A Set Theory with Support for Partial Functions

Labyrinth of Thought discusses the emergence and development of set theory and the set-theoretic approach to mathematics during the period 1850-1940. Rather than focusing on the pivotal figure of Georg Cantor, it analyzes his work and the emergence of transfinite set theory within the broader context of the rise of modern mathematics. The text has a tripartite structure. Part 1, The Emergence of Sets within Mathematics, surveys the initial motivations for a mathematical notion of a set within several branches of the discipline (geometry, algebra, algebraic number theory, real and complex analysis), emphasizing the role played by Riemann in fostering acceptance of the set-theoretic approach. In Part 2, Entering the Labyrinth, attention turns to the earliest theories of sets, their evolution, and their reception by the mathematical community; prominent are the epoch-making contributions of Cantor and Dedekind, and the complex interactions between them. Part 3, In Search of an Axiom System, studies the four-decade period from the discovery of set-theoretic paradoxes to Godel’s independence results, an era during which set theory gradually became assimilated into mainstream mathematics; particular attention is given to the interactions between axiomatic set theory and modern systems of formal logic, especially the interplay between set theory and type theory. A new Epilogue for this second edition offers further reflections on the foundations of set theory, including the dichotomy conception and the well-known iterative conception. The author paints on a grand scale. He sees clearly, and he sees whole. The result is a spacious canvas full of intriguing scenes and portraits from the history of set theory, seamlessly juxtaposed to form a fascinating and accurate picture of a vast area of modern mathematics. It is a must-have book for anyone who wishes to gain a balanced picture of this history. It is written in clear and elegant language for the learner, while experts in the area will enjoy seeing this beautiful presentation of what they already know, perhaps arguing about some of the author’s conclusions and choices of material. (Roger Cooke, University of Vermont)

A simple type theory with partial functions and subtypes

This paper proves that the computational power of quantum interactive proof systems, with a double-exponentially small gap in acceptance probability between the completeness and soundness cases, is precisely characterized by EXP, the class of problems solvable in exponential time by deterministic Turing machines. This fact, and our proof of it, has implications concerning quantum and classical interactive proof systems in the setting of unbounded error that include the following: * Quantum interactive proof systems are strictly more powerful than their classical counterparts in the unbounded-error setting unless PSPACE=EXP, as even unbounded error classical interactive proof systems can be simulated in PSPACE. * The recent proof of Jain, Ji, Upadhyay, and Watrous (STOC 2010) establishing QIP=PSPACE relies heavily on the fact that the quantum interactive proof systems defining the class QIP have bounded error. Our result implies that some nontrivial assumption on the error bounds for quantum interactive proofs is unavoidable to establish this result (unless PSPACE=EXP). * To prove our result, we give a quantum interactive proof system for EXP with perfect completeness and soundness error 1-2^{-2^poly}, for which the soundness error bound is provably tight. This establishes another respect in which quantum and classical interactive proof systems differ, because such a bound cannot hold for any classical interactive proof system: distinct acceptance probabilities for classical interactive proof systems must be separated by a gap that is at least (single-)exponentially small. We also study the computational power of a few other related unbounded-error complexity classes.

Assuming the Axiom of Constructibility, necessary and sufficient conditions are given for the vanishing of Ext\ for rings A of global dimension 1.Using Martin's Axiom, the necessity of these conditions is shown not to be a theorem of ZFC.Applications are given to abelian group theory, including a partial solution (assuming V = L) to a problem of Baer on the splitting of abelian groups.Some independence results in abelian group theory are also proved.Introduction.In a remarkable recent paper [22], S. Shelah proved that Whitehead's problem is unsolvable in ordinary set theory (Zermelo-Frankel set theory with the Axiom of Choice).More precisely, he proved that two different answers to the problem are obtained when set theory is extended by adding two different axioms: the Axiom of Constructibility and Martin's Axiom.(See Theorem 4.1.)Shelah's arguments were specific to Whitehead's problem and nonhomological in character.In this paper we extend Shelah's techniques in order to obtain some general results about the vanishing of ExtA (for modules over a ring A of global dimension 1) in different models of set theory, and we apply them to some specific problems in abelian group theory.Assuming the Axiom of Constructibility we obtain necessary and sufficient conditions for the vanishing of ExtA (Theorems 1.2 and 1.5).To state the result in.a special case: if C and A are abelian groups of cardinality N,, Ext(C, A) -0 if and only if C is the union of an increasing chain of countable subgroups [Cv\v < «,} such that the chain is smooth (i.e., C0 -U"<oC, for aii iimit ordinals a) and satisfies Ext(C0, A) = 0 and E\i(Cv+x/Cv,A) = 0 for all v < K,.Although the sufficiency of this condition is an easy consequence of classical homological algebra (see Theorem 1.2), the proof of necessity makes use of the Axiom of Constructibility (Theorem 1.5).The necessity of this condition is not provable in ordinary set theory; in particular the condition is not necessary when Martin's Axiom is assumed ( §3).

\n The notion of surreal number was introduced by J.H. Conway in the mid 1970\\'s: the surreal numbers constitute a linearly ordered (proper) class No containing the class of all ordinal numbers (On) that, working within the background set theory NBG, can be defined by a recursion on the class On. Since then, have appeared many constructions of this class and was isolated a full axiomatization of this notion that been subject of interest due to large number of interesting properties they have, including model-theoretic ones. Such constructions suggests strong connections between the class No of surreal numbers and the classes of all sets and all ordinal numbers. In an attempt to codify the universe of sets directly within the surreal number class, we have founded some clues that suggest that this class is not suitable for this purpose. The present work is an attempt to obtain an \\"algebraic (set) theory for surreal numbers\\" along the lines of the Algebraic Set Theory - a categorial set theory introduced in the 1990\\'s: to establish abstract and general links between the class of all surreal numbers and a universe of \\"surreal sets\\" similar to the relations between the class of all ordinals (On) and the class of all sets (V), that also respects and expands the links between the linearly ordered class of all ordinals and of all surreal numbers. We have introduced the notion of (partial) surreal algebra (SUR-algebra) and we explore some of its category theoretic properties, including (relatively) free SUR-algebras (SA, ST). We have established links, in both directions, between SUR-algebras and ZF-algebras (the keystone of Algebraic Set Theory). We develop the first steps of a certain kind of set theory based (or ranked) on surreal numbers, that expands the relation between V and On.\n

Mathematical theories of uncertainty in which measures of uncertainty are now well established are fuzzy set theory, evidence theory, and possibility theory, in addition to classical set theory and probability theory. Some of these theories are connected. Classical set theory is subsumed under fuzzy set theory. Probability theory and possibility theory are branches of evidence theory, while evidence theory is, in turn, a branch of fuzzy measure theory. Although information aspects of fuzzy measure theory have not been investigated as yet, the theory is briefly introduced in this chapter because it represents a broad framework for future research.

Large sets in intuitionistic set theory

Rough function is one of interesting new directions of the classical Pawlak rough set theory. This paper presents a generalized rough function with one variable under the framework of classical Pawlak rough set theory. In the definition of generalized rough function, we extend the domain of the classical rough function from the general natural number set to integer set, and then the rough function with one variable defined on an integer set is obtained. After that, we investigate the basic properties of the proposed rough function with one variable base on the classical function analysis theory in detail. Furthermore, some interesting results about the rough function with one variable over integer set are given and the relationship between the existing literatures is discussed. The contribution of this is twofold: one is present a new concept of rough function with one variable over the integer set. Another is of enriches the basis of theory of classical Pawlak rough set and also extends the areas of the applications.

Metapredicativity is a new general term in proof theory which describes the analysis and study of formal systems whose proof-theoretic strength is beyond the Feferman-Schutte ordinal Γ0 but which are nevertheless amenable to purely predicative methods. Typical examples of formal systems which are apt for scaling the initial part of metapredicativity are the transfinitely iterated fixed point theories IDα whose detailed proof-theoretic analysis is given by Jager, Kahle, Setzer and Strahm in [18]. In this paper we assume familiarity with [18]. For natural extensions of Friedman’s ATR that can be measured against transfinitely iterated fixed point theories the reader is referred to Jager and Strahm [20]. In the mid seventies, Feferman [3, 4] introduced systems of explicit mathematics in order to provide an alternative foundation of constructive mathematics. More precisely, it was the origin of Feferman’s program to give a logical account of Bishop-style constructive mathematics. Right from the beginning, systems of explicit mathematics turned out to be of general interest for proof theory, mainly in connection with the proof-theoretic analysis of subsystems of first and second order arithmetic and set theory, cf. e.g. Jager [15] and Jager and Pohlers [19]. More recently, systems of explicit mathematics have been used to develop a general logical framework for functional programming and type theory, where it is possible to derive correctness and termination properties of functional programs. Important references in this connection are Feferman [6, 7, 9] and Jager [17]. Universes are a frequently studied concept in constructive mathematics at least since the work of Martin-Lof, cf. e.g. Martin-Lof [23] or Palmgren [27] for

Descriptive set theory has been one of the main areas of research in set theory for almost a century. This text attempts to present a largely balanced approach, which combines many elements of the different traditions of the subject. It includes a wide variety of examples, exercises (over 400), and applications, in order to illustrate the general concepts and results of the theory. This text provides a first basic course in classical descriptive set theory and covers material with which mathematicians interested in the subject for its own sake or those that wish to use it in their field should be familiar. Over the years, researchers in diverse areas of mathematics, such as logic and set theory, analysis, topology, probability theory, etc., have brought to the subject of descriptive set theory their own intuitions, concepts, terminology and notation.

Tolerance-based intuitionistic fuzzy-rough set approach for attribute reduction

This paper proposes a possibility of developing an axiomatic set theory, as first-order theory within the framework of fuzzy logic in the style of Hajek's Basic fuzzy logic BL. In classical Zermelo-Fraenkel set theory, we use an analogy of the construction of a Boolean-valued universe-over a particular algebra of truth values-we show the nontriviality of our theory. We present a list of problems and research tasks.

In the article [4], a new constructive set theoretic principle called Refinement was introduced and analysed. While it seemed to be significantly weaker than its alternative, the more established axiom of Fullness (a constructive version of the Powerset axiom from classical set theory), it was shown to suffice to imply many of the mathematically important consequences. In this article, we will define for each set A a set of truth values which measures the complexity of the equality relation on A. Using these sets we will show that Refinement is actually equivalent to Fullness on the basis of the other axioms of constructive Zermelo‐Fraenkel set theory (© 2010 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

This chapter is an overview of current research regarding measures of uncertainty and information in five mathematical theories: classical set theory, fuzzy set theory, probability theory, possibility theory, and the Dempster-Shafer theory of evidence. Three types of uncertainty are involved: nonspecificity, entropy-like uncertainty, and fuzziness.

Set theory for fuzzy sets of higher level