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The Iterative Conception of Set
, 2009
"... 2 The TwoConstructor case 5 2.1 Set Equality in the TwoConstructor Case............. 6 3 More Wands 9 ..."
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2 The TwoConstructor case 5 2.1 Set Equality in the TwoConstructor Case............. 6 3 More Wands 9
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 inclusi ..."
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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 :
From bounded arithmetic to second order arithmetic via automorphisms
 Logic in Tehran, Lect. Notes Log
"... Abstract. In this paper we examine the relationship between automorphisms of models of I∆0 (bounded arithmetic) and strong systems of arithmetic, such as P A, ACA0 (arithmetical comprehension schema with restricted induction), and Z2 (second order arithmetic). For example, we establish the following ..."
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Abstract. In this paper we examine the relationship between automorphisms of models of I∆0 (bounded arithmetic) and strong systems of arithmetic, such as P A, ACA0 (arithmetical comprehension schema with restricted induction), and Z2 (second order arithmetic). For example, we establish the following characterization of P A by proving a “reversal ” of a theorem of Gaifman: Theorem. The following are equivalent for completions T of I∆0: (a) T ⊢ P A; (b) Some model M = (M, · · ·) of T has a proper end extension N which satisfies I∆0 and for some automorphism j of N, M is precisely the fixed point set of j. Our results also shed light on the metamathematics of the QuineJensen system NF U of set theory with a universal set. 1.
On Modal µCalculus and NonWellFounded Set Theory
"... A finitary characterization for nonwellfounded sets with finite transitive closure is established in terms of modal µcalculus. This result generalizes the standard approach in the literature where a finitary characterization is only provided for wellfounded sets with finite transitive closure ..."
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A finitary characterization for nonwellfounded sets with finite transitive closure is established in terms of modal µcalculus. This result generalizes the standard approach in the literature where a finitary characterization is only provided for wellfounded sets with finite transitive closure. The proof relies on the concept of automaton, leading then to new interlinks between automata theory and nonwellfounded sets.
Enriched stratified systems for the foundations of category
"... This is the fourth in a series of intermittent papers on the foundations of category theory stretching back over more than thirtyfive years. The first three were “Settheoretical foundations of category theory ” [1969], “Categorical foundations and foundations of category theory ” [1977], and much ..."
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This is the fourth in a series of intermittent papers on the foundations of category theory stretching back over more than thirtyfive years. The first three were “Settheoretical foundations of category theory ” [1969], “Categorical foundations and foundations of category theory ” [1977], and much more recently, “Typical ambiguity: Trying to have
Set Theory
, 2010
"... Set Theory deals with the fundamental concepts of sets and functions used everywhere in mathematics. Cantor initiated the study of set theory with his investigations on the cardinality of sets of real numbers. In particular, he proved that there are different infinite cardinalities: the quantity of ..."
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Set Theory deals with the fundamental concepts of sets and functions used everywhere in mathematics. Cantor initiated the study of set theory with his investigations on the cardinality of sets of real numbers. In particular, he proved that there are different infinite cardinalities: the quantity of natural numbers is strictly smaller than the quantity of real numbers. Cantor formalized and studied the notions of ordinal and cardinal numbers. Set theory considers a universe of sets which is ordered by the membership or element relation ∈. All other mathematical objects are coded into this universe and studied within this framework. In this way, set theory is one of the foundations of mathematics. This text contains all information relevant for the exams. Furthermore, the exercises in this text are those which will be demonstrated in the tutorials. Each sheet of exercises contains some important ones marked with a star and some other ones. You have to hand in an exercise marked with a star in Weeks 3 to 6, Weeks 7 to 9 and Weeks 10 to 12; each of them gives one mark. Furthermore, you can hand in any further exercises, but they are only checked for correctness. There will be two mid term exams and a final exam; the mid term exams count 15 marks each and the final exam counts 67 marks.
BADIOU AND THE CONSEQUENCES OF FORMALISM
"... ABSTRACT: I consider the relationship of Badiou’s schematism of the event to critical thought following the linguistic turn as well as to the mathematical formalisms of set theory. In Being and Event, Badiou uses formal argumentation to support his sweeping rejection of the linguistic turn as well a ..."
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ABSTRACT: I consider the relationship of Badiou’s schematism of the event to critical thought following the linguistic turn as well as to the mathematical formalisms of set theory. In Being and Event, Badiou uses formal argumentation to support his sweeping rejection of the linguistic turn as well as much of contemporary critical thought. This rejection stems from his interpretation of set theory as barring thought from the 'OneAll ' of totality; but I argue that, by interpreting it differently, we can understand this implication in a way that is in fact consistent with the critical and linguistic methods Badiou wishes to reject.
M. H. Newman’s Typability Algorithm for LambdaCalculus
, 2006
"... This article is essentially an extended review with historical comments. It looks at an algorithm published in 1943 by M. H. A. Newman, which decides whether a lambdacalculus term is typable without actually computing its principal type. Newman’s algorithm seems to have been completely neglected by ..."
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This article is essentially an extended review with historical comments. It looks at an algorithm published in 1943 by M. H. A. Newman, which decides whether a lambdacalculus term is typable without actually computing its principal type. Newman’s algorithm seems to have been completely neglected by the typetheorists who invented their own rather different typability algorithms over 15 years later.
Set Theory
"... Set Theory deals with the fundamental concepts of sets and functions used everywhere in mathematics. Cantor initiated the study of set theory with his investigations on the cardinality of sets of real numbers. In particular, he proved that there are different infinite cardinalities: the quantity of ..."
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Set Theory deals with the fundamental concepts of sets and functions used everywhere in mathematics. Cantor initiated the study of set theory with his investigations on the cardinality of sets of real numbers. In particular, he proved that there are different infinite cardinalities: the quantity of natural numbers is strictly smaller than the quantity of real numbers. Cantor formalized and studied the notions of ordinal and cardinal numbers. Set theory considers a universe of sets which is ordered by the membership or element relation ∈. All other mathematical objects are coded into this universe and studied within this framework. In this way, set theory is one of the foundations of mathematics. All of the information that will be covered by the exams can be found in this text, as well as most of the exercises that will be discussed in the tutorials. The grading scheme is as follows. • One final exam, worth 65%. • Two midterms, each worth 15%, for a total of 30%. • Three homework problems (explained below), each worth 1%, for a total of 3%. • Presentation of one problem in a tutorial, worth 2%. Each week, when exercises for the tutorials are handed out, some of them will be starred (∗). Each student must submit a solution to one starred problem assigned before the first midterm, one assigned between the first and second midterms and one assigned after the second midterm. These solutions must be carefully written up and submitted to J. Franklin by the tutorial for which they have been assigned. The other problems will not be graded. If you wish to know whether your solution is correct, you are welcome to submit these assignments to J. Franklin as well.
The quantifier complexity of NF
"... Various issues concerning the quantifier complextity (i.e., number of alternations of like quantifiers) of Quine’s theory NF, its axiomatizations, and some of its subtheories, are discussed. 1 ..."
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Various issues concerning the quantifier complextity (i.e., number of alternations of like quantifiers) of Quine’s theory NF, its axiomatizations, and some of its subtheories, are discussed. 1