINAPOVA ◽  
SPENCER UNGER
Keyword(s):  
Large Cardinals ◽  
Strong Limit ◽  
Tree Property ◽  
Image Position

AbstractWe show that from large cardinals it is consistent to have the tree property simultaneously at${\aleph _{{\omega ^2} + 1}}$and${\aleph _{{\omega ^2} + 2}}$with${\aleph _{{\omega ^2}}}$strong limit.

scholarly journals GENERIC LARGE CARDINALS AND SYSTEMS OF FILTERS

2017 ◽  
Vol 82 (3) ◽  
pp. 860-892 ◽  
Author(s):  
GIORGIO AUDRITO ◽  
SILVIA STEILA
Keyword(s):  
Large Cardinals ◽  
Image Position ◽  
Normal Ideals

AbstractWe introduce the notion of ${\cal C}$-system of filters, generalizing the standard definitions of both extenders and towers of normal ideals. This provides a framework to develop the theory of extenders and towers in a more general and concise way. In this framework we investigate the topic of definability of generic large cardinals properties.

On splitting stationary subsets of large cardinals

1977 ◽  
Vol 42 (2) ◽  
pp. 203-214 ◽  
Author(s):  
James E. Baumgartner ◽  
Alan D. Taylor ◽  
Stanley Wagon
Keyword(s):  
Large Cardinals ◽  
Normal Ideal ◽  
Weakly Compact ◽  
Uncountable Cardinal ◽  
Stationary Set ◽  
Stationary Subsets ◽  
Open Question ◽  
Mahlo Cardinals ◽  
Normal Ideals

AbstractLet κ denote a regular uncountable cardinal and NS the normal ideal of nonstationary subsets of κ. Our results concern the well-known open question whether NS fails to be κ+-saturated, i.e., are there κ+ stationary subsets of κ with pairwise intersections nonstationary? Our first observation is:Theorem. NS isκ+-saturated iff for every normal ideal J on κ there is a stationary set A ⊆ κsuch that J = NS∣A = {X ⊆ κ: X ∩ A ∈ NS}.Turning our attention to large cardinals, we extend the usual (weak) Mahlo hierarchy to define “greatly Mahlo” cardinals and obtain the following:Theorem. If κ is greatly Mahlo then NS is notκ+-saturated.Theorem. If κ is ordinal Π11-indescribable (e.g., weakly compact), ethereal (e.g., subtle), or carries aκ-saturated ideal, thenκis greatly Mahlo. Moreover, there is a stationary set of greatly Mahlo cardinals below any ordinal Π11-indescribable cardinal.These methods apply to other normal ideals as well; e.g., the subtle ideal on an ineffable cardinal κ is not κ+-saturated.

scholarly journals Precipitous towers of normal filters

1997 ◽  
Vol 62 (3) ◽  
pp. 741-754 ◽  
Author(s):  
Douglas R. Burke
Keyword(s):  
Large Cardinals ◽  
Generic Extension ◽  
Normal Filter ◽  
Important Idea ◽  
Stationary Set ◽  
Image Position ◽  
Woodin Cardinal

In this paper we investigate towers of normal filters. These towers were first used by Woodin (see [15]). Woodin proved that if δ is a Woodin cardinal and P is the full stationary tower up to δ (P<δ) or the countable version (Q<δ), then the generic ultrapower is closed under < δ sequences (so the generic ultrapower is well-founded) ([14]). We show that if ℙ is a tower of height δ, δ supercompact, and the filters generating ℙ are the club filter restricted to a stationary set, then the generic ultrapower is well-founded (ℙ is precipitous). We also give some examples of non-precipitous towers. We also show that every normal filter can be extended to a V-ultrafilter with well-founded ultrapower in some generic extension of V (assuming large cardinals). Similarly for any tower of inaccessible height. This is accomplished by showing that there is a stationary set that projects to the filter or the tower and then forcing with P<δ below this stationary set.An important idea in our proof of precipitousness (Theorem 6.4) has the following form in Woodin's proof. If are maximal antichains (i Є ω and δ Woodin) then there is a κ < δ such that each Ai ∩ Vκ is semiproper, i.e.,contains a club (relative to ∣ a∣ < κ).

scholarly journals JOINT DIAMONDS AND LAVER DIAMONDS

2019 ◽  
Vol 84 (3) ◽  
pp. 895-928
Author(s):  
MIHA E. HABIČ
Keyword(s):  
Large Cardinals ◽  
Supercompact Cardinals ◽  
The Family ◽  
Image Position

AbstractThe concept of jointness for guessing principles, specifically ${\diamondsuit _\kappa }$ and various Laver diamonds, is introduced. A family of guessing sequences is joint if the elements of any given sequence of targets may be simultaneously guessed by the members of the family. While equivalent in the case of ${\diamondsuit _\kappa }$, joint Laver diamonds are nontrivial new objects. We give equiconsistency results for most of the large cardinals under consideration and prove sharp separations between joint Laver diamonds of different lengths in the case of θ-supercompact cardinals.

LARGE CARDINALS AND LIGHTFACE DEFINABLE WELL-ORDERS, WITHOUT THE GCH

2015 ◽  
Vol 80 (1) ◽  
pp. 251-284
Author(s):  
SY-DAVID FRIEDMAN ◽  
PETER HOLY ◽  
PHILIPP LÜCKE
Keyword(s):  
Fixed Point ◽  
Measurable Cardinal ◽  
Large Cardinals ◽  
Inaccessible Cardinal ◽  
Proper Class ◽  
Supercompact Cardinals ◽  
Image Position ◽  
Class Forcing ◽  
Continuum Function ◽  
The Continuum

AbstractThis paper deals with the question whether the assumption that for every inaccessible cardinal κ there is a well-order of H(κ+) definable over the structure $\langle {\rm{H}}({\kappa ^ + }), \in \rangle$ by a formula without parameters is consistent with the existence of (large) large cardinals and failures of the GCH. We work under the assumption that the SCH holds at every singular fixed point of the ℶ-function and construct a class forcing that adds such a well-order at every inaccessible cardinal and preserves ZFC, all cofinalities, the continuum function, and all supercompact cardinals. Even in the absence of a proper class of inaccessible cardinals, this forcing produces a model of “V = HOD” and can therefore be used to force this axiom while preserving large cardinals and failures of the GCH. As another application, we show that we can start with a model containing an ω-superstrong cardinal κ and use this forcing to build a model in which κ is still ω-superstrong, the GCH fails at κ and there is a well-order of H(κ+) that is definable over H(κ+) without parameters. Finally, we can apply the forcing to answer a question about the definable failure of the GCH at a measurable cardinal.

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