"The fuzzy set theory is taking the same logical approach as what people have been doing with the classical set theory: in the classical set theory, as soon as the two-valued characteristic function has been defined and adopted, rigorous mathematics follows; in the fuzzy set case, as soon as a multi-valued characteristic function" (the membership function) has been chosen and fixed, a rigorous mathematical theory can be fully developed." (Guanrong Chen & Trung Tat Pham, "Introduction to Fuzzy Sets, Fuzzy Logic, and Fuzzy Control Systems", 2001)
"A recurring concern has been whether set theory, which speaks of infinite sets, refers to an existing reality, and if so how does one ‘know’ which axioms to accept. It is here that the greatest disparity of opinion exists" (and the greatest possibility of using different consistent axiom systems)." (Paul Cohen, "Skolem and pessimism about proof in mathematics". Philosophical Transactions of the Royal Society A 363 (1835), 2005)
"Does set theory, once we get beyond the integers, refer to an existing reality, or must it be regarded, as formalists would regard it, as an interesting formal game? [...] A typical argument for the objective reality of set theory is that it is obtained by extrapolation from our intuitions of finite objects, and people see no reason why this has less validity. Moreover, set theory has been studied for a long time with no hint of a contradiction. It is suggested that this cannot be an accident, and thus set theory reflects an existing reality. In particular, the Continuum Hypothesis and related statements are true or false, and our task is to resolve them." (Paul Cohen, "Skolem and pessimism about proof in mathematics", Philosophical Transactions of the Royal Society A 363 (1835), 2005)
"Set theory is unusual in that it deals with remarkably simple but apparently ineffable objects. A set is a collection, a class, an ensemble, a batch, a bunch, a lot, a troop, a tribe. To anyone incapable of grasping the concept of a set, these verbal digressions are apt to be of little help. […] A set may contain finitely many or infinitely many members. For that matter, a set such as {} may contain no members whatsoever, its parentheses vibrating around a mathematical black hole. To the empty set is reserved the symbol Ø, the figure now in use in daily life to signify access denied or don’t go, symbolic spillovers, I suppose, from its original suggestion of a canceled eye." (David Berlinski, "Infinite Ascent: A short history of mathematics", 2005)
"The Continuum Hypothesis is the assertion that there are no cardinalities strictly between the cardinality of the integers and the cardinality of the continuum (the cardinality of the reals). [...] In logical terms, we say that the Continuum Hypothesis is independent from the other axioms of set theory, in particular it is independent from the Axiom of Choice." (Steven G Krantz, "Essentials of Topology with Applications”, 2009)
"In each branch of mathematics it is essential to recognize when two structures are equivalent. For example two sets are equivalent, as far as set theory is concerned, if there exists a bijective function which maps one set onto the other. Two groups are equivalent, known as isomorphic, if there exists a a homomorphism of one to the other which is one-to-one and onto. Two topological spaces are equivalent, known as homeomorphic, if there exists a homeomorphism of one onto the other." (Sydney A Morris, "Topology without Tears", 2011)
"Typical symbols used in mathematics are operationals, groupings, relations, constants, variables, functions, matrices, vectors, and symbols used in set theory, logic, number theory, probability, and statistics. Individual symbols may not have much effect on a mathematician’s creative thinking, but in groups they acquire powerful connections through similarity, association, identity, resemblance and repeated imagery. ¿ey may even create thoughts that are below awareness." (Joseph Mazur. "Enlightening symbols: a short history of mathematical notation and its hidden powers", 2014)
"A very basic observation concerning a fundamental property of the world we live in is the existence of objects that can be distinguished from each other. For the definition of a set, it is indeed of crucial importance that things have individuality, because in order to decide whether objects belong to a particular set they must be distinguishable from objects that are not in the set. Without having made the basic experience of individuality of objects, it would be difficult to imagine or appreciate the concept of a set." (Alfred S Posamentier & Bernd Thaller, "Numbers: Their tales, types, and treasures", 2015)
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