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Finite Trees And The Necessary Use Of Large Cardinals
, 1998
"... this paper, a tree T = (V,) is a partial ordering with a minimum element, where V is finite, and the ancestors of any x V are linearly ordered under . The minimum element of T is called the root of T, and is written r(T). A tree is said to be trivial if and only if it has exactly one vertex, which m ..."
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Cited by 3 (1 self)
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this paper, a tree T = (V,) is a partial ordering with a minimum element, where V is finite, and the ancestors of any x V are linearly ordered under . The minimum element of T is called the root of T, and is written r(T). A tree is said to be trivial if and only if it has exactly one vertex, which must be its root. V = V(T) represents the set of all vertices of the tree T = (V,). In a tree T, if x < y and for no z is x < z < y, then we say that y is a child of x and x is the parent of y. Every vertex has at most one parent. However, vertices may have zero or more children. We write p(x,T) for the parent of x in T. We use Ch(T) = V(T)\{r(T)} for the set of all children of T. We write T 1 T 2 if and only if i) r(T 1 ) = r(T 2 ); ii) for all x Ch(T 1 ), p(x,T 1 ) = p(x,T 2 ). This is a partial ordering on trees. Note that if T 1 T 2
POSITIVE PARTITION RELATIONS FOR Pκ(λ)
"... Abstract. Let κ a regular uncountable cardinal and λ a cardinal> κ, and suppose λ <κ is less than the covering number for category cov(Mκ,κ). Then (a) I + κ κ,λ−→(I + κ,λ, ω + 1)2, (b) I + κ κ,λ−→[I + κ,λ]2 κ + if κ is a limit cardinal, and (c) I + κ κ,λ−→(I + κ,λ)2 if κ is weakly compact. 0. ..."
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Abstract. Let κ a regular uncountable cardinal and λ a cardinal> κ, and suppose λ <κ is less than the covering number for category cov(Mκ,κ). Then (a) I + κ κ,λ−→(I + κ,λ, ω + 1)2, (b) I + κ κ,λ−→[I + κ,λ]2 κ + if κ is a limit cardinal, and (c) I + κ κ,λ−→(I + κ,λ)2 if κ is weakly compact. 0.
Acknowledgements
, 2003
"... I am grateful for having had the opportunity to work under the supervision of Professor Jouko Väänänen. He suggested to me the subject that was to be the starting point of this work that constitutes a part of my doctoral dissertation. I would also like to thank all the other members of the Helsinki ..."
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I am grateful for having had the opportunity to work under the supervision of Professor Jouko Väänänen. He suggested to me the subject that was to be the starting point of this work that constitutes a part of my doctoral dissertation. I would also like to thank all the other members of the Helsinki Logic Group, who have contributed to a pleasant and inspiring working environment. Especially I wish to thank Docent Tapani Hyttinen for always taking the time to discuss and giving his guidance. Finally I wish to thank Professor Boban Veličković and Docent Kerkko Luosto
The composition of large cardinal axioms
, 2010
"... In his seminal study of the ineffability properties of cardinals, James Baumgartner discovered that the nineffable subsets of a cardinal κ can be characterized as the result of “composing ” the related property of nsubtlety with the classical property of Π12indescribability. An analogous charact ..."
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In his seminal study of the ineffability properties of cardinals, James Baumgartner discovered that the nineffable subsets of a cardinal κ can be characterized as the result of “composing ” the related property of nsubtlety with the classical property of Π12indescribability. An analogous characterization was achieved by Kanamori with versions of these properties stronger than Vopenka’s Principle. We generalize these theorems to show how a large class of large cardinal axioms can be composed with indescribability, and find a new instance using the Pκλ versions of strongly nineffable and strongly nsubtle, introduced recently by Abe. In the final section, we show that each notion of nsubtlety is itself characterized as a (slightly different) kind of “composition”, this time of stationary sets and what we call “prensubtle ” sets. 1