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Adding closed cofinal sequences to large cardinals

1982
*
Annals of Mathematical Logic
*

Magidor has a forcing notion generalizir~g Prikry's which adds a

doi:10.1016/0003-4843(82)90023-7
fatcat:tnjlaak7o5cr3jqrro6ypk7qna
*closed**cofinal**sequence*of ordinals through a*large**cardinal*. The*cardinal*remains uncountable but its regularity is still destroyed. ... We obtain a forcing notion which adds a*closed**cofinal**sequence*of ordinals (and more complex objects) through a*large**cardinal*~, of order WPe ~, and keeps K regular~ In fact u remains measurable after ... I would also like*to*express my gratitude*to*the lntel Corporation for granting me the use of their computer systems with which this document was prepared. ...##
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Restrictions on forcings that change cofinalities

2015
*
Archive for Mathematical Logic
*

We show that such forcing which changes

doi:10.1007/s00153-015-0454-7
fatcat:phczjjnt4ffmjolwembtuvtsdi
*cofinality*of a regular*cardinal*, cannot be too nice and must cause some "damage"*to*the structure of*cardinals*and stationary sets. ... In this paper we investigate some properties of forcing which can be considered "nice" in the context of singularizing regular*cardinals**to*have an uncountable*cofinality*. ... In Theorem 5 we saw that we cannot change the*cofinalities*of infinitely many*cardinals**to*a constant*cofinality*without*adding*ω-*sequences*, or changing the*cofinalities*of all the*cardinals*in the pcf ...##
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The strong tree property and weak square
[article]

2016
*
arXiv
*
pre-print

We show that it is consistent, relative

arXiv:1601.07824v2
fatcat:asjqzfkvsfbqbhi5vpryjtjsim
*to*ω many supercompact*cardinals*, that the super tree property holds at _n for all 2 ≤ n < ω but there are weak square and a very good scale at _ω. ... Let µ be a singular*cardinal*of countable*cofinality*, and let µ n | n < ω be an increasing*sequence*of regular*cardinals**cofinal*at µ. ... Let µ be a singular*cardinal*of*cofinality*ω. Let µ i | i < ω be an increasing*sequence*of regular*cardinals**cofinal*in µ. We define an ordering on i<ω µ i as follows. ...##
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Kleinberg sequences and partition cardinals below δ51

2002
*
Fundamenta Mathematicae
*

The author computes the Kleinberg

doi:10.4064/fm171-1-4
fatcat:q3klbyfy4nh7fkqh2ynrmor33e
*sequences*derived from the three different normal ultrafilters on δ 1 3 . ... Eugene Kleinberg linked the theory of partition*cardinals**to*the Axiom of Determinacy*AD*by showing that the first ω +1 infinite*cardinals*satisfy certain*large**cardinal*properties defined via partition ... This is not just a feature of choiceless set theory: In the Příkrý (ZFC)model obtained by generically*adding*a*cofinal*ω-*sequence**to*a measurable*cardinal*, the former measurable*cardinal*is a Rowbottom ...##
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A cardinal preserving extension making the set of points of countable V cofinality nonstationary

2007
*
Archive for Mathematical Logic
*

Assuming

doi:10.1007/s00153-007-0048-0
fatcat:x3j2eydb5bfkbmc3xodmhxuapa
*large**cardinals*we produce a forcing extension of V which preserves*cardinals*, does not add reals, and makes the set of points of countable V*cofinality*in κ + nonstationary. ... Finally we show that our*large**cardinal*assumption is optimal. Israel ... Let θ i , i < ω, be a*sequence*of*cardinals*,*cofinal*in κ, so that o(θ i+1 ) > θ i for each i. Let P i be the forcing of Magidor [4] ,*to*change the*cofinality*of θ i+1*to*θ i . ...##
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A very weak square principle

1997
*
Journal of Symbolic Logic (JSL)
*

This principle is strong enough

doi:10.2307/2275738
fatcat:szasnnvmj5cd7aigaszj46pn5a
*to*include many of the known applications of □, but weak enough that it is consistent with the existence of very*large**cardinals*. ... It deals with a principle dubbed "Not So Very Weak Square", which appears*close**to*Very Weak Square but turns out not*to*be equivalent. ... Standard*large**cardinal*theory then implies that if we take H ⊂ R*to*be V [G]generic with m ∈ H, then j can be extended*to*aĵ : V [G] → M [H]. ...##
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Destructibility of the tree property at _ω+1
[article]

2019
*
arXiv
*
pre-print

On the other hand we discuss some cases in which the tree property is indestructible under small or

arXiv:1603.05526v3
fatcat:yhgqjwsr4jad5e7co2dbkbarhi
*closed*forcings. ... Acknowledgments We would like*to*thank the anonymous referee for improving the readability and accuracy of this paper. ... Thus, one can complete the partial square*to*a full ℵω ,<ℵω by*adding**cofinal*ω-*sequences*. ...##
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On the non-extendibility of strongness and supercompactness through strong compactness

2002
*
Fundamenta Mathematicae
*

that the

doi:10.4064/fm174-1-5
fatcat:obq75vulxncjldovbjc3r4nmvu
*cardinals**to*which non-reflecting stationary sets of ordinals of*cofinality*ω are*added*are also strong. ... as defined by Magidor in [15] which adds Prikry*sequences**to**cardinals*above δ". ...##
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Stationary sets added when forcing squares

2018
*
Archive for Mathematical Logic
*

The purpose of this paper is

doi:10.1007/s00153-018-0613-8
fatcat:6q5jex2nyjhijclnoenyskxaru
*to*demonstrate the difficulty in such results. ... We prove that the poset S(κ, < λ), which adds a κ,<λ -*sequence*by initial segments, will also add nonreflecting stationary sets. ... supercompact*cardinals*: It is consistent for κ*to*be a singular*cardinal*of countable*cofinality*and for * κ*to*hold while every*sequence*S i : i < λ of stationary subsets of κ + containing ordinals of ...##
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Page 2100 of Mathematical Reviews Vol. , Issue 97D
[page]

1997
*
Mathematical Reviews
*

Finally, noting that the forcing for adjoining a non-reflecting stationary subset of A* is only A-strategically

*closed*, not A-*closed*, the iterated forcing is checked for preserving*large**cardinals*. ... The iteration of*adding*a non-reflecting stationary subset and then a*closed*unbounded subset through its complement is importantly observed*to*be just*adding*a Cohen subset, a known sort of property from ...##
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Global singularization and the failure of SCH

2010
*
Annals of Pure and Applied Logic
*

Building on the results in [7], we will show that if V satisfies GCH and F is an Easton function from the regular

doi:10.1016/j.apal.2009.11.002
fatcat:epbfoytlhrcsrfsvxlfeu7sanm
*cardinals*into*cardinals*satisfying some mild restrictions, then there exists a*cardinal*-preserving ... forcing extension V * where F is realised on all V -regular*cardinals*and moreover, all F ... Great thanks are due*to*my advisor Sy D. Friedman who gave me guidance and a lot of help. I also wish*to*thank James Cummings, Moti Gitik and Menachem Magidor for the personal discussions. ...##
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Some Open Problems in Mutual Stationarity Involving Inner Model Theory: A Commentary

2005
*
Notre Dame Journal of Formal Logic
*

Let κ α |α < δ for δ < κ 0 be an ascending

doi:10.1305/ndjfl/1125409336
fatcat:reivle6csfbqbk5puoi6qypb2e
*sequence*of uncountable regular*cardinals*. Let κ = df sup κ α . For λ a regular*cardinal*let cof λ = df {α ∈ On | cf(α) = λ}. ... theory, and the construction of global square*sequences*in core models,*to*obtain stronger results. ... This indicates that MS( ω n n<ω , ω 1 )(+CH) must be a*large**cardinal*property. ...##
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Strongly compact cardinals and ordinal definability
[article]

2021
*
arXiv
*
pre-print

We improve the

arXiv:2107.00513v1
fatcat:dg2mbfbfgjhgjcsanzu2b7r6wm
*large**cardinal*hypothesis of Woodin's HOD dichotomy theorem from an extendible*cardinal**to*a strongly compact*cardinal*. ... We show that assuming there is a strongly compact*cardinal*and the HOD hypothesis holds, there is no elementary embedding from HOD*to*HOD, settling a question of Woodin. ... As in Theorem 2.8, one can use this*sequence**to*show cf(δ) = |δ|. Let C ⊆ δ be a*closed**cofinal*set of ordertype cf(δ). Then δ = ξ∈C σ ξ . ...##
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Page 6616 of Mathematical Reviews Vol. , Issue 92m
[page]

1992
*
Mathematical Reviews
*

In one example a measurable

*cardinal*x for which 2 is*large*is made singular by*adding*a Prikry*sequence*Kp < K; < K2 <---. ... Summary: “Assuming*AD*+(V = L(R)), it is shown that for x an admissible Suslin*cardinal*, o(x) (= the order type of the stationary subsets of x) is ‘essentially’ regular and*closed*under ultrapowers in a ...##
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Gap Forcing
[article]

1999
*
arXiv
*
pre-print

less than delta and Q is forced

arXiv:math/9808011v2
fatcat:gzrdu5ata5at7kqf7xfnevq5tu
*to*be delta-strategically*closed*. ... Many of the most common reverse Easton iterations found in the*large**cardinal*context, such as the Laver preparation, admit a gap at some small delta in the sense that they factor as P*Q, where P has size ... no fresh binary*sequences*are*added*. ...
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