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Foundational and mathematical uses of higher types
 REFLECTIONS ON THE FOUNDATIONS OF MATHEMATICS: ESSAY IN HONOR OF SOLOMON FEFERMAN
, 1999
"... In this paper we develop mathematically strong systems of analysis in higher types which, nevertheless, are prooftheoretically weak, i.e. conservative over elementary resp. primitive recursive arithmetic. These systems are based on noncollapsing hierarchies ( n WKL+ ; n WKL+ ) of principles ..."
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In this paper we develop mathematically strong systems of analysis in higher types which, nevertheless, are prooftheoretically weak, i.e. conservative over elementary resp. primitive recursive arithmetic. These systems are based on noncollapsing hierarchies ( n WKL+ ; n WKL+ ) of principles which generalize (and for n = 0 coincide with) the socalled `weak' König's lemma WKL (which has been studied extensively in the context of second order arithmetic) to logically more complex tree predicates. Whereas the second order context used in the program of reverse mathematics requires an encoding of higher analytical concepts like continuous functions F : X ! Y between Polish spaces X;Y , the more exible language of our systems allows to treat such objects directly. This is of relevance as the encoding of F used in reverse mathematics tacitly yields a constructively enriched notion of continuous functions which e.g. for F : IN ! IN can be seen (in our higher order context)
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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Cited by 8 (2 self)
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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.
Graph Coloring Compactness Theorems Equivalent to BPI
 BPI, Scientiae Mathematicae Japonicae
, 2002
"... We introduce compactness theorems for generalized colorings and derive several particular compactness theorems from them. It is proved that the theorems and many of their consequences are equivalent in ZF set theory to BPI, the Prime Ideal Theorem for Boolean algebras. ..."
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We introduce compactness theorems for generalized colorings and derive several particular compactness theorems from them. It is proved that the theorems and many of their consequences are equivalent in ZF set theory to BPI, the Prime Ideal Theorem for Boolean algebras.
Gfree Colorability and the Boolean Prime Ideal Theorem
 Scientiae Mathematicae Japonicae
, 2003
"... We show that the compactness of Gfree kcolorability is equivalent to the Boolean prime ideal theorem for any graph G with more than two vertices and any k 2.. ..."
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We show that the compactness of Gfree kcolorability is equivalent to the Boolean prime ideal theorem for any graph G with more than two vertices and any k 2..
COMPACTNESS IN COUNTABLE TYCHONOFF PRODUCTS AND CHOICE
, 1999
"... Abstract. We study the relationship between the countable axiom of choice and the Tychonoff product theorem for countable families of topological spaces. ..."
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Abstract. We study the relationship between the countable axiom of choice and the Tychonoff product theorem for countable families of topological spaces.
The Independence of the Prime Ideal Theorem from the OrderExtension Principle
"... It is shown that the boolean prime ideal theorem BPIT : every boolean algebra has a prime ideal, does not follow from the orderextension principle OE: every partial ordering can be extended to a linear ordering. The proof uses a FraenkelMostowski model, where the family of atoms is indexed by a ..."
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It is shown that the boolean prime ideal theorem BPIT : every boolean algebra has a prime ideal, does not follow from the orderextension principle OE: every partial ordering can be extended to a linear ordering. The proof uses a FraenkelMostowski model, where the family of atoms is indexed by a countable universalhomogeneous boolean algebra whose boolean partial ordering has a `generic' extension to a linear ordering. To illustrate the technique for proving that the orderextension principle holds in the model we also study Mostowski's ordered model, and give a direct verification of OE there. The key technical point needed to verify OE in each case is the existence of a support structure. 1 Introduction The orderextension principle OE states that any partial ordering can be extended to a linear (total) ordering. Here by an extension of a partial ordering (X; ) we just mean a linear ordering ¯ of X such that (8x; y 2 X)(x y ) x ¯ y). A straightforward application of Zorn's Le...
ZORN’S LEMMA AND SOME APPLICATIONS
"... Zorn’s lemma is a result in set theory which appears in proofs of some nonconstructive existence theorems throughout mathematics. We will state Zorn’s lemma below and use it in later sections to prove some results in linear algebra, ring theory, and group theory. In an appendix, we will give an app ..."
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Zorn’s lemma is a result in set theory which appears in proofs of some nonconstructive existence theorems throughout mathematics. We will state Zorn’s lemma below and use it in later sections to prove some results in linear algebra, ring theory, and group theory. In an appendix, we will give an application to metric spaces. The statement of Zorn’s lemma
ISSN: 09316558A REPRESENTATIVE INDIVIDUAL FROM ARROVIAN AGGREGATION OF PARAMETRIC INDIVIDUAL UTILITIES
, 2009
"... ABSTRACT. This article investigates the representativeagent hypothesis for a population which faces a collective choice from a given finitedimensional space of alternatives. Each individual’s preference ordering is assumed to admit a utility representaion through an element of an exogenously given ..."
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ABSTRACT. This article investigates the representativeagent hypothesis for a population which faces a collective choice from a given finitedimensional space of alternatives. Each individual’s preference ordering is assumed to admit a utility representaion through an element of an exogenously given set of admissible utility functions. In addition, we assume that (i) the class of admissible utility functions allows for a smooth parametrization and only consists of strictly concave functions, (ii) the population is infinite, and (iii) the social welfare function satisfies Arrovian rationality axioms. We prove that there exists an admissible utility function r, called representative utility function, such that any alternative which maximizes r also maximizes the social welfare function. Given the structural similarities among the admissible utility functions (due to parametrization), we argue that the representative utility function can be interpreted as belonging to an — actual or invisible — individual. The existence proof for the representative utility function utilizes a special nonstandard model of the reals, viz. the ultrapower of the reals with respect to the ultrafilter of decisive coalitions; this construction explicitly determines the parameter vector of the representative utility function.
, such that whenever n k CkXk = ∑
"... Abstract. We shall prove a version of Gauß’s Lemma that recursively constructs ..."
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Abstract. We shall prove a version of Gauß’s Lemma that recursively constructs