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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 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.  ...

### Restrictions on forcings that change cofinalities

Yair Hayut, Asaf Karagila
2015 Archive for Mathematical Logic
We show that such forcing which changes 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  ...

### The strong tree property and weak square [article]

Yair Hayut, Spencer Unger
2016 arXiv   pre-print
We show that it is consistent, relative 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.  ...

### Kleinberg sequences and partition cardinals below δ51

Benedikt Löwe
2002 Fundamenta Mathematicae
The author computes the Kleinberg 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  ...

### A cardinal preserving extension making the set of points of countable V cofinality nonstationary

Moti Gitik, Itay Neeman, Dima Sinapova
2007 Archive for Mathematical Logic
Assuming 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  , to change the cofinality of θ i+1 to θ i .  ...

### A very weak square principle

Matthew Foreman, Menachem Magidor
1997 Journal of Symbolic Logic (JSL)
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.  ...  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].  ...

### Destructibility of the tree property at _ω+1 [article]

Yair Hayut, Menachem Magidor
2019 arXiv   pre-print
On the other hand we discuss some cases in which the tree property is indestructible under small or 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.  ...

### On the non-extendibility of strongness and supercompactness through strong compactness

Arthur W. Apter
2002 Fundamenta Mathematicae
that the cardinals to which non-reflecting stationary sets of ordinals of cofinality ω are added are also strong.  ...  as defined by Magidor in  which adds Prikry sequences to cardinals above δ".  ...

### Stationary sets added when forcing squares

Maxwell Levine
2018 Archive for Mathematical Logic
The purpose of this paper is 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  ...

### 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  ...

### Global singularization and the failure of SCH

2010 Annals of Pure and Applied Logic
Building on the results in , we will show that if V satisfies GCH and F is an Easton function from the regular 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.  ...

### Some Open Problems in Mutual Stationarity Involving Inner Model Theory: A Commentary

P. D. Welch
2005 Notre Dame Journal of Formal Logic
Let κ α |α < δ for δ < κ 0 be an ascending 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.  ...

### Strongly compact cardinals and ordinal definability [article]

Gabriel Goldberg
2021 arXiv   pre-print
We improve the 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 .  ...

### 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  ...

### Gap Forcing [article]

Joel David Hamkins
1999 arXiv   pre-print
less than delta and Q is forced 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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