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Set Theory (set + theory)

Distribution by Scientific Domains

Information Science and Computing	58%
Humanities and Social Sciences	16%
Engineering	11%
Business, Economics, Finance and Accounting	8%
Earth and Environmental Science	5%

Kinds of Set Theory

fuzzy set theory

rough set theory

Selected Abstracts

Functional Interpretations of Constructive Set Theory in All Finite Types

DIALECTICA, Issue 2 2008
Justus Diller
Gödel's dialectica interpretation of Heyting arithmetic HA may be seen as expressing a lack of confidence in our understanding of unbounded quantification. Instead of formally proving an implication with an existential consequent or with a universal antecedent, the dialectica interpretation asks, under suitable conditions, for explicit ,interpreting' instances that make the implication valid. For proofs in constructive set theory CZF - , it may not always be possible to find just one such instance, but it must suffice to explicitly name a set consisting of such interpreting instances. The aim of eliminating unbounded quantification in favor of appropriate constructive functionals will still be obtained, as our ,-interpretation theorem for constructive set theory in all finite types CZF, - shows. By changing to a hybrid interpretation ,q, we show closure of CZF, - under rules that , in stronger forms , have already been studied in the context of Heyting arithmetic. In a similar spirit, we briefly survey modified realizability of CZF, - and its hybrids. Central results of this paper have been proved by Burr 2000a and Schulte 2006, however, for different translations. We use a simplified interpretation that goes back to Diller and Nahm 1974. A novel element is a lemma on absorption of bounds which is essential for the smooth operation of our translation. [source]

Overlap in bibliographic databases

JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATION SCIENCE AND TECHNOLOGY, Issue 12 2003
William W. Hood
Bibliographic databases contain surrogates to a particular subset of the complete set of literature; some databases are very narrow in their scope, while others are multidisciplinary. These databases overlap in their coverage of the literature to a greater or lesser extent. The topic of Fuzzy Set Theory is examined to determine the overlap of coverage in the databases that index this topic. It was found that about 63% of records in the data set are unique to only one database, and the remaining 37% are duplicated in from two to 12 different databases. The overlap distribution is found to conform to a Lotka-type plot. The records with maximum overlap are identified; however, further work is needed to determine the significance of the high level of overlap in these records. The unique records are plotted using a Bradford-type form of data presentation and are found to conform (visually) to a hyperbolic distribution. The extent and causes of intra-database duplication (records duplicated in the one database) are also examined. Finally, the overlap in the top databases in the dataset were examined, and a high correlation was found between overlapping records, and overlapping DIALOG OneSearch categories. [source]

A note on Bar Induction in Constructive Set Theory

MLQ- MATHEMATICAL LOGIC QUARTERLY, Issue 3 2006
Michael Rathjen
Abstract Bar Induction occupies a central place in Brouwerian mathematics. This note is concerned with the strength of Bar Induction on the basis of Constructive Zermelo-Fraenkel Set Theory, CZF. It is shown that CZF augmented by decidable Bar Induction proves the 1-consistency of CZF. This answers a question of P. Aczel who used Bar Induction to give a proof of the Lusin Separation Theorem in the constructive set theory CZF. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

Using Complex Set Theory for Tonal Analysis: an Introduction to the Tonalities Project

MUSIC ANALYSIS, Issue 2-3 2004
Anthony Pople
First page of article [source]

Should the Logic of Set Theory be Intuitionistic?

PROCEEDINGS OF THE ARISTOTELIAN SOCIETY (HARDBACK), Issue 1 2001
Alexander Paseau
First page of article [source]

Development of applications with fuzzy objects in modern programming platforms

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, Issue 11 2005
F. Berzal
Most of the applications that are currently being developed use object-oriented programming technology, as is the case of those applications built with the Java or C, languages. Data management has not kept out of this trend, and object-oriented and object-relational database management systems have arisen as a result. Soft-computing applications need to manage imperfect data and Fuzzy Sets Theory has proven to be a good choice for accomplishing the task of imperfect data management. In this article we present a framework that allows the programmers of soft-computing applications to deal with fuzzy objects in a transparent and intuitive way. This framework can be used to develop an object-oriented code in those systems that conform with current hip object-oriented languages, so that imperfect information can be managed. © 2005 Wiley Periodicals, Inc. Int J Int Syst 20: 1117,1136, 2005. [source]

Discovering Maximal Generalized Decision Rules Through Horizontal and Vertical Data Reduction

COMPUTATIONAL INTELLIGENCE, Issue 4 2001
Xiaohua Hu
We present a method to learn maximal generalized decision rules from databases by integrating discretization, generalization and rough set feature selection. Our method reduces the data horizontally and vertically. In the first phase, discretization and generalization are integrated and the numeric attributes are discretized into a few intervals. The primitive values of symbolic attributes are replaced by high level concepts and some obvious superfluous or irrelevant symbolic attributes are also eliminated. Horizontal reduction is accomplished by merging identical tuples after the substitution of an attribute value by its higher level value in a pre-defined concept hierarchy for symbolic attributes, or the discretization of continuous (or numeric) attributes. This phase greatly decreases the number of tuples in the database. In the second phase, a novel context-sensitive feature merit measure is used to rank the features, a subset of relevant attributes is chosen based on rough set theory and the merit values of the features. A reduced table is obtained by removing those attributes which are not in the relevant attributes subset and the data set is further reduced vertically without destroying the interdependence relationships between classes and the attributes. Then rough set-based value reduction is further performed on the reduced table and all redundant condition values are dropped. Finally, tuples in the reduced table are transformed into a set of maximal generalized decision rules. The experimental results on UCI data sets and a real market database demonstrate that our method can dramatically reduce the feature space and improve learning accuracy. [source]

Consistency of a Shared Versioned Model for Distributed Cooperation

COMPUTER-AIDED CIVIL AND INFRASTRUCTURE ENGINEERING, Issue 6 2005
B. Firmenich
As a rule the ultimate solution needs many iteration steps. Available CAD-systems support synchronous and distributed work on a document base. Therefore, cooperation between the engineers can be obtained only by the exchange of documents. It is generally known that an overall consistency of the planning material is not adequately addressed by this approach. In this article a solution approach focused upon consistency of the shared planning material in a distributed CAD environment is presented. Because of the nature of the planning process, version management is applied on an object basis. Project data are stored as object versions and relationships. The operations for the distributed cooperation are identified and their impact on the project data is described formally, using logical expressions and set theory. A formulation based upon an algebra of sets is presented. [source]

Functional Interpretations of Constructive Set Theory in All Finite Types

Fuzzy adjunctions and fuzzy morphological operations based on implications

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, Issue 12 2009
Y. Shi
The concept of adjunction plays an important role in mathematical morphology. If the morphological operations, dilation and erosion form an adjunction in a complete lattice, then they, as well as the closing and opening constructed by them, will fulfill certain required properties in an algebraic context. In the context of fuzzy mathematical morphology, which is an extension of binary morphology to gray-scale morphology based on fuzzy set theory, we use conjunctions and implications to define fuzzy dilations and fuzzy erosions. In this paper, we investigate when these pairs of dilations and erosions form a fuzzy adjunction, which is also defined by an implication. We find that the so-called adjointness between a conjunction and an implication plays an important role here. Finally, we develop a theorem stating that a conjunction that is adjoint with an implication cannot only be generated by an R-implication but also by other implications. This allows the easy construction of fuzzy adjunctions. © 2009 Wiley Periodicals, Inc. [source]

Intelligent social network analysis using granular computing

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, Issue 11 2008
Ronald R. Yager
An introduction to some basic ideas of graph (relational network) theory is first provided. We then discuss some concepts from granular computing in particular the fuzzy set paradigm of computing with words. The natural connection between graph theory and granular computing, particularly fuzzy set theory, is pointed out. This connection is grounded in the fact that these are both set-based technologies. Our objective here is to take a step toward the development of intelligent social network analysis using granular computing. In particular one can start by expressing in a human-focused manner concepts associated with social networks then formalize these concepts using fuzzy sets and then evaluate these concepts with respect to social networks that have been represented using set-based relational network theory. We capture this approach in what we call the paradigm for intelligent social network analysis, PISNA. Using this paradigm, we provide definitions of a number of concepts related to social networks. © 2008 Wiley Periodicals, Inc. [source]

An algorithm for modelling key terms

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, Issue 1 2008
D. Cai
The ability to formally analyse and represent semantic relations of terms is a major challenge for many areas of computing science and an intriguing problem for other sciences. In applications of evidence theory to, for instance, information retrieval, the problem of analysis and representation becomes apparent because evidence theory is based on set theory and individual key terms have to be modelled as subsets of the frame of discernment. How to find the frame and model the key terms is a challenge. The problem leads to other practical problems, as pointed out repeatedly in the literature. In this study, we focus on such a problem, present a method for simplifying and normalizing a thesaurus, and propose an algorithm for establishing the frame of discernment and for modelling individual key terms as a subset of the frame. The key aim of this study is to treat semantic relations of terms by means of a normalized thesaurus. © 2008 Wiley Periodicals, Inc. [source]

An introduction of the condition class space with continuous value discretization and rough set theory

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, Issue 2 2006
Malcolm J. Beynon
The granularity of an information system has an incumbent effect on the efficacy of the analysis from many machine learning algorithms. An information system contains a universe of objects characterized and categorized by condition and decision attributes. To manage the concomitant granularity, a level of continuous value discretization (CVD) is often undertaken. In the case of the rough set theory (RST) methodology for object classification, the granularity contributes to the grouping of objects into condition classes with the same condition attribute values. This article exposits the effect of a level of CVD on the subsequent condition classes constructed, with the introduction of the condition class space,the domain within which the condition classes exist. This domain elucidates the association of the condition classes to the related decision outcomes,reflecting the inexactness incumbent when a level of CVD is undertaken. A series of measures is defined that quantify this association. Throughout this study and without loss of generality, the findings are made through the RST methodology. This further offers a novel exposition of the relationship between all the condition attributes and the RST-related reducts (subsets of condition attributes). © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 173,191, 2006. [source]

,-Cut fuzzy control charts for linguistic data

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, Issue 12 2004
Murat Gülbay
The major contribution of fuzzy set theory is its capability of representing vague data. Fuzzy logic offers a systematic base in dealing with situations that are ambiguous or not well defined. In the literature, there exist some fuzzy control charts developed for linguistic data that are mainly based on membership and probabilistic approaches. In this article, ,-cut control charts for attributes are developed. This approach provides the ability of determining the tightness of the inspection by selecting a suitable ,-level: The higher , the tighter inspection. The article also presents a numerical example and interprets and compares other results with the approaches developed previously. © 2004 Wiley Periodicals, Inc. Int J Int Syst 19: 1173,1195, 2004. [source]

Ranking of alternatives with ordered weighted averaging operators

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, Issue 5 2004
M. Teresa Lamata
Multiattribute decision making is an important part of the decision process for both individual and group problems. We incorporate the fuzzy set theory and the basic nature of subjectivity due to ambiguity to achieve a flexible decision approach suitable for uncertain and fuzzy environments. Let us consider the analytic hierarchy process (AHP) in which the labels are structured as fuzzy numbers. To obtain the scoring that corresponds to the best alternative or the ranking of the alternatives, we need to use a total order for the fuzzy numbers involved in the problem. In this article, we consider a definition of such a total order, which is based on two subjective aspects: the degree of optimism/pessimism reflected with the ordered weighted averaging (OWA) operators. A numerical example is given to illustrate the approach. © 2004 Wiley Periodicals, Inc. [source]

Handling indefinite and maybe information in logical fuzzy relational databases

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, Issue 3 2004
Nan-Chen Hsieh
In this article, fuzzy set theory uses an extension of the classical logical relational database model. A logical fuzzy relational database model was developed with the aim of manipulating imprecise information and adding deduction capabilities to the database system. The essence of this work is the detailed discussion on fuzzy definite, fuzzy indefinite, and fuzzy maybe information and the development of an information theoretical approach of query evaluation on the logical fuzzy relational database. We define redundancies among fuzzy tuples and the operator of their removal. A complete set of fuzzy relational operations in relational algebra and the calculus of linguistically quantified propositions are included also. © 2004 Wiley Periodicals, Inc. [source]

Approaches to knowledge reductions in inconsistent systems

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, Issue 9 2003
Wen-Xiu Zhang
This article deals with approaches to knowledge reductions in inconsistent information systems (ISs). The main objective of this work was to introduce a new kind of knowledge reduction called a maximum distribution reduct, which preserves all maximum decision classes. This type of reduction eliminates the harsh requirements of the distribution reduct and overcomes the drawback of the possible reduct that the derived decision rules may be incompatible with the ones derived from the original system. Then, the relationships among the maximum distribution reduct, the distribution reduct, and the possible reduct were discussed. The judgement theorems and discernibility matrices associated with the three reductions were examined, from which we can obtain approaches to knowledge reductions in rough set theory (RST). © 2003 Wiley Periodicals, Inc. [source]

Graph-based interaction in a fuzzy object oriented database

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, Issue 7 2001
Gloria Bordogna
In this paper, the definition of graph-based operations to select and browse a fuzzy object oriented database which manages both crisp and fuzzy information is proposed. The underlying data model is a fuzzy graph-based model (FOOD), previously defined by Bordogna et al., within fuzzy set theory and possibility theory. The evaluation mechanism of the graph-based operations is formalized in terms of graph transformations and fuzzy pattern matching. © 2001 John Wiley & Sons, Inc. [source]

Fuzzy reasoning based on the extension principle,

INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, Issue 4 2001
Yang Xu
According to the operation of decomposition (also known as representation theorem) (Negoita CV, Ralescu, DA. Kybernetes 1975;4:169,174) in fuzzy set theory, the whole fuzziness of an object can be characterized by a sequence of local crisp properties of that object. Hence, any fuzzy reasoning could also be implemented by using a similar idea, i.e., a sequence of precise reasoning. More precisely, we could translate a fuzzy relation "If A then B" of the Generalized Modus Ponens Rule (the most common and widely used interpretation of a fuzzy rule, A,,B, are fuzzy sets in a universe of discourse X, and of discourse Y, respectively) into a corresponding precise relation between a subset of P(X) and a subset of P(Y), and then extend this corresponding precise relation to two kinds of transformations between all L -type fuzzy subsets of X and those of Y by using Zadeh's extension principle, where L denotes a complete lattice. In this way, we provide an alternative approach to the existing compositional rule of inference, which performs fuzzy reasoning based on the extension principle. The approach does not depend on the choice of fuzzy implication operator nor on the choice of a t-norm. The detailed reasoning methods, applied in particular to the Generalized Modus Ponens and the Generalized Modus Tollens, are established and their properties are further investigated in this paper. © 2001 John Wiley & Sons, Inc. [source]

Using Cpk index with fuzzy numbers to evaluate service quality

INTERNATIONAL TRANSACTIONS IN OPERATIONAL RESEARCH, Issue 6 2002
Hong Tau Lee
Service quality is measured by customers' satisfaction. Traditionally, the degree of satisfaction is calculated from the data obtained from questionnaires that have been filled by customers directly. The percentile of each different level of a customer's satisfaction is employed to summarize and compare the quality of service provided by different enterprises. This approach does not consider the consistency of the customers' perceptions, thus making comparison difficult. This paper introduces the concept of a process capability index that considers both the average and the consistency of the data simultaneously. Evaluations of service quality are usually vague and linguistic. We use the fuzzy numbers of linguistic variables developed in fuzzy set theory to modify the process capability index, and then apply it to evaluate the quality of a service. The average and consistency of the data obtained from a service quality evaluation are thus considered simultaneously, making the comparison of the performance of service quality easier. Moreover, the value of the index can be applied to help to point out the direction for improving the performance of service quality whenever it is lower than some default value. [source]

Using soft computing techniques for improving foot trajectories in walking machines

JOURNAL OF FIELD ROBOTICS (FORMERLY JOURNAL OF ROBOTIC SYSTEMS), Issue 7 2001
Elena Garcia
Walking machines have been investigated during the last 40 years and some basic techniques of this field are already well known. However, some aspects still need to be optimized. For instance, speed seems to be one of the major shortcomings of legged robots; thus, improving leg speed has been chosen as the main aim of this work. Although some algorithms for optimizing trajectory control of robot manipulators already exist, we propose a more computationally efficient method that employs fuzzy set theory to involve real dynamic effects over leg motion instead of an inaccurate mathematical model. In this article, we improve leg speed by automatically tuning the acceleration of legs. For this purpose, we define fuzzy rules based on experiments and we find the optimal acceleration for every given trajectory. A simple fuzzy inference system is used to compute the required acceleration. It is based on five rules using three linguistic variables. Final results show that foot acceleration tuning for straight trajectory generation is a suitable method for achieving accurate, smooth and fast foot movements. Also it is shown that under some conditions average leg speed can be increased up to 100% using the control methods herein proposed. © 2001 John Wiley & Sons, Inc. [source]

Characterizing, Propagating, and Analyzing Uncertainty in Life-Cycle Assessment: A Survey of Quantitative Approaches

JOURNAL OF INDUSTRIAL ECOLOGY, Issue 1 2007
Shannon M. Lloyd
Summary Life-cycle assessment (LCA) practitioners build models to quantify resource consumption, environmental releases, and potential environmental and human health impacts of product systems. Most often, practitioners define a model structure, assign a single value to each parameter, and build deterministic models to approximate environmental outcomes. This approach fails to capture the variability and uncertainty inherent in LCA. To make good decisions, decision makers need to understand the uncertainty in and divergence between LCA outcomes for different product systems. Several approaches for conducting LCA under uncertainty have been proposed and implemented. For example, Monte Carlo simulation and fuzzy set theory have been applied in a limited number of LCA studies. These approaches are well understood and are generally accepted in quantitative decision analysis. But they do not guarantee reliable outcomes. A survey of approaches used to incorporate quantitative uncertainty analysis into LCA is presented. The suitability of each approach for providing reliable outcomes and enabling better decisions is discussed. Approaches that may lead to overconfident or unreliable results are discussed and guidance for improving uncertainty analysis in LCA is provided. [source]

Measuring the effects of human impact on vegetation by integrating phytosociology and remote sensing in a fuzzy set approach

JOURNAL OF VEGETATION SCIENCE, Issue 5 2003
Enrico Feoli
Anon. 1954 , Abstract. The response of fragmented vegetation to human impact has been analysed in degraded and degrading areas in Tanzania (Lake Manyara). Phytosociology was integrated with GIS and remote sensing data as follows: (1) a land cover/land use map was obtained by analysing remote sensing data and conducting field verification; (2) phytosociological relevés were randomly sampled in woodland vegetation patches using the map; (3) the pattern of land cover/land use around the relevés was described; (4) gradients of land use intensity (human impact) were obtained based on the descriptions; (5) the response of vegetation types to impacts of gradients was calculated using fuzzy set theory. Two complementary gradients of human impact were defined based on land cover/land use analysis of the remotely sensed data: one related to cultivation intensity and the other to grazing intensity. Response functions of vegetation types (defined by numerical classification) to these gradients demonstrated that the vegetation types are strongly related to the degree of human impact and that the corresponding vegetation patches show different degrees of permeability to the species of the surrounding landscape. [source]

On some peculiar aspects of the constructive theory of point-free spaces

MLQ- MATHEMATICAL LOGIC QUARTERLY, Issue 4 2010
Giovanni Curi
Abstract This paper presents several independence results concerning the topos-valid and the intuitionistic (generalised) predicative theory of locales. In particular, certain consequences of the consistency of a general form of Troelstra's uniformity principle with constructive set theory and type theory are examined (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

The Relation Reflection Scheme

MLQ- MATHEMATICAL LOGIC QUARTERLY, Issue 1 2008
Peter Aczel
Abstract We introduce a new axiom scheme for constructive set theory, the Relation Reflection Scheme (RRS). Each instance of this scheme is a theorem of the classical set theory ZF. In the constructive set theory CZF,, when the axiom scheme is combined with the axiom of Dependent Choices (DC), the result is equivalent to the scheme of Relative Dependent Choices (RDC). In contrast to RDC, the scheme RRS is preserved in Heyting-valued models of CZF, using set-generated frames. We give an application of the scheme to coinductive definitions of classes. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

Largest fixed points of set continuous operators and Boffa's Anti-Foundation

MLQ- MATHEMATICAL LOGIC QUARTERLY, Issue 4 2005
Hisato Muraki
Abstract In Aczel [1], the existence of largest (written "greatest" in Barwise and Moss [2]) fixed points of set continuous operators is proved assuming the schema version of dependent choices in Zermelo-Fraenkel set theory without the axiom of Foundation. In the present paper, we study whether the existence of largest fixed points of set continuous operators is provable without the schema version of dependent choices, using Boffa's weak antifoundation axioms. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

On topological set theory

MLQ- MATHEMATICAL LOGIC QUARTERLY, Issue 3 2005
Thierry Libert
Abstract This paper is concerned with topological set theory, and particularly with Skala's and Manakos' systems for which we give a topological characterization of the models. This enables us to answer natural questions about those theories, reviewing previous results and proving new ones. One of these shows that Skala's set theory is in a sense compatible with any ,normal' set theory, and another appears on the semantic side as a ,Cantor theorem' for the category of Alexandroff spaces. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

Allen Forte in Music Analysis

MUSIC ANALYSIS, Issue 1-2 2007
Arnold Whittall
ABSTRACT This article explores the engagement of an American music theorist with a British journal. Music Analysis excludes the word ,theory' from its title to avoid confusion with American journals. But many American theorists have welcomed the alternative it represents, and of these Allen Forte is pre-eminent. His contributions began with Vol. 2/iii (October 1983) and have continued through to the present issue. As this survey suggests, those contributions which are transcripts of spoken presentations emphasise that analysis is not just abstraction but ,a human thing'. Yet, whether Forte is writing primarily about a topic such as pitch-class set theory or about admired composers , Musorgsky, Debussy, Messiaen , he offers a distinctive blend of general and particular, placing compositional specifics within wider perspectives which invariably stimulate the reader not just to think but to listen, and re-listen. [source]

Set theoretic formulation of performance reliability of multiple response time-variant systems due to degradations in system components

QUALITY AND RELIABILITY ENGINEERING INTERNATIONAL, Issue 2 2007
Young Kap Son
Abstract This paper presents a design stage method for assessing performance reliability of systems with multiple time-variant responses due to component degradation. Herein the system component degradation profiles over time are assumed to be known and the degradation of the system is related to component degradation using mechanistic models. Selected performance measures (e.g. responses) are related to their critical levels by time-dependent limit-state functions. System failure is defined as the non-conformance of any response and unions of the multiple failure regions are required. For discrete time, set theory establishes the minimum union size needed to identify a true incremental failure region. A cumulative failure distribution function is built by summing incremental failure probabilities. A practical implementation of the theory can be manifest by approximating the probability of the unions by second-order bounds. Further, for numerical efficiency probabilities are evaluated by first-order reliability methods (FORM). The presented method is quite different from Monte Carlo sampling methods. The proposed method can be used to assess mean and tolerance design through simultaneous evaluation of quality and performance reliability. The work herein sets the foundation for an optimization method to control both quality and performance reliability and thus, for example, estimate warranty costs and product recall. An example from power engineering shows the details of the proposed method and the potential of the approach. Copyright © 2006 John Wiley & Sons, Ltd. [source]

Foundations of Mathematics: Metaphysics, Epistemology, Structure

THE PHILOSOPHICAL QUARTERLY, Issue 214 2004
Stewart Shapiro
Since virtually every mathematical theory can be interpreted in set theory, the latter is a foundation for mathematics. Whether set theory, as opposed to any of its rivals, is the right foundation for mathematics depends on what a foundation is for. One purpose is philosophical, to provide the metaphysical basis for mathematics. Another is epistemic, to provide the basis of all mathematical knowledge. Another is to serve mathematics, by lending insight into the various fields. Another is to provide an arena for exploring relations and interactions between mathematical fields, their relative strengths, etc. Given the different goals, there is little point to determining a single foundation for all of mathematics. [source]