scholarly journals Group radicals and strongly compact cardinals

2013 ◽  
Vol 366 (4) ◽  
pp. 1857-1877 ◽  
Author(s):  
Joan Bagaria ◽  
Menachem Magidor
Keyword(s):  
Strongly Compact Cardinals

scholarly journals Strongly compact cardinals and the continuum function

2021 ◽  
pp. 103013
Author(s):  
Arthur W. Apter ◽  
Stamatis Dimopoulos ◽  
Toshimichi Usuba
Keyword(s):  
Continuum Function ◽  
Strongly Compact Cardinals ◽  
The Continuum

On measurable limits of compact cardinals

1999 ◽  
Vol 64 (4) ◽  
pp. 1675-1688
Author(s):  
Arthur W. Apter
Keyword(s):  
Supercompact Cardinal ◽  
Compact Cardinal ◽  
Supercompact Cardinals ◽  
Strongly Compact Cardinals

AbstractWe extend earlier work (both individual and joint with Shelah) and prove three theorems about the class of measurable limits of compact cardinals, where a compact cardinal is one which is either strongly compact or supercompact. In particular, we construct two models in which every measurable limit of compact cardinals below the least supercompact limit of supercompact cardinals possesses non-trivial degrees of supercompactness. In one of these models, every measurable limit of compact cardinals is a limit of supercompact cardinals and also a limit of strongly compact cardinals having no non-trivial degree of supercompactness. We also show that it is consistent for the least supercompact cardinal κ to be a limit of strongly compact cardinals and be so that every measurable limit of compact cardinals below κ has a non-trivial degree of supercompactness. In this model, the only compact cardinals below κ with a non-trivial degree of supercompactness are the measurable limits of compact cardinals.

scholarly journals COMPACT CARDINALS AND EIGHT VALUES IN CICHOŃ’S DIAGRAM

2018 ◽  
Vol 83 (2) ◽  
pp. 790-803 ◽  
Author(s):  
JAKOB KELLNER ◽  
ANDA RAMONA TĂNASIE ◽  
FABIO ELIO TONTI
Keyword(s):  
Cichoń’S Diagram ◽  
Image Position ◽  
Strongly Compact Cardinals

AbstractAssuming three strongly compact cardinals, it is consistent that$${\aleph _1} < add\left( {\cal N} \right) < cov\left( {\cal N} \right) &#x003C; \mathfrakb &#x003C; \mathfrakd < non\left( {\cal N} \right) < cof\left( {\cal N} \right) < {2^{{\aleph _0}}}.$$Under the same assumption, it is consistent that$${\aleph _1} < add\left( {\cal N} \right) < cov\left( {\cal N} \right) < non\left( {\cal M} \right) < cov\left( {\cal M} \right) < non\left( {\cal N} \right) < cof\left( {\cal N} \right) < {2^{{\aleph _0}}}.$$

scholarly journals TAMENESS FROM LARGE CARDINAL AXIOMS

2014 ◽  
Vol 79 (4) ◽  
pp. 1092-1119 ◽  
Author(s):  
WILL BONEY
Keyword(s):  
Large Cardinals ◽  
Large Cardinal ◽  
Strongly Compact Cardinal ◽  
Weakly Compact ◽  
Compact Cardinal ◽  
Dual Property ◽  
First Time ◽  
Strongly Compact Cardinals ◽  
Do So

AbstractWe show that Shelah’s Eventual Categoricity Conjecture for successors follows from the existence of class many strongly compact cardinals. This is the first time the consistency of this conjecture has been proven. We do so by showing that every AEC withLS(K) below a strongly compact cardinalκis <κ-tame and applying the categoricity transfer of Grossberg and VanDieren [11]. These techniques also apply to measurable and weakly compact cardinals and we prove similar tameness results under those hypotheses. We isolate a dual property to tameness, calledtype shortness, and show that it follows similarly from large cardinals.

scholarly journals THE STRONG TREE PROPERTY AT SUCCESSORS OF SINGULAR CARDINALS

2014 ◽  
Vol 79 (01) ◽  
pp. 193-207 ◽  
Author(s):  
LAURA FONTANELLA
Keyword(s):  
Singular Limit ◽  
Inaccessible Cardinal ◽  
Tree Property ◽  
Supercompact Cardinals ◽  
Singular Cardinals ◽  
Image Position ◽  
Strongly Compact Cardinals

Abstract An inaccessible cardinal is strongly compact if, and only if, it satisfies the strong tree property. We prove that if there is a model of ZFC with infinitely many supercompact cardinals, then there is a model of ZFC where ${\aleph _{\omega + 1}}$ has the strong tree property. Moreover, we prove that every successor of a singular limit of strongly compact cardinals has the strong tree property.

scholarly journals ARONSZAJN TREES AND FAILURE OF THE SINGULAR CARDINAL HYPOTHESIS

2009 ◽  
Vol 09 (01) ◽  
pp. 139-157 ◽  
Author(s):  
ITAY NEEMAN
Keyword(s):  
Large Cardinals ◽  
Singular Cardinal ◽  
Strong Limit ◽  
Tree Property ◽  
Singular Cardinal Hypothesis ◽  
Aronszajn Trees ◽  
Strongly Compact Cardinals

The tree property at κ+ states that there are no Aronszajn trees on κ+, or, equivalently, that every κ+ tree has a cofinal branch. For singular strong limit cardinals κ, there is tension between the tree property at κ+ and failure of the singular cardinal hypothesis at κ; the former is typically the result of the presence of strongly compact cardinals in the background, and the latter is impossible above strongly compacts. In this paper, we reconcile the two. We prove from large cardinals that the tree property at κ+ is consistent with failure of the singular cardinal hypothesis at κ.

Strongly compact cardinals, elementary embeddings and fixed points

1984 ◽  
Vol 49 (3) ◽  
pp. 808-812
Author(s):  
Yoshihiro Abe
Keyword(s):  
Fixed Points ◽  
Elementary Embedding ◽  
Inner Model ◽  
Power Set ◽  
Elementary Embeddings ◽  
Basic Facts ◽  
The Universe ◽  
Normal Ultrafilter ◽  
Strongly Compact Cardinals

J. Barbanel [1] characterized the class of cardinals fixed by an elementary embedding induced by a normal ultrafilter on Pκλ assuming that κ is supercompact. In this paper we shall prove the same results from the weaker hypothesis that κ is strongly compact and the ultrafilter is fine.We work in ZFC throughout. Our set-theoretic notation is quite standard. In particular, if X is a set, ∣X∣ denotes the cardinality of X and P(X) denotes the power set of X. Greek letters will denote ordinals. In particular γ, κ, η and γ will denote cardinals. If κ and λ are cardinals, then λ<κ is defined to be supγ<κγγ. Cardinal exponentiation is always associated from the top. Thus, for example, 2λ<κ means 2(λ<κ). V denotes the universe of all sets. If M is an inner model of ZFC, ∣X∣M and P(X)M denote the cardinality of X in M and the power set of X in M respectively.We review the basic facts on fine ultrafilters and the corresponding elementary embeddings. (For detail, see [2].)Definition. Assume κ and λ are cardinals with κ ≤ λ. Then, Pκλ = {X ⊂ λ∣∣X∣ < κ}.It is important to note that ∣Pκλ∣ = λ< κ.

scholarly journals Colimits of accessible categories

2013 ◽  
Vol 155 (1) ◽  
pp. 47-50
Author(s):  
R. PARÉ ◽  
J. ROSICKÝ
Keyword(s):  
Accessible Categories ◽  
Directed Colimit ◽  
Strongly Compact Cardinals

AbstractWe show that any directed colimit of accessible categories and accessible full embeddings is accessible and, assuming the existence of arbitrarily large strongly compact cardinals, any directed colimit of accessible categories and accessible embeddings is accessible.

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