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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 :
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
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
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. 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.
Is it too much to ask, to ask . . .
"... Most of the time our quantifications generalise over a restricted domain. Thus in the last sentence, ‘most of the time ’ is arguably not a generalisation over all times in the history of the universe but is restricted to a subgroup of times, those at which humans exist and utter quantified phrases ..."
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Most of the time our quantifications generalise over a restricted domain. Thus in the last sentence, ‘most of the time ’ is arguably not a generalisation over all times in the history of the universe but is restricted to a subgroup of times, those at which humans exist and utter quantified phrases and sentences, say. Indeed the example illustrates the point that quantificational phrases often carry an explicit restriction with them: ‘some people’, ‘all dogs’. Even then, context usually restricts to a subdomain of the class specified by the count noun. Although teenagers like to have fun by being, they mistakenly think, overly literal — ‘Everyone is tired, let’s get to bed’: ‘everyone: you mean every person in the entire universe?’ — competent language users have to be sensitive to context virtually all the time. But is it always the case that generalisation is over a restricted domain? On the face of it, to claim this is paradoxical. If we say: (1) For every generalisation and every domain D, if D is a domain
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. 1
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.