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Indiscernible Sequences for Extenders, and the Singular Cardinal Hypothesis
[article]

1995
*
arXiv
*
pre-print

We prove several results giving lower bounds

arXiv:math/9507214v1
fatcat:r5mqjv5l5jguvhpehzoqpnmwbi
*for**the*large*cardinal*strength of a failure of*the**singular**cardinal**hypothesis*. ... In order to prove this theorem we give a detailed analysis of*the**sequences*of*indiscernibles*which come from applying*the*covering lemma to nonoverlapping*sequences*of*extenders*. ...*For*other applications, such as*the**singular**cardinal**hypothesis*, this is not possible. ...##
###
Indiscernible sequences for extenders, and the singular cardinal hypothesis

1996
*
Annals of Pure and Applied Logic
*

We prove several results giving lower bounds

doi:10.1016/s0168-0072(96)00007-3
fatcat:yg6k4hlffbgspndpisa4cd2mtm
*for**the*large*cardinal*strength of a failure of*the**singular**cardinal**hypothesis*.*The*main result is*the*following theorem: Theorem. ...*The*applications to*the**singular**cardinal**hypothesis*are given in Section 3,*and*some open problems are stated in Section 4. ...*For*other applications, such as*the**singular**cardinal**hypothesis*, this is not possible. ...##
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0# and inner models

2002
*
Journal of Symbolic Logic (JSL)
*

*The*principal tools are

*the*Covering Theorem

*for*L

*and*

*the*technique of reverse Easton iteration. Let I denote

*the*class of Silver

*indiscernibles*

*for*L

*and*〈i α ∣ α ϵ ORD〉 its increasing enumeration. ... Also fix an inner model M of GCH not containing 0#

*and*let ω α denote

*the*ω α of

*the*model M[0#],

*the*least inner model containing M as a submodel

*and*0# as an element. ... By

*hypothesis*,

*the*M-

*cardinality*of κ is greater than ω M 1 . ...

##
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On the singular cardinal hypothesis

1992
*
Transactions of the American Mathematical Society
*

We use core model theory to obtain

doi:10.1090/s0002-9947-1992-1073778-4
fatcat:uvgourgbezed5cuc6d5ekasxbi
*the*following lower bounds to*the*consistency strength*for**the*failure of*the**Singular**Cardinal**Hypothesis*: Suppose that k is a*singular*strong limit*cardinal*such that ... Then there is an inner model K such that o(k) = «c++ in K if k has uncountable cofinality,*and*Va < k3u < k o(k) > u in K otherwise. ... Introduction*The**Singular**Cardinal**Hypothesis*(SCH) asserts that if k is any*singular*strong limit*cardinal*then 2K = k+ . ...##
###
On the Singular Cardinal Hypothesis

1992
*
Transactions of the American Mathematical Society
*

We use core model theory to obtain

doi:10.2307/2153949
fatcat:pr6olkkxabgudcq4k22lmvwbqm
*the*following lower bounds to*the*consistency strength*for**the*failure of*the**Singular**Cardinal**Hypothesis*: Suppose that k is a*singular*strong limit*cardinal*such that ... Then there is an inner model K such that o(k) = «c++ in K if k has uncountable cofinality,*and*Va < k3u < k o(k) > u in K otherwise. ... Introduction*The**Singular**Cardinal**Hypothesis*(SCH) asserts that if k is any*singular*strong limit*cardinal*then 2K = k+ . ...##
###
On gaps under GCH type assumptions

2003
*
Annals of Pure and Applied Logic
*

We prove equiconsistency results concerning gaps between a

doi:10.1016/s0168-0072(02)00031-3
fatcat:dwhz5qakujdtrowbvaff4f7c4m
*singular*strong limit*cardinal*κ of cofinality ℵ 0*and*its power under assumptions that 2 κ = κ +δ+1*for*δ < κ*and*some weak form of*the**Singular*...*Cardinal**Hypothesis*below κ. ... If 2 κ ≥ κ +δ*and**the**Singular**Cardinal**Hypothesis*holds below κ at least*for**cardinals*of cofinality cf δ, then in*the*core model either (i) o(κ) ≥ κ +δ+1 + 1 or (ii) {α < κ | o(α) ≥ α +δ+1 + 1} is unbounded ...##
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On the Singular Cardinal Hypothesis
[article]

1992
*
arXiv
*
pre-print

We use

arXiv:math/9204202v1
fatcat:ypqxzjj6yvhp7eerw5cps22vze
*the*core model*for**sequences*of measures to prove a new lower bound*for**the*consistency strength of*the*failure of*the*SCH: THEOREM (i) If there is a*singular*strong limit*cardinal*κ such that 2 ... (ii) If there is a*singular*strong limit*cardinal*κ of uncountable cofinality such that 2^κ > κ^+ then there is an inner model with o(κ) = κ^++. ... -Introduction*The**Singular**Cardinal**Hypothesis*(SCH) asserts that if κ is any*singular*strong limit*cardinal*then 2 κ = κ + . ...##
###
On Gaps under GCH Type Assumptions
[article]

2000
*
arXiv
*
pre-print

This basically completes

arXiv:math/9908118v2
fatcat:hhf6hiuw6rfdzop5fh7b7ktxee
*the*study of consistency strength of various gaps between a strong limit*singular**cardinal*of cofinality omega*and*its power under GCH type assumptions below. ...*The*results of*the*previous version are impoved. ... We are grateful to Saharon Shelah*for*many helpful conversations*and**for*explanations that he gave on*the*pcf-theory. ...##
###
Definable Singularity

1991
*
Transactions of the American Mathematical Society
*

In

doi:10.2307/2001849
fatcat:aoq6umocvjeb7iymrv5vgb3ps4
*the*course of*the*proof we develop a simplified statement of*the*covering lemma*for**sequences*of measures which avoids*the*use of mice. ...*The*main result of this paper is a characterization of*singular**cardinals*in terms of*the*core model, assuming that there is no model of 3/c o(k) = k++ . ... Is there, under any large*cardinal**hypothesis*, a class model M of ZFC which contains a*singular**cardinal*k such that there is no witness in M to*the**singularity*of k which is, in some reasonable sense, ...##
###
Definable singularity

1991
*
Transactions of the American Mathematical Society
*

In

doi:10.1090/s0002-9947-1991-1036006-0
fatcat:zlqpeedwxfgbjaqpwl4apz2cnu
*the*course of*the*proof we develop a simplified statement of*the*covering lemma*for**sequences*of measures which avoids*the*use of mice. ...*The*main result of this paper is a characterization of*singular**cardinals*in terms of*the*core model, assuming that there is no model of 3/c o(k) = k++ . ... Is there, under any large*cardinal**hypothesis*, a class model M of ZFC which contains a*singular**cardinal*k such that there is no witness in M to*the**singularity*of k which is, in some reasonable sense, ...##
###
On possible non-homeomorphic substructures of the real line

2002
*
Proceedings of the American Mathematical Society
*

This requires large

doi:10.1090/s0002-9939-02-06385-2
fatcat:qf4jie2dujbizoxe4hklhgg2by
*cardinals*,*and*we obtain an exact consistency strength: Theorem 1. ... We consider*the*problem, raised by Kunen*and*Tall, of whether*the*real continuum can have non-homeomorphic versions in different submodels of*the*universe of all sets. ... To make c*singular**and*Jónsson requires*the*consistency of larger*cardinals*. Theorem 2.6. ...##
###
Exact saturation in pseudo-elementary classes for simple and stable theories
[article]

2022
*
arXiv
*
pre-print

PC-exact saturation at

arXiv:2009.08365v2
fatcat:76x6ezeazjd7xjvkwgaj4wehbu
*singular**cardinals*, satisfying mild set-theoretic hypotheses, which had previously been open even*for**the*random graph. ... We also consider*the*local analogue of PC-exact saturation, showing that local PC-exact saturation*for**singular**cardinals*of countable cofinality characterizes supershort theories. ...*For*example if I is an*indiscernible*set, then*the*type omitted is that of a new element in*the**sequence*. Question 6.2. Which NIP theories have local exact saturation at*singular**cardinals*as above? ...##
###
Exact saturation in simple and NIP theories
[article]

2015
*
arXiv
*
pre-print

A theory T is said to have exact saturation at a

arXiv:1510.02741v1
fatcat:or5j4wpbqngftosyqyaaubrupa
*singular**cardinal*κ if it has a κ-saturated model which is not κ^+-saturated. ... Also, an NIP theory has exact saturation if*and*only if it is not distal. This gives a new characterization of distality. ... Let d = (I 1 + J, I 2 ), which we identify with*the*corresponding cut in*the**extended**sequence*I ′ = I ∪ J. ...##
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Perfect trees and elementary embeddings

2008
*
Journal of Symbolic Logic (JSL)
*

This technique is crucial both in

doi:10.2178/jsl/1230396754
fatcat:xhob6ismunaqdejnxygkvcetje
*the*study of large*cardinal*preservation*and*of internal consistency. ... In easy cases, such as when forcing to make*the*GCH hold while preserving a measurable*cardinal*(via a reverse Easton iteration of α-Cohen forcing*for*successor*cardinals*α),*the*generic G* is simply generated ... By adding a Prikry*sequence*s through κ over V [G][g], we obtain a failure of*the**singular**cardinal**hypothesis*, as in V [G][g][s], κ is a*singular*strong limit*cardinal*of cofinality ω*and*2 κ = κ ++ . ...##
###
The maximality of the core model
[article]

1997
*
arXiv
*
pre-print

If T is an iteration tree on K

arXiv:math/9702206v1
fatcat:lty5f32ab5e3taljmj6wqoswam
*and*F is a countably certified*extender*that coheres with*the*final model of T, then F is on*the**extender**sequence*of*the*final model of T. ... Other results in this paper, when combined with work of Woodin, imply: o If square-kappa-finite fails*and*kappa is a*singular*, strong limit*cardinal*, then Inductive Determinacy holds. o If square-kappa-finite ...*The*research of*the*second author was partially supported by*the*National Science Foundation. ...
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