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Saturated filters at successors of singulars, weak reflection and yet another weak club principle [article]

Mirna Džamonja, Saharon Shelah
1996 arXiv   pre-print
We give a sufficient condition that the club filter of lambda concentrating on the points of cofinality kappa is not lambda^+-saturated.  ...  The condition is phrased in terms of a notion that we call weak reflection. We discuss various properties of weak reflection  ...  Saturated filters on the successor of a singular cardinal of uncountable cofinality. Suppose that µ is a cardinal with the property µ > cf(µ) = κ > ℵ 0 , and that λ = µ + .  ... 
arXiv:math/9601219v1 fatcat:vclf3mpjyrenpnwfq4j43t5r4e

Saturated filters at successors of singulars, weak reflection and yet another weak club principle

Mirna Džamonja, Saharon Shelah
1996 Annals of Pure and Applied Logic  
We introduce a weak version of the &-principle, which we call &*_, and show that if it holds on a stationary subset S of i, then no normal filter on S is I+-saturated.  ...  We give a sufficient condition that the club filter of 1 concentrating on the points of cofinality K is not A+-saturated.' The condition is phrased in terms of a notion that we call weak reflection.  ...  The most sincere thanks to James Cummings for his numerous comments and corrections, and to Uri Abraham, Menachem Magidor and other participants of the Logic Seminar in Jerusalem, Fall 1994, for their  ... 
doi:10.1016/0168-0072(95)00040-2 fatcat:3mqh3qtxvnfwtlf5rxwf43ii2y

Page 2099 of Mathematical Reviews Vol. , Issue 97D [page]

1997 Mathematical Reviews  
yet another weak club principle.  ...  Marion Scheepers (\-BOISE; Boise, ID) 974:03062 03E05 03E35 Diamonja, Mirna (1-W1; Madison, WI); Shelah, Saharon (IL-HEBR-IM; Jerusalem) Saturated filters at successors of singular, weak reflection and  ... 

Jensen's diamond principle and its relatives [article]

Assaf Rinot
2010 arXiv   pre-print
We survey some recent results on the validity of Jensen's diamond principle at successor cardinals.  ...  We also discuss weakening of this principle such as club guessing, and anti-diamond principles such as uniformization. A collection of open problems is included.  ...  failure of instances of the uniformization property at successor of singular cardinals.  ... 
arXiv:0911.2151v2 fatcat:jktk44uuofbqdizhaws7qyqlre

Simultaneous reflection and impossible ideals

Todd Eisworth
2012 Journal of Symbolic Logic (JSL)  
We prove that if holds for a singular cardinal μ, then any collection of fewer than cf(μ) stationary subsets of μ + must reflect simultaneously.  ...  hold at successors of singular cardinals.  ...  Lemma 3.3 also tells us a lot about weak-saturation properties of club-guessing ideals on successors of singular cardinals in ZFC. For example, suppose µ is singular and τ < µ is a regular cardinal.  ... 
doi:10.2178/jsl.7704160 fatcat:wfxkrknyxzgizpvrluln27mjni

Martin's maximum and the non-stationary ideal [article]

Shimon Garti, Yair Hayut, Haim Horowitz, Menachem Magidor
2018 arXiv   pre-print
We analyze the non-stationary ideal and the club filter at aleph_1 under MM.  ...  For the main result we used MM, but actually almost weak Laverness follows from the ℵ 1 -reflection principle and the fact that NS ω 1 is Kunen.  ...  An ideal I over a successor cardinal κ + is Laver iff I is uniform (i.e., contains all the bounded subsets of κ + ), κ + -complete and (κ ++ , κ ++ , κ)saturated.  ... 
arXiv:1708.08049v2 fatcat:mucuektxlvd73llt3vcpsz4yyi

Club-guessing, stationary reflection, and coloring theorems

Todd Eisworth
2010 Annals of Pure and Applied Logic  
We obtain very strong coloring theorems at successors of singular cardinals from failures of certain instances of simultaneous reflection of stationary sets.  ...  of S reflects at some α. (3) S is non-reflecting if S does not reflect at any α.  ...  Weak saturation provides a measure of how closely the filter I * dual to I comes to being an ultrafilter.  ... 
doi:10.1016/j.apal.2010.02.004 fatcat:4nyd4dr6snd5vlyf5eqjlzcz6e

Club-guessing, stationary reflection, and coloring theorems [article]

Todd Eisworth
2009 arXiv   pre-print
We obtain strong coloring theorems at successors of singular cardinals from failures of certain instances of simultaneous reflection of stationary sets.  ...  Along the way, we establish new results in club-guessing and in the general theory of ideals.  ...  Weak saturation provides a measure of how closely the filter I * dual to I comes to being an ultrafilter.  ... 
arXiv:0905.3754v1 fatcat:xqzl2glyznaebl3eblh7bf4qqy

Was Ulam right? I: Basic theory and subnormal ideals [article]

Tanmay Inamdar, Assaf Rinot
2021 arXiv   pre-print
We introduce various colouring principles which generalise the so-called "onto mapping principle" of Sierpinski to larger cardinals and general ideals.  ...  We prove that these principles capture the notion of an Ulam matrix and allow to characterise large cardinals, most notably weakly compact and ineffable cardinals.  ...  Some of the results of this paper were announced by the first author at the Israel Mathematical Union Annual Meeting special session in set theory and logic in July 2021.  ... 
arXiv:2107.05136v2 fatcat:3e67a7qi2jdydauffnscenmsqq

Knaster and friends II: The C-sequence number [article]

Chris Lambie-Hanson, Assaf Rinot
2019 arXiv   pre-print
We prove a number of ZFC and independence results about the C-sequence number and its relationship with large cardinals, stationary reflection, and square principles.  ...  We then introduce and study the more general C-sequence spectrum and uncover some tight connections between the C-sequence spectrum and the strong coloring principle U(...), introduced in Part I of this  ...  Acknowledgments The results of this paper were presented by the first author at the CUNY Set Theory Seminar in March 2019 and by the second author at the The 15th International Luminy Workshop in Set Theory  ... 
arXiv:1912.03342v1 fatcat:4jq2h2473vf4badqsxbcl7niwa

On the consistency strength of 'Accessible' Jonsson Cardinals and of the Weak Chang Conjecture

Hans-Dieter Donder, Peter Koepke
1983 Annals of Pure and Applied Logic  
Assume that A is a Jonsson cardinal which is 'accessible' in the sense that at least one of (l)-(4) holds: (1) A is a successor cardinal; (2) A = oE and 6 <A ; (3) A is singular of uncountable cofinality  ...  For A = P+ a successor cardinal we consider the weak Chang Conjecture, WCC(A), which is a consequence of the Chang transfer property (A+, A) j (A, p). III.  ...  Assume that A is a Jonsson cardinal which is 'accessible' in the sense that at least one of (l)-(4) holds: (1) A is a successor cardinal; (2) A = oE and 6 <A ; (3) A is singular of uncountable cofinality  ... 
doi:10.1016/0168-0072(83)90020-9 fatcat:3hohxm6apjajbekcqogkusvm3m

Fake reflection [article]

Gabriel Fernandes, Miguel Moreno, Assaf Rinot
2020 arXiv   pre-print
We prove the independence of filter reflection from ZFC, and present applications of filter reflection to the study of canonical equivalence relations of the higher Cantor and Baire spaces.  ...  We introduce a generalization of stationary set reflection which we call "filter reflection", and show it is compatible with the axiom of constructibility as well as with strong forcing axioms.  ...  At the end of the preparation of this article, the second author was visiting the Kurt Gödel Research Center supported by the Vilho, Yrjö and Kalle Väisälä Foundation of the Finnish Academy of Science  ... 
arXiv:2003.08340v1 fatcat:tb76yd4fqzehrkwvcdcoc3kljm

Iterated Forcing and Elementary Embeddings [chapter]

James Cummings
2009 Handbook of Set Theory  
. • We use On for the class of ordinals, Card for the class of cardinals, Lim for the class of limit ordinals, Reg for the class of regular cardinals and Sing for the class of singular ordinals. • A tree  ...  is normal if and only if it is nonempty, has a unique minimal element, and has the properties that every element has two immediate successors and that every element of limit height is determined uniquely  ...  the theorem by Magidor gives another example of shooting clubs to make a natural filter (defined in this case via stationary reflection) become the club filter.  ... 
doi:10.1007/978-1-4020-5764-9_13 fatcat:heb4rvf4y5ck7pkftccvxftmxu

On what I do not understand (and have something to say): Part I [article]

Saharon Shelah
1999 arXiv   pre-print
As we discuss many problems, history and side references are erratic, usually kept at a minimum ("see ... " means: see the references there and possibly the paper itself).  ...  The base were lectures in Rutgers Fall'97 and reflect my knowledge then. The other half, concentrating on model theory, will subsequently appear.  ...  A related ZFC result is that for a singular cardinal µ, the restriction of the club filter D µ + ↾ S µ + cf(µ) is not µ ++ -saturated (see Gitik and Shelah [45] ).  ... 
arXiv:math/9906113v1 fatcat:blohf2hvnfajbnibte3iuzsovy

Singular Cardinals [chapter]

Menachem Kojman
2012 Handbook of the History of Logic  
Ulam discovered that in yet another branch of mathematicsmeasure theory-the relation between a cardinal and its cofinality is useful.  ...  The main gain in weakening Jensen's diamond principle in this way is, as Shelah proved, that if κ + < λ and both κ and λ are regular, then a club-guessing sequence on S λ κ exists in ZFC.  ...  ., 15 almost free group, 23 Axiom of Choice, 6  ... 
doi:10.1016/b978-0-444-51621-3.50007-4 fatcat:fr3pdoy3nbdn5pc4pz52reng24
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