"The axiom of choice has many important consequences in set theory. It is used in the proof that every infinite set has a denumerable subset, and in the proof that every set has at least one well-ordering. From the latter, it follows that the power of every set is an aleph. Since any two alephs are comparable, so are any two transfinite powers of sets. The axiom of choice is also essential in the arithmetic of transfinite numbers." (R Bunn, "Developments in the Foundations of Mathematics, 1870-1910", 1980)
"Philosophical objections may be raised by the logical implications of building a mathematical structure on the premise of fuzziness, since it seems" (at least superficially) necessary to require that an object be or not be an element of a given set. From an aesthetic viewpoint, this may be the most satisfactory state of affairs, but to the extent that mathematical structures are used to model physical actualities, it is often an unrealistic requirement. [...] Fuzzy sets have an intuitively plausible philosophical basis. Once this is accepted, analytical and practical considerations concerning fuzzy sets are in most respects quite orthodox." (James Bezdek, 1981)
"A fractal is by definition a set for which the Hausdorff-Besicovitch dimension strictly exceeds the topological dimension." (Benoît Mandelbrot, "The Fractal Geometry of Nature", 1982)
"It is a remarkable empirical fact that mathematics can be based on set theory. More precisely, all mathematical objects can be coded as sets (in the cumulative hierarchy built by transfinitely iterating the power set operation, starting with the empty set). And all their crucial properties can be proved from the axioms of set theory. [...] At first sight, category theory seems to be an exception to this general phenomenon. It deals with objects, like the categories of sets, of groups etc. that are as big as the whole universe of sets and that therefore do not admit any evident coding as sets. Furthermore, category theory involves constructions, like the functor category, that lead from these large categories to even larger ones. Thus, category theory is not just another field whose set-theoretic foundation can be left as an exercise. An interaction between category theory and set theory arises because there is a real question: What is the appropriate set-theoretic foundation for category theory?" (Andreas Blass, "The interaction between category theory and set theory", Mathematical Applications of Category Theory, 1983)
"The progress of mathematics can be viewed as a movement from the infinite to the finite. At the start, the possibilities of a theory, for example, the theory of enumeration appear to be boundless. Rules for the enumeration of sets subject to various conditions, or combinatorial objects as they are often called, appear to obey an indefinite variety of and seem to lead to a welter of generating functions. We are at first led to suspect that the class of objects with a common property that may be enumerated is indeed infinite and unclassifiable." (Gian-Carlo Rota, [Preface to Combinatorial Enumeration by I.P. Goulden and D.M. Jackson], 1983)
"[...] problem solving generally proceeds by selective search through large sets of possibilities, using rules of thumb" (heuristics) to guide the search. Because the possibilities in realistic problem situations are generally multitudinous, trial-and-error search would simply not work; the search must be highly selective." (Herbert A Simon et al, "Decision Making and Problem Solving", Interfaces Vol. 17 (5), 1987)
"Fractal geometry is concerned with the description, classification, analysis, and observation of subsets of metric spaces (X, d). The metric spaces are usually, but not always, of an inherently 'simple' geometrical character; the subsets are typically geometrically 'complicated'. There are a number of general properties of subsets of metric spaces, which occur over and over again, which are very basic, and which form part of the vocabulary for describing fractal sets and other subsets of metric spaces. Some of these properties, such as openness and closedness, which we are going to introduce, are of a topological character. That is to say, they are invariant under homeomorphism." (Michael Barnsley, "Fractals Everwhere", 1988)
"In deterministic fractal geometry the focus is on those subsets of a space which are generated by, or possess invariance properties under, simple geometrical transformations of the space into itself. A simple geometrical transformation is one which is easily conveyed or explained to someone else. Usually they can be completely specified by a small set of parameters." (Michael Barnsley, "Fractals Everwhere", 1988)
"In deterministic geometry, structures are defined, communicated, and analysed, with the aid of elementary transformations such as affine transfor- transformations, scalings, rotations, and congruences. A fractal set generally contains infinitely many points whose organization is so complicated that it is not possible to describe the set by specifying directly where each point in it lies. Instead, the set may be defined by "the relations between the pieces." It is rather like describing the solar system by quoting the law of gravitation and stating the initial conditions. Everything follows from that. It appears always to be better to describe in terms of relationships." (Michael Barnsley, "Fractals Everwhere", 1988)
"In fractal geometry we are especially interested in the geometry of sets, and in the way they look when they are represented by pictures. Thus we use the restrictive condition of metric equivalence to start to define mathematically what we mean when we say that two sets are alike. However, in dynamical systems theory we are interested in motion itself, in the dynamics, in the way points move, in the existence of periodic orbits, in the asymptotic behavior of orbits, and so on. These structures are not damaged by homeomorphisms, as we will see, and hence we say that two dynamical systems are alike if they are related via a homeomorphism." (Michael Barnsley, "Fractals Everwhere", 1988)
"Set theory is peculiarly important [...] because mathematics can be exhibited as involving nothing but set-theoretical propositions about set-theoretical entities." (David M Armstrong, "A Combinatorial Theory of Possibility", 1989)
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