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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 :
Natural Logicism via the Logic of Orderly Pairing by
, 2008
"... Schumm, Timothy Smiley and Matthias Wille. Comments by two anonymous referees have also led to significant improvements. The aim here is to describe how to complete the constructive logicist program, in the author’s book AntiRealism and Logic, of deriving all the PeanoDedekind postulates for arith ..."
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Schumm, Timothy Smiley and Matthias Wille. Comments by two anonymous referees have also led to significant improvements. The aim here is to describe how to complete the constructive logicist program, in the author’s book AntiRealism and Logic, of deriving all the PeanoDedekind postulates for arithmetic within a theory of natural numbers that also accounts for their applicability in counting finite collections of objects. The axioms still to be derived are those for addition and multiplication. Frege did not derive them in a fully explicit, conceptually illuminating way. Nor has any neoFregean done so. These outstanding axioms need to be derived in a way fully in keeping with the spirit and the letter of Frege’s logicism and his doctrine of definition. To that end this study develops a logic, in the GentzenPrawitz style of natural deduction, for the operation of orderly pairing. The logic is an extension of free firstorder logic with identity. Orderly pairing is treated as a primitive. No notion of set is presupposed, nor any settheoretic notion of membership. The formation of ordered pairs, and the two projection operations yielding their left and right coordinates, form a coeval family of logical notions. The challenge is to furnish them with introduction and elimination rules that capture their exact meanings, and no more. Orderly pairing as a logical primitive is then used in order to introduce addition and multiplication in a conceptually satisfying way within a constructive logicist theory of the natural numbers. Because of its reliance, throughout, on senseconstituting rules of natural deduction, the completed account can be described as ‘natural logicism’. 2 1 Introduction: historical
BERNAYS AND SET THEORY
"... Abstract. We discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higherorder reflection principles. Paul Isaak Bernays (1888–1977) is an important figure in the development of mathematical logic, being the main bridge between Hilbert and Göd ..."
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Abstract. We discuss the work of Paul Bernays in set theory, mainly his axiomatization and his use of classes but also his higherorder reflection principles. Paul Isaak Bernays (1888–1977) is an important figure in the development of mathematical logic, being the main bridge between Hilbert and Gödel in the intermediate generation and making contributions in proof theory, set theory, and the philosophy of mathematics. Bernays is best known for the twovolume 1934,1939 Grundlagen der Mathematik [39, 40], written solely by him though Hilbert was retained as first author. Going into many reprintings and an eventual second edition thirty years later, this monumental work provided a magisterial exposition of the work of the Hilbert school in the formalization of firstorder logic and in proof theory and the work of Gödel on incompleteness and its surround, including the first complete proof of the Second Incompleteness Theorem. 1 Recent reevaluation of Bernays ’ role actually places him at the center of the development of mathematical logic and Hilbert’s program. 2 But starting in his forties, Bernays did his most individuated, distinctive mathematical work in set theory, providing a timely axiomatization and later applying higherorder reflection principles, and produced a stream of
Alternative Set Theories
, 2006
"... By “alternative set theories ” we mean systems of set theory differing significantly from the dominant ZF (ZermeloFrankel set theory) and its close relatives (though we will review these systems in the article). Among the systems we will review are typed theories of sets, Zermelo set theory and its ..."
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By “alternative set theories ” we mean systems of set theory differing significantly from the dominant ZF (ZermeloFrankel set theory) and its close relatives (though we will review these systems in the article). Among the systems we will review are typed theories of sets, Zermelo set theory and its variations, New Foundations and related systems, positive set theories, and constructive set theories. An interest in the range of alternative set theories does not presuppose an interest in replacing the dominant set theory with one of the alternatives; acquainting ourselves with foundations of mathematics formulated in terms of an alternative system can be instructive as showing us what any set theory (including the usual one) is supposed to do for us. The study of alternative set theories can dispel a facile identification of “set theory ” with “ZermeloFraenkel set theory”; they are not the same thing. Contents 1 Why set theory? 2 1.1 The Dedekind construction of the reals............... 3 1.2 The FregeRussell definition of the natural numbers....... 4
Some formal considerations on the generation of hierarchically structured expressions 1
"... In this note we define a machine that generates nests. The basic relations commonly attributed to linguistic expressions in configurational syntactic models as well as the device of chains postulated in current transformational grammar to represent distance relations can be naturally derived from th ..."
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In this note we define a machine that generates nests. The basic relations commonly attributed to linguistic expressions in configurational syntactic models as well as the device of chains postulated in current transformational grammar to represent distance relations can be naturally derived from the assumption that the combinatorial syntactic procedure is a nesting machine. Accordingly, the core of the transformational generative syntactic theory of language can be solidly constructed on the basis of nests, in the same terms as the general theory of order, an important methodological step that provides a rigorization of Chomsky’s minimalist intuition that the simplest way to generate hierarchically organized linguistic expressions is by postulating a combinatorial operation called Merge, which can be internal or external. Importantly, there is reason to think that nests are a useful representative tool in other domains besides language where either some recursive algorithm or evolutionary process is at work, which suggests the unifying force of the mathematical abstraction this note is based on.
On Multidominance and Linearization Mark
"... This article centers around two questions: What is the relation between movement and structure sharing, and how can complex syntactic structures be linearized? It is shown that regular movement involves internal remerge, and sharing or ‘sideward movement ’ external remerge. Without ad hoc restrictio ..."
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This article centers around two questions: What is the relation between movement and structure sharing, and how can complex syntactic structures be linearized? It is shown that regular movement involves internal remerge, and sharing or ‘sideward movement ’ external remerge. Without ad hoc restrictions on the input, both options follow from Merge. They can be represented in terms of multidominance. Although more structural freedom ensues than standardly thought, the grammar is not completely unconstrained: Arguably, proliferation of roots is prohibited. Furthermore, it is explained why external remerge has somewhat different consequences than internal remerge. For instance, apparent nonlocal behavior is attested. At the PF interface, the linearization of structures involving remerge is nontrivial. A central problem is identified, apart from the general issue why remerged material is only pronounced once: There are seemingly contradictory linearization demands for internal and external remerge. This can be resolved by taking into account the different structural configurations. It is argued that the linearization is a PF procedure involving a recursive structure scanning algorithm that makes use of the inherent asymmetry between sister nodes imposed by the operation of Merge. Keywords: linearization; movement; multidominance; PF interface; (re) merge
Three Kinds of Meanings
"... Abstract: It is argued that there are at least three distinct kinds of meaning that have wide currency across many different kinds of language use. The first kind consists of formal definitions of terms in mathematics and science. These definitions are usually clearly distinguished, as such, in the ..."
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Abstract: It is argued that there are at least three distinct kinds of meaning that have wide currency across many different kinds of language use. The first kind consists of formal definitions of terms in mathematics and science. These definitions are usually clearly distinguished, as such, in the discourse context in which they occur. The second kind consists of dictionary definitions, familiar to all of us. The third kind, that of associative meanings, is not as widely recognized as the first two, but associative meanings are at the center of our cognitive and emotional experience. Baldly stated, the thesis defended is that associations provide the computational method of computing meaning as we speak, listen, read or write about'iour thoughts and feelings. This claim is supported by a variety of research in psychology and neuroscience. For much of the use of this third kind of meaning, the familiar analyticsynthetic philosophical distinction is artificial and awkward. 1. Meaning given by formal definition I first consider definitions formalized within a theory in the ordinary
Internal and External Remerge On Movement, Multidominance, and the Linearization of Syntactic Objects *
"... Abstract. This article centers around two questions: (i) what is the relation between movement and structure sharing?, and (ii) how can complex syntactic structures be linearized? It is shown that regular movement involves internal remerge, and sharing or ‘sideward movement ’ external remerge. Witho ..."
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Abstract. This article centers around two questions: (i) what is the relation between movement and structure sharing?, and (ii) how can complex syntactic structures be linearized? It is shown that regular movement involves internal remerge, and sharing or ‘sideward movement ’ external remerge. Without ad hoc restrictions on the input, both options follow from Merge. They can be represented in terms of multidominance. Although more structural freedom ensues than standardly thought, the grammar is not completely unconstrained: arguably, proliferation of roots is prohibited. Furthermore, it is explained why external remerge has somewhat different consequences than internal remerge. For instance, apparent nonlocal behavior is attested. At the PF interface, the linearization of structures involving remerge is nontrivial. A central problem is identified, apart from the general issue why remerged material is only pronounced once: there are seemingly contradictory linearization demands for internal and external remerge. This can be resolved by taking into account the different structural configurations. It is argued that the linearization is a PF procedure involving a recursive structure scanning algorithm that makes use of the inherent asymmetry between sister nodes imposed by the operation of Merge.