... at: yanruixia@gmail.com © Emerald Group Publishing Limited 2012 Information processing Set theory Mathematics Fuzzy set Vague set Rough set Let U be a non‐empty set and R an equivalent relation on U. (U, R)={E₁...

...), surface‐based computing models (Liu, 2000), Boolean circuits (Amos and Dunne, 1997), and sticker models (Roweis et al., 1998). Zhixiang Yin can be contacted at: Zxyin66@163.com © Emerald Group Publishing Limited 2012 Programming and algorithm theory Set theory DNA computing...

... is not an extension of IFT. Cybernetics Logic Set theory Topology Fuzzy logic Philosophy A NS x(T,I,F) is called paraconsistent if inf(T)+inf(I)+inf(F)>1 (Smarandache, 2005a). In this case the pair (X,τ) is called an IFTS and any IFS...

... Group Publishing Limited 2009 Cybernetics Systems theory Dynamics Set theory Nature provides us with a wide variety of symbolic strings ranging from the sequences generated by the symbolic dynamics of nonlinear systems (Ebeling and Nicolis, 1991, 1992 ; Metropolis et al...

... Group Publishing Limited 2009 All IFS is a NS. Logic Fuzzy logic Topology Set theory In various recent papers, Smarandache (1998, 2002, 2003, 2005a, b) generalizes intuitionistic fuzzy sets (IFSs) and other kinds of sets to neutrosophic sets (NSs). In Smarandache (2005a...

.... Francisco Gallego Lupiáñez can be contacted at: fg_lupianez@mat.ucm.es © Emerald Group Publishing Limited 2009 Set theory Topology Cybernetics Fuzzy logic In various recent papers, Smarandache (2002, 2003, 2005) generalizes intuitionistic fuzzy sets (IFSs) and other kinds of sets...

... minimization Set theory Learning Statistical learning theory (SLT) started by Vapnik (1998) in 1960s and expanded in 1990s deals with the laws of statistical learning carried out for small data samples. It offers an efficient solution to machine learning methods such as neural networks designed...

... field of pansystem, and studies granularity from the view of panweighted field of pansystems. Haixia Li can be contacted at: polyxia@163.com © Emerald Group Publishing Limited 2008 Cybernetics Modelling Problem solving Set theory People often resolve problems by apperceiving...

...). All IFS is a NS. Francisco Gallego Lupiáñez can be contacted at: fg_lupianez@mat.ucm.es © Emerald Group Publishing Limited 2008 Cybernetics Set theory Topology In various recent papers, Smarandache (2002, 2003, 2005) generalizes intuitionistic fuzzy sets (IFSs...

... Systems theory Set theory Pansystems methodology (PM), put forward by Professor Wu, X. M. in 1976, proposes several basic pansystems relations such as micro‐macro, body‐shadow, motion‐rest, observo‐control, shengke, pan‐order, parallel‐serial, whole‐parts and, etc. Based on these pansystems...

...Yi Lin; Wujia Zhu; Yi Lin; Ningsheng Gong; Guoping Du Purpose The paper's purpose is to show a class of well‐used methods in set theory is invalid. Design/methodology/approach A conceptual approach is taken. Findings Based on the concept of Cauchy theater introduced elsewhere, it is proven...

... in naive set theory and modern axiomatic set theory, are also self‐contradicting non‐sets. Originality/value The concept of general Cauchy theater is introduced. Based on the preparation above, let us now prove the following theorem: Each uncountable, actual infinite set in the framework...

... in the system P^PIN, denoted symbolically ⊢A. Yi Lin can be contacted at: Jeffrey.forrest@sru.edu © Emerald Group Publishing Limited 2008 Cybernetics Set theory A truth–value assignment ν is a function with such a domain that contains every set...

... in the countable set theory was discussed in Zhu et al. by employing the method of analysis of allowing two different kinds of infinities. What was obtained is that various countable infinite sets, studied in the naïve and modern axiomatic set theories, are all incorrect concepts containing self...

... infinities. Originality/value The authors prove the relevant results on the reliability and the completeness of the logical system. Yi Lin can be contacted at: Jeffrey.forrest@sru.edu © Emerald Group Publishing Limited 2008 Cybernetics Set theory Based on all the concepts...

.../value This work is the non‐logical axiomatic part of the mathematical system of potential infinities: the axiomatic set theoretic system. At the end, the problem of consistency of this axiomatic set theory is discussed. © Emerald Group Publishing Limited 2008 Cybernetics Set theory All...

... the thinking logic of actual infinities in the naive and modern axiomatic set theory, and that Cauchy and Weierstrass completely applied that of potential infinities in the theory of limits. However, when it explores in depth the essential intensions of both potential and actual infinities, and after...

... Set theory Mathematics Number theory Let us first illustrate the concepts of “the number of natural numbers” and “the magnitude of a natural number.” In general, when a concept is defined, one has to make use of some more basic and more general concepts than the one to be introduced...

... and development of modern axiomatic set theory have provided a method to explain Russell's paradox. On the other hand, even though it has not been proven theoretically that there will not appear new paradoxes in modern axiomatic set theory, it has been indeed a century that no one has found a new paradox...

... Cantor's naive set theory to Zermelo's modern axiomatic set theory, actual infinities are strongly emphasized. On the surface, these theories seem to be established in the spirit of carrying out Law (I) of dividing into two. However, as a matter of fact, these theories did not completely exclude potential...