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Blowing up the power of a singular cardinal of uncountable cofinality with collapses [article]

Sittinon Jirattikansakul
2022 arXiv   pre-print
We prove that given a singular cardinal κ of cofinality η in the ground model, which is a limit of suitable large cardinals, and η^+=ℵ_γ, then there is a forcing extension which preserves cardinals and  ...  The Singular Cardinal Hypothesis (SCH) is one of the most classical combinatorial principles in set theory. It says that if κ is singular strong limit, then 2^κ=κ^+.  ...  Our forcing construction is a combination of Gitik's construction [3] which blows up the powerset of a singular cardinal with any cofinality, and his recent work on collpasing generators [4] .  ... 
arXiv:2011.00409v3 fatcat:5elf75ws4bgrrbylwobl2isfxy

The power set function [article]

Moti Gitik
2002 arXiv   pre-print
We survey old and recent results on the problem of finding a complete set of rules describing the behavior of the power function, i.e. the function which takes a cardinal κ to the cardinality of its power  ...  Restrictions on the power of singular cardinals The Singular Cardinal Problem (SCP) is the problem of finding a complete set of rules describing the behavior of the power function on singular cardinals  ...  Under this assumption it is possible to blow up the power of κ arbitrary high preserving GCH below κ. Also, it is possible to turn κ into the first fixed point of the ℵ function, see [6] .  ... 
arXiv:math/0212405v1 fatcat:izwlmtt5frfthn7dimwpnm7bm4

Singular Cardinals and the PCF Theory

Thomas Jech
1995 Bulletin of Symbolic Logic  
The work on the singular cardinals problem has led to many often surprising results, culminating in a beautiful theory of Saharon Shelah called the pcf theory ("pcf" stands for "possible cofinalities")  ...  In the simplest nontrivial case, 2κrepresents the cardinal number of the power setP(κ), the set of all subsets ofκ.  ...  The general idea in all these proofs is as follows: start with a large cardinal K, blow up 2", change the cofinality of K by adjoining a new set of cardinals C cofinal in K , and destroy all cardinals  ... 
doi:10.2307/421130 fatcat:2e4jbjbaebbcrgljim4lzuk5oy

Page 4052 of Mathematical Reviews Vol. , Issue 97G [page]

1997 Mathematical Reviews  
From the introduction: “Suppose that « is a singular cardinal of cofinality w. We would like to blow up its power.  ...  Howard (1-EMI; Ypsilanti, MI) 97g:03054 03E35 03540 03ES0 03E55 Gitik, Moti (I1L-TLAV; Tel Aviv) Blowing up the power of a singular cardinal. Ann. Pure Appl. Logic 80 (1996), no. 1, 17-33.  ... 

Applications of pcf for mild large cardinals to elementary embeddings [article]

Moti Gitik, Saharon Shelah
2013 arXiv   pre-print
The following pcf results are proved: 1. Assume that kappa > aleph_0 is a weakly compact cardinal. Let mu > 2^kappa be a singular cardinal of cofinality kappa.  ...  Let mu be a strong limit cardinal and theta a cardinal above mu. Suppose that at least one of them has an uncountable cofinality.  ...  Use the extender based Magidor to blow up the power of τ to η + simultaneously changing the cofinality of τ to κ.  ... 
arXiv:1307.5977v1 fatcat:giqyzvwejngazgdrytajqp2pki

Applications of pcf for mild large cardinals to elementary embeddings

Moti Gitik, Saharon Shelah
2013 Annals of Pure and Applied Logic  
The following pcf results are proved: 1. Assume that κ > ℵ 0 is a weakly compact cardinal. Let µ > 2 κ be a singular cardinal of cofinality κ.  ...  Let µ be a strong limit cardinal and θ a cardinal above µ. Suppose that at least one of them has an uncountable cofinality.  ...  Use the extender based Magidor to blow up the power of τ to η + simultaneously changing the cofinality of τ to κ.  ... 
doi:10.1016/j.apal.2013.03.002 fatcat:bzmw3hmodjhcppvtbpxsk5lgda

More on uniform ultrafilters over a singular cardinal

Moti Gitik
2020 Fundamenta Mathematicae  
Morgan [1] to singular cardinals of uncountable cofinality. In [4] , it was shown that already at κ = ℵ ω it is possible to have the ultrafilter number smaller than 2 κ .  ...  Shelah [6] initiated the study of the character and the ultrafilter number of uniform ultrafilters over a singular cardinal.  ...  We would like to thank the referee of the paper for his remarks and corrections. The research was partially supported by Israel Science Foundation Grant no. 1216/18.  ... 
doi:10.4064/fm845-5-2020 fatcat:boxb6ar5knaqto4krk67fmmb2a

Dense non-reflection for stationary collections of countable sets

David Asperó, John Krueger, Yasuo Yoshinobu
2009 Annals of Pure and Applied Logic  
If λ is singular with countable cofinality, then dense non-reflection in Pω 1 (λ) follows from the existence of squares.  ...  We present several forcing posets for adding a non-reflecting stationary subset of Pω 1 (λ), where λ ≥ ω 2 .  ...  This can be achieved by a natural variation of the forcing in Section 1. However, this forcing will blow up the power set of all cardinals between γ and λ to at least λ.  ... 
doi:10.1016/j.apal.2009.06.002 fatcat:ba5kcnnwprasnlgvyflpug7v6i

Page 7313 of Mathematical Reviews Vol. , Issue 97M [page]

1997 Mathematical Reviews  
component to blow up the size of a generating family for the cub filter F, on certain cardinals 4 < « from 4* to A** (here assuming GCH).  ...  The fol- lowing gives the precise formulation: Suppose that « is a regular uncountable cardinal, and S and 7 are stationary subsets of «.  ... 

THE EIGHTFOLD WAY

JAMES CUMMINGS, SY-DAVID FRIEDMAN, MENACHEM MAGIDOR, ASSAF RINOT, DIMA SINAPOVA
2018 Journal of Symbolic Logic (JSL)  
with the Prikry-collapse poset for turning a large cardinal into ${\aleph _\omega }$.  ...  The proofs combine the techniques of adding and then destroying a nonreflecting stationary set or a ${\kappa ^{ + + }}$-Souslin tree, variants of Mitchell's forcing to obtain the tree property, together  ...  Acknowledgements The results in this paper were conceived and proved during three visits to the American Institute of Mathematics, as part of the Institute's SQuaRE collaborative research program.  ... 
doi:10.1017/jsl.2017.69 fatcat:e6uqlibcpvenpbafcmy7okq2cy

Stationary Reflection and the failure of SCH [article]

Omer Ben-Neria, Yair Hayut, Spencer Unger
2019 arXiv   pre-print
In this paper we prove that from large cardinals it is consistent that there is a singular strong limit cardinal ν such that the singular cardinal hypothesis fails at ν and every collection of fewer than  ...  For uncountable cofinality, this situation was not previously known to be consistent.  ...  The second study is Gitik's recent work [5] for blowing up the power of a singular cardinal using a Mitchell order increasing sequence of overlapping extenders.  ... 
arXiv:1908.11145v1 fatcat:7ljh5z34lbfxbnit7xb75t3rq4

ON THE SPLITTING NUMBER AT REGULAR CARDINALS

OMER BEN-NERIA, MOTI GITIK
2015 Journal of Symbolic Logic (JSL)  
Let κ, λ be regular uncountable cardinals such that λ &gt; κ + is not a successor of a singular cardinal of low cofinality.  ...  We construct a generic extension with s(κ) = λ starting from a ground model in which o(κ) = λ and prove that assuming ¬0¶, s(κ) = λ implies that o(κ) ≥ λ in the core model.  ...  What is the consistency strength of the statement that κ is a measurable and s(κ) = κ ++ ? In the model of Kamo, κ remains a measurable (and even a supercompact). Question 3.  ... 
doi:10.1017/jsl.2015.22 fatcat:rg2b4upgp5hb3psm3arw7dzyja

Global singularization and the failure of SCH

Radek Honzik
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 cardinals into cardinals satisfying some mild restrictions, then there exists a cardinal-preserving  ...  We say that κ is µ-hypermeasurable (or µ-strong) for a cardinal µ ≥ κ + if there is an Such a j is called a witnessing embedding.  ...  The work was supported by a postdoctoral grant of the Grant Agency of the Czech Republic 201/09/P115, and by a Short Visit Grant 2903 from the European Science Foundation: New frontiers of infinitymathematical  ... 
doi:10.1016/j.apal.2009.11.002 fatcat:epbfoytlhrcsrfsvxlfeu7sanm

A general tool for consistency results related to I1

Vincenzo Dimonte, Liuzhen Wu
2015 European Journal of Mathematics  
Examples of such properties are the first failure of GCH, a very good scale and the negation of the approachability property, or the tree property at λ + and λ ++ .  ...  In this paper we provide a general tool to prove the consistency of I1(λ) with various combinatorial properties at λ typical at settings with 2 λ > λ + , that does not need a profound knowledge of the  ...  The typical way to make SCH fail at a singular cardinal (i.e. blowing up the cardinality of its powerset) is to start with κ measurable, blowing up its power and then adding an ω-sequence cofinal to κ  ... 
doi:10.1007/s40879-015-0092-y fatcat:raiak3jltveh3cd3cim62udwna

A general tool for consistency results related to I1 [article]

Vincenzo Dimonte, Liuzhen Wu
2015 arXiv   pre-print
Examples of such properties are the first failure of GCH, a very good scale and the negation of the approachability property, or the tree property at λ^+ and λ^++.  ...  In this paper we provide a general tool to prove the consistency of I1(λ) with various combinatorial properties at λ typical at settings with 2^λ>λ^+, that does not need a profound knowledge of the forcing  ...  The typical way to make SCH fail at a singular cardinal (i.e. blowing up the cardinality of its powerset) is to start with κ measurable, blowing up its power and then adding an ω-sequence cofinal to κ  ... 
arXiv:1510.03287v1 fatcat:d44lvu4hvzg2nfccqtuuzilo4y
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