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Approachability at the second successor of a singular cardinal

Moti Gitik, John Krueger
2009 Journal of Symbolic Logic (JSL)  
Using this result, we construct models in which the approachability property fails at the successor of a singular cardinal.  ...  We also construct models in which the properties of being internally club and internally approachable are distinct for sets of size the successor of a singular cardinal.  ...  Approachability at the Second Successor of a Singular We now show how to obtain a failure of the approachability property AP µ + , where µ is a singular cardinal.  ... 
doi:10.2178/jsl/1254748688 fatcat:halc7ora7feidflmrpmz4ntpie

Contributions to the Theory of Large Cardinals through the Method of Forcing

Alejandro Poveda
2021 Bulletin of Symbolic Logic  
relationship between the Singular Cardinal Hypothesis (SCH) with other relevant combinatorial principles at the level of successors of singular cardinals.  ...  , the Gitik-Sharon poset, or the Extender Based Prikry forcing, among many others.Our motivation in this part of the dissertation is to prove an iteration theorem at the level of the successor of a singular  ...  Our motivation in this part of the dissertation is to prove an iteration theorem at the level of the successor of a singular cardinal.  ... 
doi:10.1017/bsl.2021.22 fatcat:yynwd7jnyfc2vmkgazoqknnvua

Transfer Principles in Henselian Valued Fields

Pierre Touchard
2021 Bulletin of Symbolic Logic  
The burden is a cardinal related to the model theoretic complexity and a notion of dimension associated to $\text {NTP}_2$ theories.  ...  Second, we show a transfer principle for the property that all types realized in a given elementary extension are definable.  ...  Our motivation in this part of the dissertation is to prove an iteration theorem at the level of the successor of a singular cardinal.  ... 
doi:10.1017/bsl.2021.31 fatcat:cw6v3vw4fbbfjn5w5shhs5xteu

Bounded stationary reflection

James Cummings, Chris Lambie-Hanson
2015 Proceedings of the American Mathematical Society  
We prove that, assuming large cardinals, it is consistent that there are many singular cardinals µ such that every stationary subset of µ + reflects but there are stationary subsets of µ + that do not  ...  reflect at ordinals of arbitrarily high cofinality.  ...  We now define a forcing notion to shoot a club through the set of approachable points of the successor of a singular cardinal.  ... 
doi:10.1090/proc12743 fatcat:tr2oesfgkrbsfgdl7lztai2sym

JSL volume 74 issue 4 Cover and Front matter

2009 Journal of Symbolic Logic (JSL)  
fields, by ARNO FEHM and WULF-DIETER GEYER 1206 Approachability at the second successor of a singular cardinal, by MOTI GITIK and JOHN KRUEGER 1211 Heirs of box types in polynomially bounded  ...  SOULDATOS 1273 Successors of singular cardinals and coloring theorems II, by TODD EISWORTH and SAHARON SHELAH 1287 The modal /^-calculus hierarchy over restricted classes of transition systems,  ... 
doi:10.1017/s0022481200003030 fatcat:xn6gdbq5m5eczef34cjuzwlo5y

Page 4136 of Mathematical Reviews Vol. , Issue 93h [page]

1993 Mathematical Reviews  
-supercompact then REF fails, and it fails at the successor of a singular cardinal. Using some of the technology developed by M. Magidor and the author [“When does almost free imply free?”, J. Amer.  ...  The authors also obtain from a model of ZFC containing a regular limit of strongly compact cardinals a model of ZF in which every successor 8 has the Specker property, i.e., 2® is a countable union of  ... 

A combinatorial forcing for coding the universe by a real when there are no sharps [article]

Saharon Shelah, Lee Stanley
1993 arXiv   pre-print
Assuming 0# does not exist, we present a combinatorial approach to Jensen's method of coding by a real.  ...  The forcing uses combinatorial consequences of fine structure (including the Covering Lemma, in various guises), but makes no direct appeal to fine structure itself.  ...  If τ ∈ D κ ∩ λ 0 , or (if κ is a limit of singular cardinals), for some successor point, λ, of D κ ∩ λ 0 , τ ∈ D λ , or (if κ is a limit of singular cardinals) for some successor point, λ, of D κ \ λ 0  ... 
arXiv:math/9311204v1 fatcat:qwhowykbhbgbvcif5s3pjkbcpe

Language, procedures, and the non-perceptual origin of number word meanings

DAVID BARNER
2017 Journal of Child Language  
Children first learn small numbers using the same logic that supports other linguistic number marking (e.g. singular/plural).  ...  Systems of numerical perception are either assumed to contain the logical foundations of arithmetic innately, or to supply the basis for their induction.  ...  At this point, these children are typically called 'Cardinal Principle Knowers' (CP-knowers), since they appear to know that the last word in a count labels the cardinality of the set as a whole.  ... 
doi:10.1017/s0305000917000058 pmid:28376934 fatcat:zzvhb2ljbjaxpdpy7e6dgvippa

Bipartite Graphs and Monochromatic Squares

Shimon Garti
2018 Order  
We prove that consistently every bipartite graph of size κ^+×κ^+ contains either a clique or an independent subset of size τ×τ for every τ∈κ^+, where κ is a successor cardinal.  ...  What can be sais about double successors of singular cardinals? Basically, there are two approaches here.  ...  The first one is to try forcing a sufficiently saturated ideal over a successor of a singular cardinal.  ... 
doi:10.1007/s11083-018-9457-4 fatcat:zimw27shpfdpfcf6flecgz5mnq

A combinatorial forcing for coding the universe by a real when there are no sharps

Saharon Shelah, Lee J. Stanley
1995 Journal of Symbolic Logic (JSL)  
Assuming 0# does not exist, we present a combinatorial approach to Jensen's method of coding by a real.  ...  The forcing uses combinatorial consequences of fine structure (including the Covering Lemma, in various guises), but makes no direct appeal to fine structure itself.  ...  The second is a consequence of our treatment of inaccessible cardinals.  ... 
doi:10.2307/2275507 fatcat:g4snturckfe7lg6w2pwgtrdeoq

A very weak square principle

Matthew Foreman, Menachem Magidor
1997 Journal of Symbolic Logic (JSL)  
In the second section we show that our principle is consistent with a supercompact cardinal.  ...  This principle is strong enough to include many of the known applications of □, but weak enough that it is consistent with the existence of very large cardinals.  ...  In this section we prove that it is consistent with a supercompact cardinal that Very Weak Square holds at the successor of every singular cardinal.  ... 
doi:10.2307/2275738 fatcat:szasnnvmj5cd7aigaszj46pn5a

Souslin trees and successors of singular cardinals

Shai Ben-David, Saharon Shelah
1986 Annals of Pure and Applied Logic  
Ben-David noticed that a strongly compact K suffices for the result. 0168-0072/86/$3.50 0 1986, Elsevier Science Publishers B.V. (North-Holland)  ...  Theorem 1 shows the consistency with the existence of a super-compact cardinal of existence of Souslin trees for any successor of a singular cardinal. The proof is via forcing.  ...  The strategy for player I will be as follows: At a successor stage i + 2 let Tpz+, be the tree of the last condition chosen by player II.  ... 
doi:10.1016/0168-0072(86)90020-5 fatcat:35pzg26ipjgqxbzrwwgp2ku3pm

Forcing Axioms, Supercompact Cardinals, Singular Cardinal Combinatorics

Matteo Viale
2008 Bulletin of Symbolic Logic  
The purpose of this communication is to present some recent advances on the consequences that forcing axioms and large cardinals have on the combinatorics of singular cardinals.  ...  I will introduce a few examples of problems in singular cardinal combinatorics which can be fruitfully attacked using ideas and techniques coming from the theory of forcing axioms and then translate the  ...  This is a square-like principle on the successor of a singular κ of countable cofinality which is consistent with κ being larger than a supercompact.  ... 
doi:10.2178/bsl/1208358846 fatcat:dw2rmhz54reldbz3dlznzghv44

Small u(kappa) at singular kappa with compactness at kappa++ [article]

Radek Honzik, Sarka Stejskalova
2019 arXiv   pre-print
We show that the tree property, stationary reflection and the failure of approachability at κ^++ are consistent with u(κ) = κ^+ < 2^κ, where κ is a singular strong limit cardinal with the countable or  ...  As a by-product, we show that if λ is a regular cardinal, then stationary reflection at λ^+ is indestructible under all λ-cc forcings (out of general interest, we also state a related result for the preservation  ...  The underlying goal is to find out whether these principles can hold at the successor or the double successor of a singular cardinal, a long interval of cardinals, or whether they determine the continuum  ... 
arXiv:1910.14391v1 fatcat:63wx2eijv5du5naokjeff56tsi

Spectra of uniformity

Hayut Yair, Karagila Asaf
2019 Commentationes Mathematicae Universitatis Carolinae  
Institute of Mathematics of the Czech Academy of Sciences provides access to digitized documents strictly for personal use. Each copy of any part of this document must contain these Terms of use.  ...  This document has been digitized, optimized for electronic delivery and stamped with digital signature within the project DML-CZ: The Czech Digital Mathematics Library http://dml.cz  ...  The authors would like to thank the anonymous referees for their thorough reviewing efforts, as well as for suggesting Question 6.3, which helped to clarify and improve this paper.  ... 
doi:10.14712/1213-7243.2019.008 fatcat:vnnpnmnntnev7ckrjumll2agcm
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