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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 ...
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
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.
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.
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 re ..."
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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.
COHEN AND SET THEORY
"... Abstract. We discuss the work of Paul Cohen in set theory and its influence, especially the background, discovery, development of forcing. Paul Joseph Cohen (1934–2007) in 1963 established the independence of the Axiom of Choice (AC) from ZF and the independence of the Continuum Hypothesis (CH) from ..."
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Abstract. We discuss the work of Paul Cohen in set theory and its influence, especially the background, discovery, development of forcing. Paul Joseph Cohen (1934–2007) in 1963 established the independence of the Axiom of Choice (AC) from ZF and the independence of the Continuum Hypothesis (CH) from ZFC. That is, he established that Con(ZF) implies Con(ZF+¬AC) and Con(ZFC) implies Con(ZFC+¬CH). Already prominent as an analyst, Cohen had ventured into set theory with fresh eyes and an openmindedness about possibilities. These results delimited ZF and ZFC in terms of the two fundamental issues at the beginnings of set theory. But beyond that, Cohen’s proofs were the inaugural examples of a new technique, forcing, which was to become a remarkably general and flexible method for extending models of set theory. Forcing has strong intuitive underpinnings and reinforces the notion of set as given by the firstorder ZF axioms with conspicuous uses of Replacement and Foundation. If Gödel’s construction of L had launched set theory as a distinctive field of mathematics, then Cohen’s forcing began its transformation into a modern, sophisticated one. The extent and breadth of the expansion of set theory henceforth dwarfed all that came before, both in terms of the numbers of people involved and the results established. With clear intimations of a new and concrete way of building models, set theorists rushed in and with forcing were soon establishing a cornucopia of relative consistency results, truths in a wider sense, with some illuminating classical problems of mathematics. Soon, ZFC became quite unlike Euclidean geometry and much like group theory, with a wide range of models of set theory being investigated for their own sake. Set theory had undergone a seachange, and with the subject so enriched, it is difficult to convey the strangeness of it. Received April 24, 2008. This is the full text of an invited address given at the annual meeting of the Association
INDESTRUCTIBILITY OF VOPĚNKA’S PRINCIPLE
"... Abstract. Vopěnka’s Principle is a natural large cardinal axiom that has recently found applications in category theory and algebraic topology. We show that Vopěnka’s Principle and Vopěnka cardinals are relatively consistent with a broad range of other principles known to be independent of stand ..."
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Abstract. Vopěnka’s Principle is a natural large cardinal axiom that has recently found applications in category theory and algebraic topology. We show that Vopěnka’s Principle and Vopěnka cardinals are relatively consistent with a broad range of other principles known to be independent of standard (ZFC) set theory, such as the Generalised Continuum Hypothesis, and the existence of a definable wellorder on the universe of all sets. We achieve this by showing that they are indestructible under a broad class of forcing constructions, specifically, reverse Easton iterations of increasingly directed closed partial orders. §1. Introduction. Vopěnka’s Principle is a large cardinal axiom that can readily be expressed in either set or categorytheoretic terms. It has received significant attention in the latter context, yielding structural results for certain important kinds of categories, as described in the final chapter of Adámek and Rosický’s book [1].