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.

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).

Recent trends and open questions in grain boundary segregation

2018 ◽  
Vol 33 (18) ◽  
pp. 2647-2660 ◽  
Author(s):  
Pavel Lejček ◽  
Monika Všianská ◽  
Mojmír Šob
Keyword(s):  
Grain Boundary ◽  
Boundary Segregation ◽  
Grain Boundary Segregation ◽  
Open Questions ◽  
Recent Trends ◽  
Image Position

Abstract

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]).

A remark on Zilber's pseudoexponentiation

2006 ◽  
Vol 71 (3) ◽  
pp. 791-798 ◽  
Author(s):  
David Marker
Keyword(s):  
Baire Category ◽  
Positive Answer ◽  
Complex Conjugation ◽  
Closed Set ◽  
Definable Set ◽  
Open Questions ◽  
Novel Approach ◽  
Definable Sets ◽  
Order Arithmetic ◽  
Image Position

When studying the model theory ofthe first observation is that the integers can be defined asSince ∂exp is subject to all of Gödel's phenomena, this is often also the last observation. After Wilkie proved that ℝexp is model complete, one could ask the same question for ∂exp, but the answer is negative.Proposition 1.1. ∂expis not model completeProof. If ∂exp is model complete, then every definable set is a projection of a closed set. Since ∂ is locally compact, every definable set is Fσ. The same is true for the complement, so every definable set is also Gδ. But, since ℤ is definable, ℚ is definable and a standard corollary of the Baire Category Theorem tells us that ℚ is not Gδ.Still, there are several interesting open questions about ∂exp.• Is ℝ definable in ∂exp?• (quasiminimality) Is every definable set countable or co-countable? (Note that this is true in the structure (∂, ℤ, +, ·) where we add a predicate for ℤ).• (Mycielski) Is there an automorphism of ∂exp other than the identity and complex conjugation?1A positive answer to the first question would tell us that ∂exp is essentially second order arithmetic, while a positive answer to the second would say that integers are really the only obstruction to a reasonable theory of definable sets.A fascinating, novel approach to ∂exp is provided by Zilber's [6] pseudoexponentiation. Let L be the language {+, · E, 0, 1}.

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.

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.

ON ISOMORPHISM CLASSES OF COMPUTABLY ENUMERABLE EQUIVALENCE RELATIONS

2019 ◽  
Vol 85 (1) ◽  
pp. 61-86 ◽  
Author(s):  
URI ANDREWS ◽  
SERIKZHAN A. BADAEV
Keyword(s):  
Explicit Construction ◽  
The Other ◽  
Equivalence Relations ◽  
Index Set ◽  
Isomorphism Classes ◽  
Closed Sets ◽  
Open Questions ◽  
Index Sets ◽  
Image Position ◽  
Computably Enumerable

AbstractWe examine how degrees of computably enumerable equivalence relations (ceers) under computable reduction break down into isomorphism classes. Two ceers are isomorphic if there is a computable permutation of ω which reduces one to the other. As a method of focusing on nontrivial differences in isomorphism classes, we give special attention to weakly precomplete ceers. For any degree, we consider the number of isomorphism types contained in the degree and the number of isomorphism types of weakly precomplete ceers contained in the degree. We show that the number of isomorphism types must be 1 or ω, and it is 1 if and only if the ceer is self-full and has no computable classes. On the other hand, we show that the number of isomorphism types of weakly precomplete ceers contained in the degree can be any member of $[0,\omega ]$. In fact, for any $n \in [0,\omega ]$, there is a degree d and weakly precomplete ceers ${E_1}, \ldots ,{E_n}$ in d so that any ceer R in d is isomorphic to ${E_i} \oplus D$ for some $i \le n$ and D a ceer with domain either finite or ω comprised of finitely many computable classes. Thus, up to a trivial equivalence, the degree d splits into exactly n classes.We conclude by answering some lingering open questions from the literature: Gao and Gerdes [11] define the collection of essentially FC ceers to be those which are reducible to a ceer all of whose classes are finite. They show that the index set of essentially FC ceers is ${\rm{\Pi }}_3^0$-hard, though the definition is ${\rm{\Sigma }}_4^0$. We close the gap by showing that the index set is ${\rm{\Sigma }}_4^0$-complete. They also use index sets to show that there is a ceer all of whose classes are computable, but which is not essentially FC, and they ask for an explicit construction, which we provide.Andrews and Sorbi [4] examined strong minimal covers of downwards-closed sets of degrees of ceers. We show that if $\left( {{E_i}} \right)$ is a uniform c.e. sequence of non universal ceers, then $\left\{ {{ \oplus _{i \le j}}{E_i}|j \in \omega } \right\}$ has infinitely many incomparable strong minimal covers, which we use to answer some open questions from [4].Lastly, we show that there exists an infinite antichain of weakly precomplete ceers.

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