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]).

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.

The Documentary Interview

1965 ◽  
Vol 8 (02) ◽  
pp. 9-14 ◽  
Author(s):  
J. Vansina
Keyword(s):  
Critical Analysis ◽  
The Political ◽  
Political Scientist ◽  
Loss Of Information ◽  
Oral Testimony ◽  
Image Position

The political scientist uses oral testimony to record events which happened sometime not too long ago. He retrieves documents about events, usually from eyewitnesses but sometimes from descendants of eyewitnesses. Whenever a witness testifies to events and his testimony is recorded, the following sequence or chain between the events and the record of them has taken place: The relation between the events and the events as described in the document has therefore undergone the following “distortions”: events--part of the events are perceived--part of the perception is stored in the memory of a man and colored by his personality--part of what is in the memory of the man is released and the release is colored by the interview. There is quite definitely a loss of information between the event and the record of it. There is also, and this is less obvious, quite an accretion to the record of the event by the reflections and the personality of the witness. The aim of the person who uses the record is to know what are the accretions and the distortions so that he would know what actually happened insofar as it is recorded. Critical analysis is the tool used to discover this. It can be made much easier if certain items of information besides the testimony and about it are available. It is therefore of great value to collect this ancillary documentation together with the oral testimony itself.

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).

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.

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.

scholarly journals DENSE IDEALS AND CARDINAL ARITHMETIC

2016 ◽  
Vol 81 (3) ◽  
pp. 789-813 ◽  
Author(s):  
MONROE ESKEW
Keyword(s):  
Large Cardinals ◽  
Open Questions ◽  
Image Position ◽  
Cardinal Arithmetic

AbstractFrom large cardinals we show the consistency of normal, fine, κ-complete λ-dense ideals on ${{\cal P}_\kappa }\left( \lambda \right)$ for successor κ. We explore the interplay between dense ideals, cardinal arithmetic, and squares, answering some open questions of Foreman.

Simple Identification of Electron Diffraction Ring Patterns

1969 ◽  
Vol 27 ◽  
pp. 160-161
Author(s):  
Carolyn Nohr ◽  
Ann Ayres
Keyword(s):  
Electron Microscope ◽  
Electron Diffraction ◽  
Diffraction Ring ◽  
Image Position

Texts on electron diffraction recommend that the camera constant of the electron microscope be determine d by calibration with a standard crystalline specimen, using the equation

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