Results 1  10
of
14
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 ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
(Show Context)
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 ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
(Show Context)
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 ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
(Show Context)
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.
The Iterative Conception of Set
, 2009
"... 2 The TwoConstructor case 5 2.1 Set Equality in the TwoConstructor Case............. 6 3 More Wands 9 ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
(Show Context)
2 The TwoConstructor case 5 2.1 Set Equality in the TwoConstructor Case............. 6 3 More Wands 9
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 ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
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
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 ..."
Abstract
 Add to MetaCart
(Show Context)
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.
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 ..."
Abstract
 Add to MetaCart
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
The Universe Among Other Things
"... Abstract. Peter Simons has argued that the expression ‘the universe ’ is not a genuine singular term: it can name neither a single, completely encompassing individual, nor a collection of individuals. (It is, rather, a semantically plural term standing equally for every existing object.) I offer rea ..."
Abstract
 Add to MetaCart
Abstract. Peter Simons has argued that the expression ‘the universe ’ is not a genuine singular term: it can name neither a single, completely encompassing individual, nor a collection of individuals. (It is, rather, a semantically plural term standing equally for every existing object.) I offer reasons for resisting Simons’s arguments on both scores. 1.
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 ..."
Abstract
 Add to MetaCart
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
THE EMPTY SET, THE SINGLETON, AND THE ORDERED PAIR AKIHIRO KANAMORI
, 2002
"... For the modern set theorist the empty set ∅, the singleton {a}, and the ordered pair 〈x, y 〉 are at the beginning of the systematic, axiomatic development of set theory, both as a field of mathematics and as a unifying framework for ongoing mathematics. These notions are the simplest building blocks ..."
Abstract
 Add to MetaCart
(Show Context)
For the modern set theorist the empty set ∅, the singleton {a}, and the ordered pair 〈x, y 〉 are at the beginning of the systematic, axiomatic development of set theory, both as a field of mathematics and as a unifying framework for ongoing mathematics. These notions are the simplest building blocks in the abstract, generative conception of sets advanced by the initial axiomatization of Ernst Zermelo [1908a] and are quickly assimilated long before the complexities of Power Set, Replacement, and Choice are broached in the formal elaboration of the ‘set of ’ {} operation. So it is surprising that, while these notions are unproblematic today, they were once sources of considerable concern and confusion among leading pioneers of mathematical logic like Frege, Russell, Dedekind, and Peano. In the development of modern mathematical logic out of the turbulence of 19th century logic, the emergence of the empty set, the singleton, and the ordered pair as clear and elementary settheoretic concepts serves as a motif that reflects and illuminates larger and more significant developments in mathematical logic: