Results 1  10
of
15
Safety Signatures for Firstorder Languages and Their Applications
 In FirstOrder Logic Revisited (Hendricks et all,, eds.), 3758, Logos Verlag
, 2004
"... ..."
Reflections on Skolem's Paradox
"... In 1922, Thoraf Skolem published a paper titled "Some remarks on Axiomatized Set Theory". The paper presents a new proof of... This dissertation focuses almost exclusively on the first half of this project  i.e., the half which tries to expose an initial tension between Cantor's theorem and the Lö ..."
Abstract

Cited by 4 (2 self)
 Add to MetaCart
In 1922, Thoraf Skolem published a paper titled "Some remarks on Axiomatized Set Theory". The paper presents a new proof of... This dissertation focuses almost exclusively on the first half of this project  i.e., the half which tries to expose an initial tension between Cantor's theorem and the LöwenheimSkolem theorem. I argue that, even on quite naive understandings of set theory and model theory, there is no such tension. Hence, Skolem's Paradox is not a genuine paradox, and there is very little reason to worry about (or even to investigate) the more extreme consequences that are supposed to follow from this paradox. The heart of my...
A Framework for Formalizing Set Theories Based on the Use of Static Set Terms
"... To Boaz Trakhtenbrot: a scientific father, a friend, and a great man. Abstract. We present a new unified framework for formalizations of axiomatic set theories of different strength, from rudimentary set theory to full ZF. It allows the use of set terms, but provides a static check of their validity ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
To Boaz Trakhtenbrot: a scientific father, a friend, and a great man. Abstract. We present a new unified framework for formalizations of axiomatic set theories of different strength, from rudimentary set theory to full ZF. It allows the use of set terms, but provides a static check of their validity. Like the inconsistent “ideal calculus ” for set theory, it is essentially based on just two settheoretical principles: extensionality and comprehension (to which we add ∈induction and optionally the axiom of choice). Comprehension is formulated as: x ∈{x  ϕ} ↔ϕ, where {x  ϕ} is a legal set term of the theory. In order for {x  ϕ} to be legal, ϕ should be safe with respect to {x}, where safety is a relation between formulas and finite sets of variables. The various systems we consider differ from each other mainly with respect to the safety relations they employ. These relations are all defined purely syntactically (using an induction on the logical structure of formulas). The basic one is based on the safety relation which implicitly underlies commercial query languages for relational database systems (like SQL). Our framework makes it possible to reduce all extensions by definitions to abbreviations. Hence it is very convenient for mechanical manipulations and for interactive theorem proving. It also provides a unified treatment of comprehension axioms and of absoluteness properties of formulas. 1
Date Declaration Set Theory Without the Axiom of Foundation
"... I declare that this essay is work done as part of the Part III Examination. It is the result of my own work, and except where stated otherwise, includes nothing which was performed in collaboration. No part of this essay has been submitted for a degree or any such qualification. ..."
Abstract
 Add to MetaCart
I declare that this essay is work done as part of the Part III Examination. It is the result of my own work, and except where stated otherwise, includes nothing which was performed in collaboration. No part of this essay has been submitted for a degree or any such qualification.
Concepts and Axioms
, 1998
"... The paper discusses the transition from informal concepts to mathematically precise notions; examples are given, and in some detail the case of lawless sequences, a concept of intuitionistic mathematics, is discussed. A final section comments on philosophical discussions concerning intuitionistic lo ..."
Abstract
 Add to MetaCart
The paper discusses the transition from informal concepts to mathematically precise notions; examples are given, and in some detail the case of lawless sequences, a concept of intuitionistic mathematics, is discussed. A final section comments on philosophical discussions concerning intuitionistic logic in connection with a "theory of meaning". What I have to tell here is not a new story, and it does not contain any really new ideas. The main difference with my earlier discussions of the same topics ([TD88, chapter16],[Tro91]) is in the emphasis. This paper starts with some examples of the transition from informal concepts to mathematically precise notions, followed by a more detailed discussion of one of these examples, the intuitionistic notion of a choice sequence, arguing for the lasting interest of this notion for the philosophy of mathematics. In a final section, I describe my own position relative to some of the philosophical discussions concerning intuitionistic logic in the wr...
Choice Sequences: a Retrospect
, 1996
"... Introduction The topic of this talk will be the lasting interest of L.E.J. Brouwer's notion of choice sequence for the philosophy of mathematics. In the past here has been done a good deal of work on choice sequences, but in the last decade the subject is a bit out of fashion, for several reasons, ..."
Abstract
 Add to MetaCart
Introduction The topic of this talk will be the lasting interest of L.E.J. Brouwer's notion of choice sequence for the philosophy of mathematics. In the past here has been done a good deal of work on choice sequences, but in the last decade the subject is a bit out of fashion, for several reasons, which I shall not go into here. In this retrospective I want to take a look with you at a special aspect of choice sequences, namely their interest as an important "casestudy" in the philosophy of mathematics. How does mathematics arrive at its concepts, and discover the principles holding for those concepts? This is a typically philosophical question, more easily posed than answered. A procedure which certainly has played a role and still plays a role might be described as informally rigorous analysis of a concept That is to say,  given an informally described, but intuitively clear concept,  one analyzes the concept as carefully as possibl
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 ..."
Abstract
 Add to MetaCart
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.
Alternative Set Theories
, 2006
"... By “alternative set theories ” we mean systems of set theory differing significantly from the dominant ZF (ZermeloFrankel set theory) and its close relatives (though we will review these systems in the article). Among the systems we will review are typed theories of sets, Zermelo set theory and its ..."
Abstract
 Add to MetaCart
By “alternative set theories ” we mean systems of set theory differing significantly from the dominant ZF (ZermeloFrankel set theory) and its close relatives (though we will review these systems in the article). Among the systems we will review are typed theories of sets, Zermelo set theory and its variations, New Foundations and related systems, positive set theories, and constructive set theories. An interest in the range of alternative set theories does not presuppose an interest in replacing the dominant set theory with one of the alternatives; acquainting ourselves with foundations of mathematics formulated in terms of an alternative system can be instructive as showing us what any set theory (including the usual one) is supposed to do for us. The study of alternative set theories can dispel a facile identification of “set theory ” with “ZermeloFraenkel set theory”; they are not the same thing. Contents 1 Why set theory? 2 1.1 The Dedekind construction of the reals............... 3 1.2 The FregeRussell definition of the natural numbers....... 4
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY
"... Ask a mathematician of the age of Gauss “What is mathematics? ” and you could expect a stock answer somewhat along the following lines: “Mathematics consists of arithmetic and geometry, arithmetic being the science of quantity, just as geometry is the science of space. ” A few philosophers (Berkeley ..."
Abstract
 Add to MetaCart
Ask a mathematician of the age of Gauss “What is mathematics? ” and you could expect a stock answer somewhat along the following lines: “Mathematics consists of arithmetic and geometry, arithmetic being the science of quantity, just as geometry is the science of space. ” A few philosophers (Berkeley or Kant, for instance) might raise questions about the nature and sources of mathematical knowledge or about the legitimacy of certain forms of mathematical reasoning, but those questions existed on the fringes of mathematics and seemed to have little to do with the core of the discipline. Then, over the course of the nineteenth century, under the pressure of developments within mathematics itself, the accepted answer dramatically broke down. In analysis, Bolzano, investigating the foundations of the calculus, gives his “purely analytic proof ” of the intermediate value theorem; Weierstrass and his students independently rediscover his results and attempt to put the calculus on a rigorous arithmetical foundation. In algebra, Gauss and Hamilton provide geometric interpretations of the complex numbers; Hamilton widens the number concept, introducing