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**1 - 7**of**7**### Information and Computation 207 (2009) 258–283 Contents lists available at ScienceDirect

"... Information and Computation journal homepage: www.elsevier.com/locate/ic ..."

### SET THEORY FROM CANTOR TO COHEN

"... Set theory is an autonomous and sophisticated field of mathematics, enormously successful not only at its continuing development of its historical heritage but also at analyzing mathematical propositions and gauging their consistency strength. But set theory is also distinguished by having begun int ..."

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Set theory is an autonomous and sophisticated field of mathematics, enormously successful not only at its continuing development of its historical heritage but also at analyzing mathematical propositions and gauging their consistency strength. But set theory is also distinguished by having begun intertwined with pronounced metaphysical attitudes, and these have even been regarded as crucial by some of its great developers. This has encouraged the exaggeration of crises in foundations and of metaphysical doctrines in general. However, set theory has proceeded in the opposite direction, from a web of intensions to a theory of extension par excellence, and like other fields of mathematics its vitality and progress have depended on a steadily growing core of mathematical proofs and methods, problems and results. There is also the stronger contention that from the beginning set theory actually developed through a progression of mathematical moves, whatever and sometimes in spite of what has been claimed on its behalf. What follows is an account of the development of set theory from its beginnings through the creation of forcing based on these contentions, with an avowedly Whiggish emphasis on the heritage that has been retained and developed by the current theory. The

### THE EMPTY SET, THE SINGLETON, AND THE ORDERED PAIR AKIHIRO KANAMORI

, 2002

"... For the modern set theorist the empty set ∅, the singleton {a}, and the ordered pair 〈x, y 〉 are at the beginning of the systematic, axiomatic development of set theory, both as a field of mathematics and as a unifying framework for ongoing mathematics. These notions are the simplest building blocks ..."

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For the modern set theorist the empty set ∅, the singleton {a}, and the ordered pair 〈x, y 〉 are at the beginning of the systematic, axiomatic development of set theory, both as a field of mathematics and as a unifying framework for ongoing mathematics. These notions are the simplest building blocks in the abstract, generative conception of sets advanced by the initial axiomatization of Ernst Zermelo [1908a] and are quickly assimilated long before the complexities of Power Set, Replacement, and Choice are broached in the formal elaboration of the ‘set of ’ {} operation. So it is surprising that, while these notions are unproblematic today, they were once sources of considerable concern and confusion among leading pioneers of mathematical logic like Frege, Russell, Dedekind, and Peano. In the development of modern mathematical logic out of the turbulence of 19th century logic, the emergence of the empty set, the singleton, and the ordered pair as clear and elementary set-theoretic concepts serves as a motif that reflects and illuminates larger and more significant developments in mathematical logic:

### ZORN’S LEMMA

"... Zermelo gave a beautiful proof in [5] that every set can be well ordered, and Kneser adapted it to give a direct proof of Zorn’s lemma in [2]. Papers such as [3], [4], and most recently, [1], describe this proof, but it still doesn’t seem to be generally known by mathematicians. ..."

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Zermelo gave a beautiful proof in [5] that every set can be well ordered, and Kneser adapted it to give a direct proof of Zorn’s lemma in [2]. Papers such as [3], [4], and most recently, [1], describe this proof, but it still doesn’t seem to be generally known by mathematicians.

### Alloy: A New Object Modelling Notation

"... Alloy is a lightweight, precise and tractable notation for object modelling. It attempts to combine the practicality of UML's static structure notation with the rigour of Z, and to be expressive enough for most object modelling problems while remaining amenable to automatic analysis. Alloy has ..."

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Alloy is a lightweight, precise and tractable notation for object modelling. It attempts to combine the practicality of UML's static structure notation with the rigour of Z, and to be expressive enough for most object modelling problems while remaining amenable to automatic analysis. Alloy has a

### Dedication

, 2004

"... This project is dedicated to the late Richard D. Yoakam: mentor and inspiration for both of my careers. Acknowledgements I would not have survived the past five years of graduate school without the support and encouragement of my family, especially my parents, Mary Jo and Thomas Conway. I would like ..."

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This project is dedicated to the late Richard D. Yoakam: mentor and inspiration for both of my careers. Acknowledgements I would not have survived the past five years of graduate school without the support and encouragement of my family, especially my parents, Mary Jo and Thomas Conway. I would like to thank Don Heider, not only for his help on this project, but for his invaluable insight and assistance during my transition from the television world to the academic world. Not only did he pay for many lunches, he patiently listened to a lot of whining and bad ideas while always keeping me on track to realize this goal. I am indebted to Don Carleton for his insight into historical research as well as opening up for me the wonderful world of the Center for American History. I would like to thank the Center for American History for helping finance a trip to videotape oral history interviews as well as carry out research at different archives. The other members of this dissertation committee, Kris Wilson, James Tankard,

### Annals of the Japan Association for Philosophy of Science Vol.21 (2013) 21～35 21 Mathematical Knowledge: Motley and Complexity of Proof

"... Modern mathematics is, to my mind, a complex edifice based on conceptual constructions. The subject has undergone something like a biological evolution, an opportunistic one, to the point that the current subject matter, methods, and proce-dures would be patently unrecognizable a century, certainly ..."

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Modern mathematics is, to my mind, a complex edifice based on conceptual constructions. The subject has undergone something like a biological evolution, an opportunistic one, to the point that the current subject matter, methods, and proce-dures would be patently unrecognizable a century, certainly two centuries, ago. What has been called “classical mathematics ” has indeed seen its day. With its richness, variety, and complexity any discussion of the nature of modern mathematics cannot but accede to the primacy of its history and practice. As I see it, the applicability of mathematics may be a driving motivation, but in the end mathematics is au-tonomous. Mathematics is in a broad sense self-generating and self-authenticating, and alone competent to address issues of its correctness and authority. What brings us mathematical knowledge? The carriers of mathematical knowl-edge are proofs, more generally arguments and constructions, as embedded in larger contexts.1 Mathematicians and teachers of higher mathematics know this, but it should be said. Issues about competence and intuition can be raised as well as factors of knowledge involving the general dissemination of analogical or inductive reasoning