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FORCING WITH SEQUENCES OF MODELS OF TWO TYPES
"... Abstract. We present an approach to forcing with finite sequences of models that uses models of two types. This approach builds on earlier work of Friedman and Mitchell on forcing to add clubs in cardinals larger than ℵ1, with finite conditions. We use the two-type approach to give a new proof of th ..."
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Abstract. We present an approach to forcing with finite sequences of models that uses models of two types. This approach builds on earlier work of Friedman and Mitchell on forcing to add clubs in cardinals larger than ℵ1, with finite conditions. We use the two-type approach to give a new proof of the consistency of the proper forcing axiom. The new proof uses a finite support forcing, as opposed to the countable support iteration in the standard proof. The distinction is important since a proof using finite supports is more amenable to generalizations to cardinals greater than ℵ1.
Diagonal Prikry extensions
- J. Symbolic Logic
"... It is a well-known phenomenon in set theory that problems in infinite combinatorics involving singular cardinals and their successors tend to be harder than the parallel problems for regular cardinals. Examples include the behaviour of cardinal exponentiation, the extent ..."
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It is a well-known phenomenon in set theory that problems in infinite combinatorics involving singular cardinals and their successors tend to be harder than the parallel problems for regular cardinals. Examples include the behaviour of cardinal exponentiation, the extent
THE TREE PROPERTY UP TO ℵω+1
"... Abstract. Assuming ω supercompact cardinals we force to obtain a model where the tree property holds both at ℵω+1, and at ℵn for all 2 ≤ n < ω. A model with the former was obtainedby Magidor–Shelahfrom a huge cardinaland ω supercompactcardinals above it, and recently by Sinapova from ω supercompa ..."
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Abstract. Assuming ω supercompact cardinals we force to obtain a model where the tree property holds both at ℵω+1, and at ℵn for all 2 ≤ n < ω. A model with the former was obtainedby Magidor–Shelahfrom a huge cardinaland ω supercompactcardinals above it, and recently by Sinapova from ω supercompact cardinals. A model with the latter was obtained by Cummings–Foreman from ω supercompact cardinals. Our model, where the two hold simultaneously, is another step toward the goal of obtaining the tree property on increasingly large intervals of successor cardinals. MSC-2010: 03E35, 03E05, 03E55.
THE TREE PROPERTY AT ℵω+1
"... Abstract. We show that given ω many supercompact cardinals, there is a generic extension in which there are no Aronszajn trees at ℵω+1. This is an improvement of the large cardinal assumptions. The previous hypothesis was a huge cardinal and ω many supercompact cardinals above it, in Magidor-Shelah ..."
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Abstract. We show that given ω many supercompact cardinals, there is a generic extension in which there are no Aronszajn trees at ℵω+1. This is an improvement of the large cardinal assumptions. The previous hypothesis was a huge cardinal and ω many supercompact cardinals above it, in Magidor-Shelah [7]. 1.
The tree property and the failure of the singular cardinal hypothesis at ℵ ω 2
- J. Symbolic Logic
"... Abstract. We show that given ω many supercompact cardinals, there is a generic extension in which the tree property holds at ℵ ω 2 +1 and the SCH fails at ℵ ω 2. 1. ..."
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Abstract. We show that given ω many supercompact cardinals, there is a generic extension in which the tree property holds at ℵ ω 2 +1 and the SCH fails at ℵ ω 2. 1.
COMBINATORICS AT ℵω
"... Abstract. We construct a model in which the singular cardinal hypothesis fails at ℵω. We use characterizations of genericity to show the existence of a projection between different Prikry type forcings. 1. ..."
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Abstract. We construct a model in which the singular cardinal hypothesis fails at ℵω. We use characterizations of genericity to show the existence of a projection between different Prikry type forcings. 1.
SINGULAR CARDINALS: FROM HAUSDORFF’S GAPS TO SHELAH’S PCF THEORY
"... The mathematical subject of singular cardinals is young and many of the mathematicians who made important contributions to it are still active. This makes writing a history of singular cardinals a somewhat riskier mission than writing the history of, say, Babylonian arithmetic. Yet exactly the discu ..."
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The mathematical subject of singular cardinals is young and many of the mathematicians who made important contributions to it are still active. This makes writing a history of singular cardinals a somewhat riskier mission than writing the history of, say, Babylonian arithmetic. Yet exactly the discussions with some of the people who created the 20th century history of singular cardinals made the writing of this article fascinating. I am indebted to Moti Gitik, Ronald Jensen, István Juhász, Menachem Magidor and Saharon Shelah for the time and effort they spent on helping me understand the development of the subject and for many illuminations they provided. A lot of what I thought about the history of singular cardinals had to change as a result of these discussions. Special thanks are due to István Juhász, for his patient reading for me from the Russian text of Alexandrov and Urysohn’s Memoirs, to Salma Kuhlmann, who directed me to the definition of singular cardinals in Hausdorff’s writing, and to Stefan Geschke, who helped me with the German texts I needed to read and
DIAGONAL EXTENDER BASED PRIKRY FORCING
"... Abstract. We present a new forcing notion combining diagonal supercompact Prikry focing with interleaved extender based forcing. In the final model the singular cardinal hypothesis fails at κ and GCH holds below κ. Moreover we define a scale at κ, which has a stationary set of bad points in the grou ..."
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Abstract. We present a new forcing notion combining diagonal supercompact Prikry focing with interleaved extender based forcing. In the final model the singular cardinal hypothesis fails at κ and GCH holds below κ. Moreover we define a scale at κ, which has a stationary set of bad points in the ground model. 1.
THE TREE PROPERTY AND THE FAILURE OF SCH AT UNCOUNTABLE COFINALITY
"... Abstract. Given a regular cardinal λ and λ many supercompact cardinals, we describe a type of forcing such that in the generic extension there is a cardinal κ with cofinality λ, the Singular Cardinal Hypothesis at κ fails, and the tree property holds at κ +. 1. ..."
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Abstract. Given a regular cardinal λ and λ many supercompact cardinals, we describe a type of forcing such that in the generic extension there is a cardinal κ with cofinality λ, the Singular Cardinal Hypothesis at κ fails, and the tree property holds at κ +. 1.
