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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.
The Mathematical Import Of Zermelo's WellOrdering Theorem
 Bull. Symbolic Logic
, 1997
"... this paper, the seminal results of set theory are woven together in terms of a unifying mathematical motif, one whose transmutations serve to illuminate the historical development of the subject. The motif is foreshadowed in Cantor's diagonal proof, and emerges in the interstices of the inclusion vs ..."
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this paper, the seminal results of set theory are woven together in terms of a unifying mathematical motif, one whose transmutations serve to illuminate the historical development of the subject. The motif is foreshadowed in Cantor's diagonal proof, and emerges in the interstices of the inclusion vs. membership distinction, a distinction only clarified at the turn of this century, remarkable though this may seem. Russell runs with this distinction, but is quickly caught on the horns of his wellknown paradox, an early expression of our motif. The motif becomes fully manifest through the study of functions f :
History of Valuation Theory  Part I
"... The theory of valuations was started in 1912 by the Hungarian mathematician Josef Kursch'ak who formulated the valuation axioms as we are used today. The main motivation was to provide a solid foundation for the theory of padic fields as defined by Kurt Hensel. In the following decades we can o ..."
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The theory of valuations was started in 1912 by the Hungarian mathematician Josef Kursch'ak who formulated the valuation axioms as we are used today. The main motivation was to provide a solid foundation for the theory of padic fields as defined by Kurt Hensel. In the following decades we can observe a quick development of valuation theory, triggered mainly by the discovery that much of algebraic number theory could be better understood by using valuation theoretic notions and methods. An outstanding figure in this development was Helmut Hasse. Independent of the application to number theory, there were essential contributions to valuation theory given by Alexander Ostrowski, published 1934. About the same time Wolfgang Krull gave a more general, universal definition of valuation which turned out to be applicable also in many other mathematical disciplines such as algebraic geometry or functional analysis, thus opening a new era of valuation theory.
On a Class of TimeVarying Behaviors
"... We study a class of timevarying systems that we encounter when we look into decomposition of behaviors. This class is the set of behaviors that are themselves of polynomials in time, with coe#cients as timeinvariant behaviors. Operators that have such behaviors as their kernels are studied. It tur ..."
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We study a class of timevarying systems that we encounter when we look into decomposition of behaviors. This class is the set of behaviors that are themselves of polynomials in time, with coe#cients as timeinvariant behaviors. Operators that have such behaviors as their kernels are studied. It turns out that autonomous behaviors allow kernel representations of this kind. We are led to the study of skew polynomial ring as the underlying ring for such operators. 1
Looking back … Max Zorn: World Renowned Mathematician and Member of the Indiana MAA Section
"... undergraduate math majors, at some point in their studies learn the importance of the axiom of choice and some of its equivalent versions. The author of this note was a graduate student studying topology in the 1970’s, when the current “mathematical humor’ ’ in my group involved riddles like the fol ..."
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undergraduate math majors, at some point in their studies learn the importance of the axiom of choice and some of its equivalent versions. The author of this note was a graduate student studying topology in the 1970’s, when the current “mathematical humor’ ’ in my group involved riddles like the following: Q: “What’s sour and yellow and equivalent to the axiom of choice?” A: “Zorn’s lemon!” While I knew that “Zorn ” was the name of a mathematician who proved an equivalent version of the Axiom of Choice, I didn’t know much aboutthe man Max Zorn until the late 1980s when a research colleague told me that Max was a professor at Indiana University, not far from RoseHulman, the school I happened to be joining as a faculty member. I never had a chance to meet Max, but during my first few years in Indiana it was a pleasure to learn more about this man, his interesting personality, and his mathematics. Max August Zorn was born in Germany on June 6, 1906. He attended Hamburg University, from which he received his Ph.D. in 1930; his thesis advisor was Emil Artin. After a few years at the University of Halle, Max and his family left Germany, and they moved to the United States in 1933. [1] It was as a Fellow at Yale University (1934 – 1936) that Max published “A Remark on Method in Transfinite Algebra ” [2], the article in which he presented the result that would become known as “Zorn’s
The Mathematical Infinite as a Matter of Method
, 2010
"... Abstract. I address the historical emergence of the mathematical infinite, and how we are to take the infinite in and out of mathematics. The thesis is that the mathematical infinite in mathematics is a matter of method. The infinite, of course, is a large topic. At the outset, one can historically ..."
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Abstract. I address the historical emergence of the mathematical infinite, and how we are to take the infinite in and out of mathematics. The thesis is that the mathematical infinite in mathematics is a matter of method. The infinite, of course, is a large topic. At the outset, one can historically discern two overlapping clusters of concepts: (1) wholeness, completeness, universality, absoluteness. (2) endlessness, boundlessness, indivisibility, continuousness. The first, the metaphysical infinite, I shall set aside. It is the second, the mathematical infinite, that I will address. Furthermore, I will address mathematical infinite by considering its historical emergence in set theory and how we are to take it in and out of mathematics. Insofar as physics and, more broadly, science deals with the mathematical infinite through mathematical language and techniques, my remarks should be subsuming and consequent. The main underlying point is that how the mathematical infinite is approached, assimilated, and applied in mathematics is not a matter of “ontological commitment”, of coming to terms with whatever that might mean, but rather of epistemological articulation, of coming to terms through knowledge. The mathematical infinite in mathematics is a matter of method. How we deal with the specific individual issues involving the infinite turns on the narrative we present about how it fits into methodological mathematical frameworks established and being established. The first section discusses the mathematical infinite in historical context, and the second, set theory and the emergence of the mathematical infinite. The third section discusses the infinite in and out of mathematics, and how it is to be taken. §1. The Infinite in Mathematics What role does the infinite play in modern mathematics? In modern mathematics, infinite sets abound both in the workings of proofs and as subject matter in statements, and so do universal statements, often of ∀ ∃ “for all there exists” form, which are indicative of direct engagement with the infinite. In many ways the role of the infinite is importantly “secondorder ” in the sense that Frege regarded number generally, in that the concepts of modern mathematics are understood as having infinite instances over a broad range. 1 But
Nullstellen and Subdirect Representation
"... David Hilbert’s solvability criterion for polynomial systems in n variables from the 1890s was linked by Emmy Noether in the 1920s to the decomposition of ideals in commutative rings, which in turn led Garret Birkhoff in the 1940s to his subdirect representation theorem for general algebras. The Hil ..."
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David Hilbert’s solvability criterion for polynomial systems in n variables from the 1890s was linked by Emmy Noether in the 1920s to the decomposition of ideals in commutative rings, which in turn led Garret Birkhoff in the 1940s to his subdirect representation theorem for general algebras. The HilbertNoetherBirkhoff linkage was brought to light in the late 1990s in talks by Bill Lawvere. The aim of this article is to analyze this linkage in the most elementary terms and then, based on our work of the 1980s, to present a general categorical framework for Birkhoff’s theorem.
iii Contents Preface
, 2011
"... These are the notes based on the course on Ramsey Theory taught at Universität Hamburg in Summer 2011. The lecture was based on the textbook “Ramsey theory ” of Graham, Rothschild, and Spencer [44]. In fact, large part of the material is taken from that book. ..."
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These are the notes based on the course on Ramsey Theory taught at Universität Hamburg in Summer 2011. The lecture was based on the textbook “Ramsey theory ” of Graham, Rothschild, and Spencer [44]. In fact, large part of the material is taken from that book.
ON THE FOUNDATIONS OF MATHEMATICAL ECONOMICS
, 2010
"... Kumaraswamy Vela Velupillai [74] presents a constructivist perspective on the foundations of mathematical economics, praising the views of Feynman in developing path integrals and Dirac in developing the delta function. He sees their approach as consistent with the Bishop constructive mathematics an ..."
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Kumaraswamy Vela Velupillai [74] presents a constructivist perspective on the foundations of mathematical economics, praising the views of Feynman in developing path integrals and Dirac in developing the delta function. He sees their approach as consistent with the Bishop constructive mathematics and considers its view on the BolzanoWeierstrass, HahnBanach, and intermediate value theorems, and then the implications of these arguments for such “crown jewels ” of mathematical economics as the existence of general equilibrium and the second welfare theorem. He also relates these ideas to the weakening of certain assumptions to allow for more general results as shown by Rosser [51] in his extension of Gödel’s incompleteness theorem in his opening section. This paper considers these arguments in reverse order, moving from the matters of economics applications to the broader issue of constructivist mathematics, concluding by considering the views of Rosser on these matters, drawing both on his writings and on personal conversations with him. Acknowledgements: I thank K. Vela Velupillai most particularly for his efforts to push me to consider these matters in the most serious manner, as well as my late father, J. Barkley Rosser [Sr.] and also his friend, the late Stephen C. Kleene, for their personal remarks on these matters to me over a long period of time. I also wish to thank Eric Bach, Ken Binmore, Herb Gintis, Jerome Keisler, Roger Koppl, David Levy, and Adrian Mathias for useful comments. The usual caveat holds. I also wish to dedicate this to K. Vela Velupillai who inspired it with his insistence that I finally deal with the work and thought of my father, J. Barkley Rosser [Sr.], as well as ShuHeng Chen, who supported him in this insistence. I thank both of them for this.