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A Model In Which GCH Holds At Successors But Fails At Limits
 Transactions of the American Mathematical Society
, 1992
"... . Starting with GCH and a P3hypermeasurable cardinal, a model is produced in which 2 = + if is a successor cardinal and 2 = ++ if is a limit cardinal. The proof uses a Reverse Easton extension followed by a modified Radin forcing. 1. INTRODUCTION 1. Historical remarks and statement o ..."
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. Starting with GCH and a P3hypermeasurable cardinal, a model is produced in which 2 = + if is a successor cardinal and 2 = ++ if is a limit cardinal. The proof uses a Reverse Easton extension followed by a modified Radin forcing. 1. INTRODUCTION 1. Historical remarks and statement of the main result. The continuum problem is an old one, dating back to Cantor and his statement of the Continuum Hypothesis in [Ca]. Put in a modern form which might have puzzled Cantor, the problem is to determine which behaviours of the continuum function 7\Gamma! 2 are consistent with ZFC. Throughout this paper ZFC will be the base set theory, though as we see below strong settheoretic hypotheses will play an essential role in the result. Before Godel progress on the continuum problem was made by the descriptive set theorists, who showed that certain easily definable sets of reals could not be counterexamples to CH. Godel [G] took the major step forward of showing that in a certain ...
The Mathematical Development Of Set Theory  From Cantor To Cohen
 The Bulletin of Symbolic Logic
, 1996
"... This article is dedicated to Professor Burton Dreben on his coming of age. I owe him particular thanks for his careful reading and numerous suggestions for improvement. My thanks go also to Jose Ruiz and the referee for their helpful comments. Parts of this account were given at the 1995 summer meet ..."
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This article is dedicated to Professor Burton Dreben on his coming of age. I owe him particular thanks for his careful reading and numerous suggestions for improvement. My thanks go also to Jose Ruiz and the referee for their helpful comments. Parts of this account were given at the 1995 summer meeting of the Association for Symbolic Logic at Haifa, in the Massachusetts Institute of Technology logic seminar, and to the Paris Logic Group. The author would like to express his thanks to the various organizers, as well as his gratitude to the Hebrew University of Jerusalem for its hospitality during the preparation of this article in the autumn of 1995.
Gap forcing: generalizing the Lévy–Solovay theorem
 Bull. Symb. Log
, 1999
"... Abstract. The LevySolovay Theorem [LevSol67] limits the kind of large cardinal embeddings that can exist in a small forcing extension. Here I announce a generalization of this theorem to a broad new class of forcing notions. One consequence is that many of the forcing iterations most commonly found ..."
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Abstract. The LevySolovay Theorem [LevSol67] limits the kind of large cardinal embeddings that can exist in a small forcing extension. Here I announce a generalization of this theorem to a broad new class of forcing notions. One consequence is that many of the forcing iterations most commonly found in the large cardinal literature create no new weakly compact cardinals, measurable cardinals, strong cardinals, Woodin cardinals, strongly compact cardinals, supercompact cardinals, almost huge cardinals, huge cardinals, and so on. Large cardinal set theorists today generally look upon small forcing—that is, forcing with a poset P of cardinality less than whatever large cardinal κ is under consideration—as benign. This outlook is largely due to the LevySolovay theorem [LevSol67], which asserts that small forcing does not affect the measurability of any cardinal. (Specifically, the theorem says that if a forcing notion P has size less than κ, then the ground model V and the forcing extension V P agree on the measurability of κ in a strong way: the ground model measures on κ all generate as filters measures in the forcing extension, the corresponding ultrapower embeddings lift uniquely from the ground model to the forcing extension and all the measures and ultrapower embeddings in the forcing extension arise in this way.) Since the LevySolovay argument generalizes to the other large cardinals whose existence is witnessed by certain kinds of measures or ultrapowers, such as strongly compact cardinals, supercompact cardinals, almost huge cardinals and so on, one is led to the broad conclusion that small forcing is harmless; one can understand the measures in a small forcing extension by their relation to the measures existing already in the ground model. Here in this Communication I would like to announce a generalization of the LevySolovay Theorem to a broad new class of forcing notions.
Singular Cardinals And The PCF Theory
 Bull. Symbolic Logic
, 1995
"... this paper we present a brief history of the singular cardinals problem, the present knowledge, and an introduction into Shelah's pcf theory. In Sections 2, 3 and 4 we introduce the reader to cardinal arithmetic and to the singular cardinals problems. Sections 5, 6, 7 and 8 describe the main results ..."
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this paper we present a brief history of the singular cardinals problem, the present knowledge, and an introduction into Shelah's pcf theory. In Sections 2, 3 and 4 we introduce the reader to cardinal arithmetic and to the singular cardinals problems. Sections 5, 6, 7 and 8 describe the main results and methods of the last 25 years and explain the role of large cardinals in the singular cardinals problem. In Section 9 we present an outline of the pcf theory.
Iterated Forcing and Elementary Embeddings
"... In this chapter we present a survey of the area of set theory in which iterated forcing interacts with elementary embeddings. The original plan was to concentrate on forcing constructions which preserve large cardinal axioms, particularly Reverse Easton iterations. However ..."
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In this chapter we present a survey of the area of set theory in which iterated forcing interacts with elementary embeddings. The original plan was to concentrate on forcing constructions which preserve large cardinal axioms, particularly Reverse Easton iterations. However
Handbook of Set Theory
"... this paper, but we state here a simplified version which generalizes the result of Dodd and Jensen by showing that a singular cardinal which is regular in K is made singular by a set which approximates a PrikryMagidor generic set (see chapter introduction, section 2.2 ) ..."
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this paper, but we state here a simplified version which generalizes the result of Dodd and Jensen by showing that a singular cardinal which is regular in K is made singular by a set which approximates a PrikryMagidor generic set (see chapter introduction, section 2.2 )
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