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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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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.
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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Cited by 5 (1 self)
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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 :
TRANSFINITE PARTITIONS OF JORDAN CURVES
, 2006
"... Abstract. The ωasymmetry induced by transfinite partitions makes it impossible for Jordan curves to have an infinite length. 1. ωasymmetry As we known from the XVIII century, ωpartitions (as we call them nowadays) of finite line segments are only possible if the successive adjacent parts of the ω ..."
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Abstract. The ωasymmetry induced by transfinite partitions makes it impossible for Jordan curves to have an infinite length. 1. ωasymmetry As we known from the XVIII century, ωpartitions (as we call them nowadays) of finite line segments are only possible if the successive adjacent parts of the ωpartition are of a decreasing length. This inevitable restriction induces a huge asymmetry in the very partition. In fact, whatever be the length of the ωpartitioned line segment and whatever be the ωpartition, all its parts, except a finite number of them, will necessarily lie within an arbitrarily small final segment. For the sake of illustration, consider an ωpartition of a 10 30 light years length segmentthe assumed diameter of the universe. Whatever be the ωpartition of this enormous line segment all its infinitely many parts, except a finite number of them, will inevitably lie within a final segment inconceivable less than, for instance, Planck length ( ∼ 10 −33 cm). There is no way of performing a more equitable partition if the partition has to be ωordered. Thus, ωpartitions are ωasymmetrical. For the same reason it is impossible to consider two proper points in the real line R separated by an infinite euclidean distance, in spite of the assumed infiniteness of the real line. The above simply unaesthetic consequences of ωpartitions become a little more controversial if the partitioned object is a closed line as a Jordan curve. The objective of the following short discussion is just to examine one of those consequences. Figure 1. A cosmic ωasymmetry. The ωorder makes it impossible a more equitable distribution of the available space. 1 2 Transfinite partitions of Jordan Curves 2. Transfinite partitions of Jordan Curves Let f(x) be a real valued function whose graph is a Jordan Curve J in the euclidean plane R2. If a and b are any two J’s points, we will write L(a, b) to denote the length of the J’s arc ãb whose endpoints are a and b. That is to say: b √ L(a, b) = 1 + (f(x)′) 2dx (1)
EXTENDING CANTOR’S PARADOX A CRITIQUE OF INFINITY AND SELFREFERENCE
, 809
"... Abstract. This paper examines infinity and selfreference from a critique perspective. Starting from an extension of Cantor Paradox that suggests the inconsistency of the actual infinite, the paper makes a short review of the controversial history of infinity and suggests several indicators of its i ..."
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Abstract. This paper examines infinity and selfreference from a critique perspective. Starting from an extension of Cantor Paradox that suggests the inconsistency of the actual infinite, the paper makes a short review of the controversial history of infinity and suggests several indicators of its inconsistency. Semantic selfreference is also examined from the same critique perspective by comparing it with selfreferent sets. The platonic scenario of infinity and selfreference is finally criticized from a biological and neurobiological perspective. 1.
THE ALEPHZERO OR ZERO DICHOTOMY (New and extended version with new arguments)
, 804
"... Abstract. This paper proves the existence of a dichotomy which being formally derived from the topological successiveness of ω ∗order leads to the same absurdity of Zeno’s Dichotomy II. It also derives a contradictory result from the first Zeno’s Dichotomy. ..."
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Abstract. This paper proves the existence of a dichotomy which being formally derived from the topological successiveness of ω ∗order leads to the same absurdity of Zeno’s Dichotomy II. It also derives a contradictory result from the first Zeno’s Dichotomy.