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Types in logic and mathematics before 1940
 Bulletin of Symbolic Logic
, 2002
"... Abstract. In this article, we study the prehistory of type theory up to 1910 and its development between Russell and Whitehead’s Principia Mathematica ([71], 1910–1912) and Church’s simply typed λcalculus of 1940. We first argue that the concept of types has always been present in mathematics, thou ..."
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Cited by 11 (6 self)
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Abstract. In this article, we study the prehistory of type theory up to 1910 and its development between Russell and Whitehead’s Principia Mathematica ([71], 1910–1912) and Church’s simply typed λcalculus of 1940. We first argue that the concept of types has always been present in mathematics, though nobody was incorporating them explicitly as such, before the end of the 19th century. Then we proceed by describing how the logical paradoxes entered the formal systems of Frege, Cantor and Peano concentrating on Frege’s Grundgesetze der Arithmetik for which Russell applied his famous paradox 1 and this led him to introduce the first theory of types, the Ramified Type Theory (rtt). We present rtt formally using the modern notation for type theory and we discuss how Ramsey, Hilbert and Ackermann removed the orders from rtt leading to the simple theory of types stt. We present stt and Church’s own simply typed λcalculus (λ→C 2) and we finish by comparing rtt, stt and λ→C. §1. Introduction. Nowadays, type theory has many applications and is used in many different disciplines. Even within logic and mathematics, there are many different type systems. They serve several purposes, and are formulated in various ways. But, before 1903 when Russell first introduced
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 :
RECURSION AND THE AXIOM OF INFINITY
, 2008
"... Abstract. This paper examines the completion of an ωordered sequence of recursive definition which on the one hand defines an increasing sequence of nested set and on the other redefines successively a numeric variable as the cardinal of the successively defined nested sets. The consequence is a co ..."
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Abstract. This paper examines the completion of an ωordered sequence of recursive definition which on the one hand defines an increasing sequence of nested set and on the other redefines successively a numeric variable as the cardinal of the successively defined nested sets. The consequence is a contradiction involving the consistency of ωorder and then that of the Axiom of Infinity. 1. Recursion and successiveness A recursive definition usually starts with a first definition (basic clause) which is followed by an infinite (usually ωordered) sequence of definitions such that each one of them defines an object in terms of the previously defined ones (inductive or recursive clause). For instance, if A = {a1, a2, a3,...} is an ωordered set, the following recursive definition: A1 = {a1} Basic clause (1) Ai+1 = Ai ∪ {ai+1}; i = 1, 2, 3,... recursive clause (2)
A1 = {a1} Basic clause (1)
, 2006
"... Abstract. This paper examines the recursive definition of an increasing sequence of nested sets by means of a testing set whose countably many successive redefinitions as the successive sets of the sequence, leads to some contradictory results involving the Axiom of Infinity. 1. Recursion and succes ..."
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Abstract. This paper examines the recursive definition of an increasing sequence of nested sets by means of a testing set whose countably many successive redefinitions as the successive sets of the sequence, leads to some contradictory results involving the Axiom of Infinity. 1. Recursion and successiveness As is well known, an increasing ωordered sequence of nested sets is one in which each set has both an immediate successor, of which it is a proper subset, and an immediate predecessor (except the first one). A common way of defining this type of sequences is by recursion. For instance, if N is the set of natural numbers, and A = {a1, a2, a3,...} and B = {b1, b2, b3,...} are two ωordered sets, the following recursive definition:
CORRECTING A MINOR ERROR IN CANTOR’S CALCULATION OF THE POWER OF THE CONTINUUM
, 2006
"... Abstract. Cantor’s algebraic calculation of the power of the continuum contains an easily repairable error related to Cantor own way of defining the addition of cardinal numbers. The appropriate correction is suggested. 1. The exponentiation of powers Cantor’s most significant contribution to the th ..."
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Abstract. Cantor’s algebraic calculation of the power of the continuum contains an easily repairable error related to Cantor own way of defining the addition of cardinal numbers. The appropriate correction is suggested. 1. The exponentiation of powers Cantor’s most significant contribution to the theory of transfinite numbers is, without a doubt, Beiträge zur Begründung der transfiniten Mengelehre 1. A memory of more than 70 pages divided into two parts which appeared in the Mathematische Annalen in the years 1895 and 1897 respectively ([4], [5]). Beiträge’s first six epigraphs are devoted to found the arithmetics of cardinals. Cantor begins by defining the concept of set and the union of disjoint sets, after which he proposes the following definition of power or cardinal number ([6], p. 86): We call by the name ”power ” or ”cardinal number ” of [the set] M the general concept which, by means of our active faculty of thought, arises from the set M when we make abstraction of the nature of its various elements m and of the order in which they are give.
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.
RECURSION AND ωORDER
, 804
"... Abstract. This paper examines the recursive definition of an increasing sequence of nested sets by means of a control set whose countably many successive redefinitions leads to a contradictory result that compromises ωorder and the Axiom of Infinity. 1. Recursion and successiveness A recursive defi ..."
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Abstract. This paper examines the recursive definition of an increasing sequence of nested sets by means of a control set whose countably many successive redefinitions leads to a contradictory result that compromises ωorder and the Axiom of Infinity. 1. Recursion and successiveness A recursive definition usually starts with a first definition (basic clause) which is followed by an infinite (usually ωordered) sequence of definitions such that each one of them defines an object in terms of the previously defined ones (inductive or recursive clause). For instance, if A = {a1, a2, a3,...} is an ωordered set, the following recursive definition: A1 = {a1} Basic clause (1) Ai+1 = Ai ∪ {ai+1}; i = 1, 2, 3,... Recursive clause (2) defines an ωordered increasing sequence 〈Ai〉i∈N of nested sets A1 ⊂ A2 ⊂ A3 ⊂... Recursive definitions as (1)(2) imply (mathematical)
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.