On Set Theory: On Sets (2010-)

"First, what are the 'graphs' studied in graph theory? They are not graphs of functions as studied in calculus and analytic geometry. They are" (usually finite) structures consisting of vertices and edges. As in geometry, we can think of vertices as points" (but they are denoted by thick dots in diagrams) and of edges as arcs connecting pairs of distinct vertices. The positions of the vertices and the shapes of the edges are irrelevant: the graph is completely specified by saying which vertices are connected by edges. A common convention is that at most one edge connects a given pair of vertices, so a graph is essentially just a pair of sets: a set of objects." (John Stillwell, "Mathematics and Its History", 2010)

"Today, most mathematicians have embraced the axiom of choice because of the order and simplicity it brings to mathematics in general. For example, the theorems that every vector space has a basis and every field has an algebraic closure hold only by virtue of the axiom of choice. Likewise, for the theorem that every sequentially continuous function is continuous. However, there are special places where the axiom of choice actually brings disorder. One is the theory of measure." (John Stillwell, "Roads to Infinity: The mathematics of truth and proof", 2010)

"A crucial difference between topology and geometry lies in the set of allowable transformations. In topology, the set of allowable transformations is much larger and conceptu￾ally much richer than is the set of Euclidean transformations. All Euclidean transformations are topological transformations, but most topological transformations are not Euclidean. Similarly, the sets of transformations that define other geometries are also topological transformations, but many topological transformations have no counterpart in these geometries. It is in this sense that topology is a generalization of geometry." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)

"A topological property is, therefore, any property that is preserved under the set of all homeomorphisms. […] Homeomorphisms generally fail to preserve distances between points, and they may even fail to preserve shapes." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011

"Although topology grew out of geometry - at least in the sense that it was initially concerned with sets of geometric points - it quickly evolved to include the study of sets for which no geometric representation is possible. This does not mean that topological results do not apply to geometric objects. They do. Instead, it means that topological results apply to a very wide class of mathematical objects, only some of which have a geometric interpretation." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)

"At first glance, sets are about as primitive a concept as can be imagined. The concept of a set, which is, after all, a collection of objects, might not appear to be a rich enough idea to support modern mathematics, but just the opposite proved to be true. The more that mathematicians studied sets, the more astonished they were at what they discovered, and astonished is the right word. The results that these mathematicians obtained were often controversial because they violated many common sense notions about equality and dimension." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)

"In dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values" (the bifurcation parameters) of a system causes a sudden 'qualitative' or topological change in its behaviour. Generally, at a bifurcation, the local stability properties of equilibria, periodic orbits or other invariant sets changes." (Gregory Faye, "An introduction to bifurcation theory", 2011)

"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)

"The details of the shapes of the neighborhoods are not important. If the two sets of neighborhoods satisfy the equivalence criterion, then any set that is open, closed, and so on with respect to one set of neighborhoods will be open, closed, and so on with respect to the other set of neighborhoods." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)

"[…] topologies are determined by the way the neighborhoods are defined. Neighborhoods, not individual points, are what matter. They determine the topological structure of the parent set. In fact, in topology, the word point conveys very little information at all." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)

"We use the term fuzzy logic to refer to all aspects of representing and manipulating knowledge that employ intermediary truth-values. This general, commonsense meaning of the term fuzzy logic encompasses, in particular, fuzzy sets, fuzzy relations, and formal deductive systems that admit intermediary truth-values, as well as the various methods based on them." (Radim Belohlavek & George J Klir, "Concepts and Fuzzy Logic", 2011)

"What distinguishes topological transformations from geometric ones is that topological transformations are more 'primitive'. They retain only the most basic properties of the sets of points on which they act." (John Tabak, "Beyond Geometry: A new mathematics of space and form", 2011)

"A function acts like a set of rules for turning some numbers into others, a machine with parts that we can manipulate to accomplish anything we can imagine." (David Perkins, "Calculus and Its Origins", 2012)

"A function is also sometimes referred to as a map or a mapping. This terminology is common in mathematics, but less so in physics or other scientific fields. The idea of a mapping is useful if one wants to think of a function as acting on an entire set of input values." (David P Feldman,"Chaos and Fractals: An Elementary Introduction", 2012)

"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)

"For group theoretic reasons the most impressive paradoxical decompositions occur in dimension at least three, but there are also interesting decompositions in lower dimensions. For example, one can partition a disc into finitely many subsets and rigidly rearrange these subsets to form a square of the same area as the original disc. Even without the Axiom of Choice it is possible to construct some counterintuitive subsets of the plane, so Choice cannot be held responsible for all that is counterintuitive in geometry. Some more radical alternatives to the Axiom of Choice obstruct such constructions more effectively." (Barnaby Sheppard, "The Logic of Infinity", 2014)

"Given any collection of infinite sets the Axiom of Choice tells us that there exists a set which has one element in common with each of the sets in the collection. Choice, which seems to be an intuitively sound principle, is equivalent to the much less plausible statement that every set has a well-ordering. Although many tried to prove Choice, they only seemed to be able to find equivalent statements which were just as difficult to prove." (Barnaby Sheppard, "The Logic of Infinity", 2014)

"[...] it is one thing to posit a set of axioms for a putative discipline. It is quite another to show that those proposed axioms are not mutually contradictory. A set of axioms that is not mutually contradictory is also called a consistent axiom system. Of course, a set of mutually contradictory axioms might be such that one can easily see a contradiction or maybe a simple argument could reveal a contradiction. More worrisome is the possibility that an argument that is very clever or very long or both is needed to reveal a contradiction. Given a mathematical theory that seems to be free of internal contradictions, the way mathematicians show it is, in fact, free of contradictions is by constructing what is called a model for the theory. This involves using another mathematical theory, say set theory, to produce a concrete mathematical object that satisfies the axioms of the theory being investigated. Once a model has been constructed, then we know that if set theory itself is free of internal contradictions, then the same is true of the theory being investigated. In practice, one does not often go all the way back to set theory to construct a model - mathematicians work with higher level constructions - but, in principle, they could start with set theory and work up from there." (Steven G Krantz & Harold R Parks, "A Mathematical Odyssey: Journey from the Real to the Complex", 2014)

"Since the membership relation is well-founded, well-founded relations can be defined on any class, however, the existence of a well-ordering of every set cannot be proved without appealing to the Axiom of Choice. Indeed, the assumption that every set has a well-ordering is equivalent to the Axiom of Choice." (Barnaby Sheppard, "The Logic of Infinity", 2014)

"Objections to the Axiom of Choice, either the strong or the weak version, are typically either philosophical, based on the intuitive temporal implausibility of making an infinite number of choices, or on the non-constructive nature of the axiom, or are based on a peculiar identification of continuum-based models of physics with the physical objects being modelled; properties of the model which are implied by the Axiom of Choice are deemed to be counterintuitive because the physical objects they model don’t have these properties. Motivated by these objections, or just for curiosity, several alternatives to Choice have been explored." (Barnaby Sheppard, "The Logic of Infinity", 2014)

"Some intuitive sets seem to have a less tangible existence than others, but it is difficult to draw a clear boundary between these different levels of concreteness. Naively we might expect to be able to include everything, to associate a set with any given property (precisely the set of all things having that property), but by a clever choice of property this leads to a contradiction. One of the tasks of set theory is to exclude these paradox-generating predicates and to describe the stable construction of new sets from old. Mathematics has a long history of creating concrete models of notions which at their birth were difficult abstract ideas."  (Barnaby Sheppard, "The Logic of Infinity", 2014)

"The most obvious variations of the Axiom of Choice are those that restrict the cardinality of the sets in question. Other variations impose relational restrictions between the sets. When the early set theorists tried to prove the Axiom of Choice they invariably ended up showing it is equivalent to some other statement that they were unable to prove. This collection of equivalent statements has grown to an enormous size. One of its striking features is that some of the statements seem intuitively obvious while others are either wildly counterintuitive or evade any kind of evaluation." (Barnaby Sheppard, "The Logic of Infinity", 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)

"Moreover, there is still another important observation that seems to be essential for the idea to group objects into a set: This is the human ability to recognize similarities in different objects. Usually, a collection, or group, consists of objects that somehow belong together, objects that share a common property. While a mathematical set could also be a completely arbitrary collection of unrelated objects, this is usually not what we want to count. We count coins or hours or people, but we usually do not mix these categories." (Alfred S Posamentier & Bernd Thaller, "Numbers: Their tales, types, and treasures", 2015)

"The invariance principle states that the result of counting a set does not depend on the order imposed on its elements during the counting process. Indeed, a mathematical set is just a collection without any implied ordering. A set is the collection of its elements - nothing more." (Alfred S Posamentier & Bernd Thaller, "Numbers: Their tales, types, and treasures", 2015)

"[…] the role that symmetry plays is not confined to material objects. Symmetries can also refer to theories and, in particular, to quantum theory. For if the laws of physics are to be invariant under changes of reference frames, the set of all such transformations will form a group. Which transformations and which groups depends on the systems under consideration." (William H Klink & Sujeev Wickramasekara, "Relativity, Symmetry and the Structure of Quantum Theory I: Galilean quantum theory", 2015)

"When we extend the system of natural numbers and counting to embrace infinite cardinals, the larger system need not have all of the properties of the smaller one. However, familiarity with the smaller system leads us to expect certain properties, and we can become confused when the pieces don’t seem to fit. Insecurity arose when the square of a complex number violated the real number principle that all squares are positive. This was resolved when we realised that the complex numbers cannot be ordered in the same way as their subset of reals." (Ian Stewart & David Tall, "The Foundations of Mathematics" 2nd Ed., 2015)

"Independence of two events is not a property of the events themselves, rather it is a property that comes from the probabilities of the events and their intersection. This is in contrast to mutually exclusive events, which have the property that they contain no elements in common. Two mutually exclusive events each with non-zero probability cannot be independent. Their intersection is the empty set, so it must have probability zero, which cannot equal the product of the probabilities of the two events!" (William M Bolstad & James M Curran, "Introduction to Bayesian Statistics" 3rd Ed., 2017)

"Making a map is the physical production including conceptualization and design. Mapping is the mental interpretation of the world and although it must precede the map, it does not necessarily result in a map artifact. Mapping defined in mathematics is the correspondence between each element of a given set with each element of another. Similarly in linguistics emphasis is on the correspondence between associated elements of different types. For designers all drawings are maps - they represent relationships between objects, places and ideas." (Winifred E Newman, "Data Visualization for Design Thinking: Applied Mapping", 2017)

"Mathematics is the domain of all formal languages, and allows the expression of arbitrary statements" (most of which are uncomputable). Computation may be understood in terms of computational systems, for instance via defining states" (which are sets of discernible differences, i.e. bits), and transition functions that let us derive new states." (Joscha Bach, "The Cortical Conductor Theory: Towards Addressing Consciousness in AI Models", 2017)

"The elements of this cloud of uncertainty (the set of all possible errors) can be described in terms of probability. The center of the cloud is the number zero, and elements of the cloud that are close to zero are more probable than elements that are far away from that center. We can be more precise in this definition by defining the cloud of uncertainty in terms of a mathematical function, called the probability distribution." (David S Salsburg, "Errors, Blunders, and Lies: How to Tell the Difference", 2017)

"The Axiom of Choice says that it is possible to make an infinite number of arbitrary choices. […] Mathematicians don’t exactly care whether or not the Axiom of Choice holds over all, but they do care whether you have to use it in any given situation or not." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)

"Quaternions are not actual extensions of imaginary numbers, and they are not taking complex numbers into a multi-dimensional space on their own. Quaternion units are instances of some number-like object type, identified collectively, but they are not numbers" (be it real or imaginary). In other words, they form a closed, internally consistent set of object instances; they can of course be plotted visually on a multi-dimensional space but this only is a visualization within their own definition." (Huseyin Ozel, "Redefining Imaginary and Complex Numbers, Defining Imaginary and Complex Objects", 2018)

On Set Theory: On Sets (2000-2009)

"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)

"An organizing frame provides a topology for the space it organizes; that is, it provides a set of organizing relations among the elements in space. When two spaces share the same organizing frame, they share the corresponding topology and so can easily be put into correspondence. Establishing a cross-space mapping between inputs becomes straightforward." (Gilles Fauconnier, "The Way We Think: Conceptual Blending and The Mind's Hidden Complexities", 2002)

"Fuzzy relations are developed by allowing the relationship between elements of two or more sets to take on an infinite number of degrees of relationship between the extremes of 'completely related' and 'not related', which are the only degrees of relationship possible in crisp relations. In this sense, fuzzy relations are to crisp relations as fuzzy sets are to crisp sets; crisp sets and relations are more constrained realizations of fuzzy sets and relations. " (Timothy J Ross & W Jerry Parkinson, "Fuzzy Set Theory, Fuzzy Logic, and Fuzzy Systems", 2002)

"Systems-people everywhere share certain attributes, but each specific System tends to attract people with specific sets of traits. […] Systems attract not only Systems-people who have qualities making for success within the System; they also attract individuals who possess specialized traits adapted to allow them to thrive at the expense of the System; i.e., persons who parasitize the System." (John Gall, "Systemantics: The Systems Bible", 2002)

"The Smale's horseshoe is the classical example of a structurally stable chaotic system: Its dynamical properties do not change under small perturbations, such as changes in control parameters. This is due to the horseshoe map being hyperbolic (i.e., the stable and unstable manifolds are transverse at each point of the invariant set)." (Robert Gilmore & Marc Lefranc, "TheTopologyof Chaos: Alice in Stretch and Squeezeland", 2002)

"Combinatorics could be described as the study of arrangements of objects according to specified rules. We want to know, first, whether a particular arrangement is possible at all, and, if so, in how many different ways it can be done. Algebraic and even probabilistic methods play an increasingly important role in answering these questions. If we have two sets of arrangements with the same cardinality, we might want to construct a natural bijection between them. We might also want to have an algorithm for constructing a particular arrangement or all arrangements, as well as for computing numerical characteristics of them; in particular, we can consider optimization problems related to such arrangements. Finally, we might be interested in an even deeper study, by investigating the structural properties of the arrangements. Methods from areas such as group theory and topology are useful here, by enabling us to study symmetries of the arrangements, as well as topological properties of certain spaces associated with them, which translate into combinatorial properties." (Cristian Lenart, "The Many Faces of Modern Combinatorics", 2003)

"Essentially, the theory of finite state machines is that of computation. It postulates two finite sets of external states called 'input states' and 'output states', one finite set of 'internal states', and two explicitly defined operations" (computations) which determine the instantaneous and temporal relations between these states." (Heinz von Foerster, "Understanding Understanding: Essays on Cybernetics and Cognition", 2003)

"Knowledge is encoded in models. Models are synthetic sets of rules, and pictures, and algorithms providing us with useful representations of the world of our perceptions and of their patterns." (Didier Sornette, "Why Stock Markets Crash - Critical Events in Complex Systems", 2003)

"A symmetry is a set of transformations applied to a structure, such that the transformations preserve the properties of the structure." (Philip Dorrell, "What is Music?: Solving a Scientific Mystery", 2004)

"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)

"Minkowski calls a spatial point existing at a temporal point a world point. These coordinates are now called 'space-time coordinates'. The collection of all imaginable value systems or the set of space-time coordinates Minkowski called the world. This is now called the manifold. The manifold is four-dimensional and each of its space-time points represents an event." (Friedel Weinert," The Scientist as Philosopher: Philosophical Consequences of Great Scientific Discoveries", 2005)

"The Nash equilibrium is often rationalized using a story about how people think and how their behaviour is related to their thoughts. Economists generally assume that, from a set of alternatives, a player will actively choose the one he likes best. This is the assumption of economic rationality, one of the core assumptions of standard game theory. Rationality alone will not predict behaviour in a game, but it leads us to single out the member of any inducement correspondence that yields the greatest payoff for the player that is making the choice. The resulting behaviour is sometimes described as 'myopic' because it fails to take into account how other players might respond to a given choice." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 2005)

"One important and standardized block of information in the formal descriptions used by game theorists is called a game form. A form specifies the payoffs associated with every possible combination of decisions. There are several widely used forms, including the strate-gic form, typically presented in a matrix, the extensive form, which is usually represented as a tree, and the characteristic function form, expressed as a function on subsets of players." (David Robinson & David Goforth, "The Topology of the 2×2 Games: A New Periodic Table", 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)

"A group is a set along with a rule that tells how to combine any two elements in the set to get another element in the set. We usually use the word composition to describe the act of combining two elements of the group to get a third." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"Each fuzzy set is uniquely defined by a membership function. […] There are two approaches to determining a membership function. The first approach is to use the knowledge of human experts. Because fuzzy sets are often used to formulate human knowledge, membership functions represent a part of human knowledge. Usually, this approach can only give a rough formula of the membership function and fine-tuning is required. The second approach is to use data collected from various sensors to determine the membership function. Specifically, we first specify the structure of membership function and then fine-tune the parameters of membership function based on the data." (Huaguang Zhang & Derong Liu, "Fuzzy Modeling and Fuzzy Control", 2006)

"Numerical invariants and invariant properties enable us to distinguish certain topological spaces. We can go further and associate with a topological space a set having an algebraic structure. The fundamental group is the most basic of such possibilities. It not only provides a useful invariant for topological spaces, but the algebraic operation of multiplication defined for this group reflects the global structure of the space." (Robert Messer & Philip Straffin, "Topology Now!", 2006)

"The set of complex numbers is another example of a field. It is handy because every polynomial in one variable with integer coefficients can be factored into linear factors if we use complex numbers. Equivalently, every such polynomial has a complex root. This gives us a standard place to keep track of the solutions to polynomial equations." (Avner Ash & Robert Gross, "Fearless Symmetry: Exposing the hidden patterns of numbers", 2006)

"Venn diagrams visually ground symbolic logic and abstract set operations. They do not ground probability. Their common overuse in introducing probability, especially in teaching, can have undesirable consequences." (R W Oldford & W H Cherry, "Picturing Probability: the poverty of Venn diagrams, the richness of Eikosograms", 2006)

"Mathematical structures are usually introduced either within the framework of set theory, or within higher order logic, that is, type theory. Both presentations turn out to be essentially equivalent; professional mathematicians and model theorists favor the first approach, while some logicians like Russell and Carnap explored the second way." (N C A da Costa & F A Doria, "Janus-Faced Physics: On Hilbert's 6th Problem" [in Cristian S Calude, "Randomness & Complexity, from Leibniz to Chaitin"] 2007)

"A little ingenuity is involved, but once a couple of tricks are learnt, it is not hard to show many sets of numbers are countable, which is the term we use to mean that the set can be listed in the same fashion as the counting numbers. Otherwise a set is called uncountable." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"The rationals therefore also form a countable set, as do the euclidean numbers, and indeed if we consider the set of all numbers that arise from the rationals through taking roots of any order, the collection produced is still countable. We can even go beyond this: the collection of all algebraic numbers, which are those that are solutions of ordinary polynomial equations∗ form a collection that can, in principle, be arrayed in an infinite list: that is to say it is possible, with a little more crafty argument, to describe a systematic listing that sweeps them all out." (Peter M Higgins, "Number Story: From Counting to Cryptography", 2008)

"The word ‘symmetry’ conjures to mind objects which are well balanced, with perfect proportions. Such objects capture a sense of beauty and form. The human mind is constantly drawn to anything that embodies some aspect of symmetry. Our brain seems programmed to notice and search for order and structure. Artwork, architecture and music from ancient times to the present day play on the idea of things which mirror each other in interesting ways. Symmetry is about connections between different parts of the same object. It sets up a natural internal dialogue in the shape." (Marcus du Sautoy, "Symmetry: A Journey into the Patterns of Nature", 2008)

"Topology makes it possible to explain the general structure of the set of solutions without even knowing their analytic expression." (Michael I. Monastyrsky, "Riemann, Topology, and Physics" 2nd Ed., 2008)

"A graph enables us to visualize a relation over a set, which makes the characteristics of relations such as transitivity and symmetry easier to understand. […] Notions such as paths and cycles are key to understanding the more complex and powerful concepts of graph theory. There are many degrees of connectedness that apply to a graph; understanding these types of connectedness enables the engineer to understand the basic properties that can be defined for the graph representing some aspect of his or her system. The concepts of adjacency and reachability are the first steps to understanding the ability of an allocated architecture of a system to execute properly." (Dennis M Buede, "The Engineering Design of Systems: Models and methods", 2009)

"One of the nice features of the metric space setting is that all topological notions can be formulated in terms of sequences. Such is not the case in an arbitrary topological space. [...] Whereas a sequence is modeled on the natural numbers, a net is modeled on a more general object called a directed set. The general feel of the subject is similar to that for sequences, but it is rather more abstract." (Steven G Krantz, "Essentials of Topology with Applications”, 2009)

"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)

"The most fundamental tool in the subject of point-set topology is the homeomorphism. This is the device by means of which we measure the equivalence of topological spaces." (Steven G Krantz, "Essentials of Topology with Applications”, 2009)

On Set Theory: On Sets (1990-1999)

"Summarizing, we have indicated why homoclinic points imply chaos. The converse is less formalizable, but I believe it is basically true. However,  there is a stronger notion of chaos, which we might call full chaos, where the set of initial conditions with sensitive dependence has positive or even full measure. In this case the corresponding homoclinic point could well be associated with a chaotic attractor, such as the Lorenz attractor. " (Steven Smale, "What is chaos?", 1990)

"And you should not think that the mathematical game is arbitrary and gratuitous. The diverse mathematical theories have many relations with each other: the objects of one theory may find an interpretation in another theory, and this will lead to new and fruitful viewpoints. Mathematics has deep unity. More than a collection of separate theories such as set theory, topology, and algebra, each with its own basic assumptions, mathematics is a unified whole." (David Ruelle, "Chance and Chaos", 1991)

"Mathematics has deep unity. More than a collection of separate theories such as set theory, topology, and algebra, each with its own basic assumptions, mathematics is a unified whole. Mathematics is a great kingdom, and that kingdom belongs to those who see." (David Ruelle, "Chance and Chaos", 1991)

"Systems theory pursues the scientific exploration and understanding of systems that exist in the various realms of experience, in order to arrive at a general theory of systems: an organized expressing of sets of interrelated concepts and principles that apply to all systems." (Béla H Bánáthy, "Systems Design of Education", 1991)

"What is an attractor? It is the set on which the point P, representing the system of interest, is moving at large times (i.e., after so-called transients have died out). For this definition to make sense it is important that the external forces acting on the system be time independent" (otherwise we could get the point P to move in any way we like). It is also important that we consider dissipative systems (viscous fluids dissipate energy by self-friction). Dissipation is the reason why transients die out. Dissipation is the reason why, in the infinite-dimensional space representing the system, only a small set" (the attractor) is really interesting." (David Ruelle, "Chance and Chaos", 1991)

"The systems' basic components are treated as sets of rules. The systems rely on three key mechanisms: parallelism, competition, and recombination. Parallelism permits the system to use individual rules as building blocks, activating sets of rules to describe and act upon the changing situations. Competition allows the system to marshal its rules as the situation demands, providing flexibility and transfer of experience. This is vital in realistic environments, where the agent receives a torrent of information, most of it irrelevant to current decisions. The procedures for adaptation - credit assignment and rule discovery - extract useful, repeatable events from this torrent, incorporating them as new building blocks. Recombination plays a key role in the discovery process, generating plausible new rules from parts of tested rules. It implements the heuristic that building blocks useful in the past will prove useful in new, similar contexts." (John H Holland, "Complex Adaptive Systems", Daedalus Vol. 121" (1), 1992)

"An attractor that consists of an infinite number of curves, surfaces, or higher-dimensional manifolds - generalizations of surfaces to multidimensional space - often occurring in parallel sets, with a gap between any two members of the set, is called a strange attractor." (Edward N Lorenz, "The Essence of Chaos", 1993)

"An infinite set is any set from which you can remove some members without reducing its size. In fact, we can start with an infinite set, remove an infinite set, and still have an infinite set left behind […]." (Alexander Humez et al, "Zero to Lazy Eight: The romance of numbers", 1993)

"Attractors are examples of invariant sets - sets that will consist of precisely the same points if each point is replaced by the point to which it is mapped, When there is more than one attractor, each basin of attraction is an invariant set, as is the basin boundary, sometimes called a separatrix. There is still another invariant set, which connects the attractors when there are more than one, and which by analogy ought to be called a 'connectrix', but generally, together with the attractors that it connects, is called the attracting set. Despite its name, the attracting set should not be confused with the set of attractors, which is sometimes only a portion of it.(Edward N Lorenz, "The Essence of Chaos", 1993)

"Fuzzy entropy measures the fuzziness of a fuzzy set. It answers the question 'How fuzzy is a fuzzy set?' And it is a matter of degree. Some fuzzy sets are fuzzier than others. Entropy means the uncertainty or disorder in a system. A set describes a system or collection of things. When the set is fuzzy, when elements belong to it to some degree, the set is uncertain or vague to some degree. Fuzzy entropy measures this degree. And it is simple enough that you can see it in a picture of a cube." (Bart Kosko, "Fuzzy Thinking: The new science of fuzzy logic", 1993)

"In any mathematical study, the first thing to do is specify the type or category of objects to be investigated. In topology, the most general possible objects of study are sets of points with just enough structure to be able to define continuous functions." (L Christine Kinsey. "Topology of Surfaces", 1993)

"Zero is the only number that is neither positive nor negative. As such, it represents a quantity: If three is the name we give to the number of items in a trilogy, a trinity, or a triad, zero is our name for the number of items in an empty, or null set, i.e., one having no members. This is not the same as saying the set doesn't exist; in fact, we can and do make valid assertions about null sets […]" (Alexander Humez et al, "Zero to Lazy Eight: The romance of numbers", 1993)

"Geometry and topology most often deal with geometrical figures, objects realized as a set of points in a Euclidean space (maybe of many dimensions). It is useful to view these objects not as rigid (solid) bodies, but as figures that admit continuous deformation preserving some qualitative properties of the object. Recall that the mapping of one object onto another is called continuous if it can be determined by means of continuous functions in a Cartesian coordinate system in space. The mapping of one figure onto another is called homeomorphism if it is continuous and one-to-one, i.e. establishes a one-to-one correspondence between points of both figures." (Anatolij Fomenko, "Visual Geometry and Topology", 1994)

"A fuzzy set can be defined mathematically by assigning to each possible individual in the universe of discourse a value representing its grade of membership in the fuzzy set. This grade corresponds to the degree to which that individual is similar or compatible with the concept represented by the fuzzy set. Thus, individuals may belong in the fuzzy act to a greater or lesser degree as indicated by a larger or smaller membership grade. As already mentioned, these membership grades are very often represented by real-number values ranging in the closed interval between 0 and 1." (George J Klir & Bo Yuan, "Fuzzy Sets and Fuzzy Logic: Theory and Applications", 1995)

"As with subtle bifurcations, catastrophes also involve a control parameter. When the value of that parameter is below a bifurcation point, the system is dominated by one attractor. When the value of that parameter is above the bifurcation point, another attractor dominates. Thus the fundamental characteristic of a catastrophe is the sudden disappearance of one attractor and its basin, combined with the dominant emergence of another attractor. Any type of attractor static, periodic, or chaotic can be involved in this. Elementary catastrophe theory involves static attractors, such as points. Because multidimensional surfaces can also attract" (together with attracting points on these surfaces), we refer to them more generally as attracting hypersurfaces, limit sets, or simply attractors." (Courtney Brown, "Chaos and Catastrophe Theories", 1995)

"To select an appropriate fuzzy implication for approximate reasoning under each particular situation is a difficult problem. Although some theoretically supported guidelines are now available for some situations, we are still far from a general solution to this problem." (George Klir, "Fuzzy sets and fuzzy logic", 1995)

"In abstract mathematics, special attention is given to particular properties of numbers. Then those properties are taken in a very pure (and primitive) form. Those properties in pure form are then assigned to a given set. Therefore, by studying in details the internal mathematical structure of a set, we should be able to clarify the meaning of original properties of the objects. Likewise, in set theory, numbers disappear and only the concept of sets and characteristic properties of sets remain." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)

"Set theory has a dual role in mathematics. In pure mathematics, it is the place where questions about infinity are studied. Although this is a fascinating study of permanent interest, it does not account for the importance of set theory in applied areas. There the importance stems from the fact that set theory provides an incredibly versatile toolbox for building mathematical models of various phenomena." (Jon Barwise & Lawrence Moss, "Vicious Circles: On the Mathematics of Non-Wellfounded Phenomena", 1996)

"Venn diagrams are widely used to solve problems in set theory and to test the validity of syllogisms in logic. […] However, it is a fact that Venn diagrams are not considered valid proofs, but heuristic tools for finding valid formal proofs." (Sun-Joo Shin, "Situation-Theoretic Account of Valid Reasoning with Venn Diagrams", [in "Logical Reasoning with Diagrams"], 1996)

"Math has its own inherent logic, its own internal truth. Its beauty lies in its ability to distill the essence of truth without the messy interference of the real world. It’s clean, neat, above it all. It lives in an ideal universe built on the geometer’s perfect circles and polygons, the number theorist’s perfect sets. It matters not that these objects don’t exist in the real world. They are articles of faith." (K C Cole, "The Universe and the Teacup: The Mathematics of Truth and Beauty", 1997)

"A formal system consists of a number of tokens or symbols, like pieces in a game. These symbols can be combined into patterns by means of a set of rules which defines what is or is not permissible (e.g. the rules of chess). These rules are strictly formal, i.e. they conform to a precise logic. The configuration of the symbols at any specific moment constitutes a 'state' of the system. A specific state will activate the applicable rules which then transform the system from one state to another. If the set of rules governing the behaviour of the system are exact and complete, one could test whether various possible states of the system are or are not permissible." (Paul Cilliers, "Complexity and Postmodernism: Understanding Complex Systems", 1998)

"The discovery of complex numbers was the last in a sequence of discoveries that gradually filled in the set of all numbers, starting with the positive integers (finger counting) and then expanding to include the positive rationals and irrational reals, negatives, and then finally the complex." (Paul J Nahin, "An Imaginary Tale: The History of √-1", 1998)

"Algebraic topology studies properties of a narrower class of spaces, - basically the classical objects of mathematics: spaces given by systems of algebraic and functional equations, surfaces lying in Euclidean space, and other sets which in mathematics are called manifolds. Examining the narrower class of spaces permits deeper penetration into their structure." (Michael I Monastyrsky, "Riemann, Topology, and Physics", 1999)

"[...] fuzzy logic [FL] has many distinct facets - facets which overlap and have unsharp boundaries. Among these facets there are four that stand out in importance. They are" (i) the logical facet;" (ii) the set-theoretic facet:" (iii) the relational facet, and" (iv) the epistemic facet. […] The logical facet of FL, FL/L, is a logical system or, more accurately, a collection of logical systems which includes as a special case both two-valued and multiple-valued systems. […] The set-theoretic facet of FL, FL/S, is concerned with classes or sets whose boundaries are not sharply defined. […] The relational facet of FL, FL/R, is concerned in the main with representation and manipulation of imprecisely defined functions and relations. […] The epistemic facet of FL, FL/E, is linked to its logical facet and is centered on applications of FL to knowledge representation, information systems, fuzzy databases and the theories of possibility imprecise probabilities." (Lotfi A Zadeh, "The Birth and Evolution of Fuzzy Logic: A personal perspective", 1999)

"One of the basic tasks of topology is to learn to distinguish nonhomeomorphic figures. To this end one introduces the class of invariant quantities that do not change under homeomorphic transformations of a given figure. The study of the invariance of topological spaces is connected with the solution of a whole series of complex questions: Can one describe a class of invariants of a given manifold? Is there a set of integral invariants that fully characterizes the topological type of a manifold? and so forth." (Michael I Monastyrsky, "Riemann, Topology, and Physics", 1999)

"The most fundamental tool in the subject of point-set topology is the homeomorphism. This is the device by means of which we measure the equivalence of topological spaces." (Steven G Krantz, "Essentials of Topology with Applications”, 2009)

On Set Theory: On Sets (1970-1979)

"Related is the idea of structural stability and certain variations. This kind of is a property of a dynamical system itself (not of a or orbit) and asserts that nearby dynamical systems have the same structure. The 'same structure' can be defined in several interesting ways, but the basic idea is that two dynamical systems have 'the same structure' if they have the same gross behavior, or the same qualitative behavior. For example, the original definition of 'same structure' of two dynamical systems was that there was an orbit preserving continuous transformation between them. This yields the definition of structural stability proper. It is a recent theorem that every compact manifold admits structurally stable systems, and almost all gradient dynamical systems are structurally stable. But while there exists a rich set of structurally stable systems, there are also important examples which are not stable, and have good but weaker stability properties." (Stephen Smale, "Personal perspectives on mathematics and mechanics", 1971)

"For a long time the aim of combinatorial analysis was to count the different ways of arranging objects under given circumstances. Hence, many of the traditional problems of analysis or geometry which are concerned at a certain moment with finite structures, have a combinatorial character. Today, combinatorial analysis is also relevant to problems of existence, estimation and structuration, like all other parts of mathematics, but exclusively for finite sets." (Louis Comtet, "Advanced Combinatorics", 1974)

"For the mathematician, the physical way of thinking is merely the starting point in a process of abstraction or idealization. Instead of being a dot on a piece of paper or a particle of dust suspended in space, a point becomes, in the mathematician's ideal way of thinking, a set of numbers or coordinates. In applied mathematics we must go much further with this process because the physical problems under consideration are more complex. We first view a phenomenon in the physical way, of course, but we must then go through a process of idealization to arrive at a more abstract representation of the phenomenon which will be amenable to mathematical analysis." (Peter Lancaster, "Mathematics: Models of the Real World", 1976)

"Combinatorial analysis, or combinatorics, is the study of how things can be arranged. In slightly less general terms, combinatorial analysis embodies the study of the ways in which elements can be grouped into sets subject to various specified rules, and the properties of those groupings. […] Combinatorial analysis often asks for the total number of different ways that certain things can be combined according to certain rules." (Martin Gardner, "Aha! Insight", 1978)

"Graph theory is the study of sets of points that are joined by lines." (Martin Gardner,"Aha! Insight", 1978)

"The theory of the nature of mathematics is extremely reactionary. We do not subscribe to the fairly recent notion that mathematics is an abstract language based, say, on set theory. In many ways, it is unfortunate that philosophers and mathematicians like Russell and Hilbert were able to tell such a convincing story about the meaning-free formalism of mathematics. [...] Mathematics is a way of preparing for decisions through thinking. Sets and classes provide one way to subdivide a problem for decision preparation; a set derives its meaning from decision making, and not vice versa." (C West Churchman et al, "Thinking for Decisions Deduction Quantitative Methods", 1975)

"In modern mathematics set theory plays the role of the frame, the skeleton which joins all its parts into a single whole but cannot be seen from the outside and does not come in direct contact with the external world. This situation can be truly understood and the formal and contentual aspects of mathematics combined only from the 'linguistic' point of view regarding mathematics." (Valentin F Turchin, "The Phenomenon of Science: a cybernetic approach to human evolution", 1977)

"To use set theory in the way it is used by modern mathematics, however, it is not at all necessary to force one's imagination and try to picture actual infinity. The 'sets' which are used in mathematics are simply symbols, linguistic objects used to construct models of reality. The postulated attributes of these objects correspond partially to intuitive concepts of aggregateness and potential infinity; therefore intuition helps to some extent in the development of set theory, but sometimes it also deceives. Each new mathematical (linguistic) object is defined as a 'set' constructed in some particular way. This definition has no significance for relating the object to the external world, that is for interpreting it: it is needed only to coordinate it with the frame of mathematics, to mesh the internal wheels of mathematical models. So the language of set theory is in fact a metalanguage in relation to the language of contentual mathematics, and in this respect it is similar to the language of logic. If logic is the theory of proving mathematical statements, then set theory is the theory of constructing mathematical linguistic objects." (Valentin F Turchin, "The Phenomenon of Science: a cybernetic approach to human evolution", 1977)

"A surface is a topological space in which each point has a neighbourhood homeomorphic to the plane, ad for which any two distinct points possess disjoint neighbourhoods. […] The requirement that each point of the space should have a neighbourhood which is homeomorphic to the plane fits exactly our intuitive idea of what a surface should be. If we stand in it at some point (imagining a giant version of the surface in question) and look at the points very close to our feet we should be able to imagine that we are standing on a plane. The surface of the earth is a good example. Unless you belong to the Flat Earth Society you believe it to,be (topologically) a sphere, yet locally it looks distinctly planar. Think more carefully about this requirement: we ask that some neighbourhood of each point of our space be homeomorphic to the plane. We have then to treat this neighbourhood as a topological space in its own right. But this presents no difficulty; the neighbourhood is after all a subset of the given space and we can therefore supply it with the subspace topology." (Mark A Armstrong, "Basic Topology", 1979)

"Having vegetated on the fringes of mathematical science for centuries, combinatorics has now burgeoned into one of the fastest growing branches of mathematics – undoubtedly so if we consider the number of publications in this field, its applications in other branches of mathematics and in other sciences, and also the interest of scientists, economists and engineers in combinatorial structures. The mathematical world was attracted by the successes of algebra and analysis and only in recent years has it become clear, due largely to problems arising from economics, statistics, electrical engineering and other applied sciences, that combinatorics, the study of finite sets and finite structures, has its own problems and principles. These are independent of those in algebra and analysis but match them in difficulty, practical and theoretical interest and beauty." (László Lovász, "Combinatorial Problems and Exercises", 1979)

"If we gather more and more data and establish more and more associations, however, we will not finally find that we know something. We will simply end up having more and more data and larger sets of correlations." (Kenneth N Waltz, "Theory of International Politics Source: Theory of International Politics", 1979)

"In set theory, perhaps more than in any other branch of mathematics, it is vital to set up a collection of symbolic abbreviations for various logical concepts. Because the basic assumptions of set theory are absolutely minimal, all but the most trivial assertions about sets tend to be logically complex, and a good system of abbreviations helps to make otherwise complex statements." (Keith Devlin, "Sets, Functions, and Logic: An Introduction to Abstract Mathematics", 1979)

"It is now generally recognized that the field of combinatorics has, over the past years, evolved into a fully-fledged branch of discrete mathematics whose potential with respect to computers and the natural sciences is only beginning to be realized. Still, two points seem to bother most authors: The apparent difficulty in defining the scope of combinatorics and the fact that combinatorics seems to consist of a vast variety of more or less unrelated methods and results. As to the scope of the field, there appears to be a growing consensus that combinatorics should be divided into three large parts: (i) Enumeration, including generating functions, inversion, and calculus of finite differences; (ii) Order Theory, including finite sets and lattices, matroids, and existence results such as Hall's and Ramsey's; (iii) Configurations, including designs, permutation groups, and coding theory." (Martin Aigner, "Combinatorial Theory", 1979)

On Set Theory: On Sets (1960-1969)

"Every mathematician agrees that every mathematician must know some set theory; the disagreement begins in trying to decide how much is some. [...] The student's task in learning set theory is to steep himself in unfamiliar but essentially shallow generalities till they become so familiar that they can be used with almost no conscious effort. In other words, general set theory is pretty trivial stuff really, but, if you want to be a mathematician, you need some, and here it is; read it, absorb it, and forget it [...] the language and notation are those of ordinary informal mathematics. A more important way in which the naive point of view predominates is that set theory is regarded as a body of facts, of which the axioms are a brief and convenient summary; in the orthodox axiomatic view the logical relations among various axioms are the central objects of study." (Paul R Halmos, "Naive Set Theory", 1960)

"A set is formed by the grouping together of single objects into a whole. A set is a plurality thought of as a unit. If these or similar statements were set down as definitions, then it could be objected with good reason that they define idem per idemi or even obscurum per obscurius. However, we can consider them as expository, as references to a primitive concept, familiar to us all, whose resolution into more fundamental concepts would perhaps be neither competent nor necessary." (Felix Hausdorff, "Set Theory", 1962)

"A special role is played in the theory of metric spaces by the class of open spheres within the class of all open sets. The main feature of their relationship is that the open sets coincide with all unions of open spheres, and it follows from this that the continuity of a mapping can be expressed either in terms of open spheres or in terms of open sets, at our convenience." (George F Simmons, "Introduction to Topology and Modern Analysis", 1963)

"In many branches of mathematics - in geometry as well as analysis - it has been found extremely convenient to have available a notion of distance which is applicable to the elements of abstract sets. A metric space (as we define it below) is nothing more than a non-empty set equipped with a concept of distance which is suitable for the treatment of convergent sequences in the set and continuous functions defined on the set." (George F Simmons, "Introduction to Topology and Modern Analysis", 1963)

"It is sometimes said that mathematics is the study of sets and functions. Naturally, this oversimplifies matters; but it does come as close to the truth as an aphorism can." (George F Simmons, "Introduction to Topology and Modern Analysis", 1963)

"The study of sets and functions leads two ways. One path goes down, into the abysses of logic, philosophy, and the foundations of mathematics. The other goes up, onto the highlands of mathematics itself, where these concepts are indispensable in almost all of pure mathematics as it is today." (George F Simmons, "Introduction to Topology and Modern Analysis", 1963)

"A fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership (characteristic) function which assigns to each object a grade of membership ranging between zero and one. The notions of inclusion, union, intersection, complement, relation, convexity, etc., are extended to such sets, and various properties of these notions in the context of fuzzy sets are established. In particular, a separation theorem for convex fuzzy sets is proved without requiring that the fuzzy sets be disjoint." (Lotfi A Zadeh, "Fuzzy Sets", 1965)

"It is paradoxical that while mathematics has the reputation of being the one subject that brooks no contradictions, in reality it has a long history of successful living with contradictions. This is best seen in the extensions of the notion of number that have been made over a period of 2500 years. From limited sets of integers, to infinite sets of integers, to fractions, negative numbers, irrational numbers, complex numbers, transfinite numbers, each extension, in its way, overcame a contradictory set of demands." (Philip J Davis,"The Mathematics of Matrices", 1965)

"The notion of a fuzzy set provides a convenient point of departure for the construction of a conceptual framework which parallels in many respects the framework used in the case of ordinary sets, but is more general than the latter and, potentially, may prove to have a much wider scope of applicability, particularly in the fields of pattern classification and information processing. Essentially, such a framework provides a natural way of dealing with problems in which the source of imprecision is the absence of sharply denned criteria of class membership rather than the presence of random variables." (Lotfi A Zadeh, "Fuzzy Sets", 1965)

"Categorical algebra has developed in recent years as an effective method of organizing parts of mathematics. Typically, this sort of organization uses notions such as that of the category G of all groups. [...] This raises the problem of finding some axiomatization of set theory - or of some foundational discipline like set theory - which will be adequate and appropriate to realizing this intent. This problem may turn out to have revolutionary implications vis-`a-vis the accepted views of the role of set theory." (Saunders Mac Lane, "Categorical algebra and set-theoretic foundations", 1967)

"[...] 'information' is not a substance or concrete entity but rather a relationship between sets or ensembles of structured variety." (Walter F Buckley, "Sociology and modern systems theory", 1967)

 "A manifold, roughly, is a topological space in which some neighborhood of each point admits a coordinate system, consisting of real coordinate functions on the points of the neighborhood, which determine the position of points and the topology of that neighborhood; that is, the space is locally cartesian. Moreover, the passage from one coordinate system to another is smooth in the overlapping region, so that the meaning of 'differentiable' curve, function, or map is consistent when referred to either system." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"General or point set topology can be thought of as the abstract study of the ideas of nearness and continuity. This is done in the first place by picking out in elementary geometry those properties of nearness that seem to be fundamental and taking them as axioms." (Andrew H Wallace, "Differential Topology: First Steps", 1968)

"In practice, let us note, the determination of sets by means of characterizing criteria runs into difficulty because of the ambiguity of our language. The task of separating the objects belonging to a set from those that do not is often made difficult by the large number of objects of intermediate type." (Naum Ya. Vilenkin, "Stories about Sets", 1968)

"Infinite sets possess remarkable properties. In studying these properties mathematicians were led to continually perfect their reasoning and to further develop mathematical logic." (Naum Ya. Vilenkin, "Stories about Sets", 1968)

"Many examples occur in the theory of sets in which the definition of the set is self-contradictory. The study of the question of the conditions under which this takes place leads to deep questions of logic. Consideration of these questions has completely changed the face of the subject." (Naum Ya. Vilenkin, "Stories about Sets", 1968)

"Set theory is concerned with abstract objects and their relation to various collections which contain them. We do not define what a set is but accept it as a primitive notion. We gain an intuitive feeling for the meaning of sets and, consequently, an idea of their usage from merely listing some of the synonyms: class, collection, conglomeration, bunch, aggregate. Similarly, the notion of an object is primitive, with synonyms element and point. Finally, the relation between elements and sets, the idea of an element being in a set, is primitive." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"The mathematical models for many physical systems have manifolds as the basic objects of study, upon which further structure may be defined to obtain whatever system is in question. The concept generalizes and includes the special cases of the cartesian line, plane, space, and the surfaces which are studied in advanced calculus. The theory of these spaces which generalizes to manifolds includes the ideas of differentiable functions, smooth curves, tangent vectors, and vector fields. However, the notions of distance between points and straight lines (or shortest paths) are not part of the idea of a manifold but arise as consequences of additional structure, which may or may not be assumed and in any case is not unique." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"The very name 'set' leads us to think that any set must contain many elements" (at least two). But this is not the case. In mathematics it is sometimes necessary to examine sets having only one element and sometimes even a set having no elements at all." (Naum Ya. Vilenkin, "Stories about Sets", 1968)

"Two kinds of sets turn up in geometry. First of all, in geometry we ordinarily talk about the properties of some set of geometric figures. For example, the theorem stating that the diagonals of a parallelogram bisect each other relates to the set of all parallelograms. Secondly, the geometric figures are themselves sets composed of the points occurring within them. We can therefore speak of the set of all points contained within a given circle, of the set of all points within a given cone, etc." (Naum Ya. Vilenkin, "Stories about Sets", 1968)

"The modern definition of a function in the context of real numbers is that it is a relationship, usually a formula, by which a correspondence is established between two sets A (the domain) and B (the range) of real numbers in such a manner that to every number in set A there corresponds only one number in set B." (Alan Jeffrey, "Mathematics for Engineers and Scientists", 1969)

On Set Theory: On Sets (1900-1959)

"A collection of terms is infinite when it contains as parts other collections which have just as many terms in it as it has. If you can take away some of the terms of a collection, without diminishing the number of terms, then there is an infinite number of terms in the collection." (Bertrand Russell. International Monthly, Vol. 4, 1901)

"Let us subjugate a collection of objects taking into account their qualities and differences each from another, then we are lead, in our mathematical perspective, to the study of integers and their connecting operations, that is, we are lead to Number Theory. If we, however, disregard the qualities of each individual object and only account for the difference between two objects insofar that they are different, then we are lead to investigations which are concerned with the position, the order, the choosing of these objects. This branch of mathematics is called Combinatorics." (Eugen Netto, "Lehrbuch der Combinatorik", 1901)

"[...] hence upon the principle that even for an infinite totality of sets there are always mappings that associate with every set one of its elements, or, expressed formally, that the product of an infinite totality of sets, each containing at least one element, itself differs from zero. This logical principle cannot, to be sure, be reduced to a still simpler one, but it is applied without hesitation everywhere in mathematical deduction." (Ernst Zermelo, 1904)

"If one intends to base arithmetic on the theory of natural numbers as finite cardinals, one has to deal mainly with the definition of finite set; for the cardinal is, according to its nature, a property of a set, and any proposition about finite cardinals can always be expressed as a proposition about finite sets. In the following I will try to deduce the most important property of natural numbers, namely the principle of complete induction, from a definition of finite set which is as simple as possible, at the same time showing that the different definitions [of finite set] given so far are equivalent to the one given here." (Ernst Zermelo, "Ueber die Grundlagen der Arithmetik", Atti del IV Congresso Internazionale dei Matematici, 1908)

"The critical mathematician has abandoned the search for truth. He no longer flatters himself that his propositions are or can be known to him or to any other human being to be true; and he contents himself with aiming at the correct, or the consistent. The distinction is not annulled nor even blurred by the reflection that consistency contains immanently a kind of truth. He is not absolutely certain, but he believes profoundly that it is possible to find various sets of a few propositions each such that the propositions of each set are compatible, that the propositions of each such set imply other propositions, and that the latter can be deduced from the former with certainty. That is to say, he believes that there are systems of coherent or consistent propositions, and he regards it his business to discover such systems. Any such system is a branch of mathematics." (Cassius J Keyser, Science, New Series, Vol. 35" (904), 1912)

"Since the examination of consistency is a task that cannot be avoided, it appears necessary to axiomatize logic itself and to prove that number theory and set theory are only parts of logic. This method was prepared long ago (not least by Frege’s profound investigations); it has been most successfully explained by the acute mathematician and logician Russell. One could regard the completion of this magnificent Russellian enterprise of the axiomatization of logic as the crowning achievement of the work of axiomatization as a whole." (David Hilbert, "Axiomatisches Denken" ["Axiomatic Thinking"], [address] 1917)

"It seems clear that [set theory] violates against the essence of the continuum, which, by its very nature, cannot at all be battered into a single set of elements. Not the relationship of an element to a set, but of a part to a whole ought to be taken as a basis for the analysis of a continuum." (Hermann Weyl, "Reimanns geometrische Ideen, ihre Auswirkungen und ihre Verknüpfung mit der Gruppentheorie", 1925)

"The word is of Arabic origin. ‘Al’ is the Arabic article the, and ‘gebar’ is the verb to set, to restitute." (Tobias Dantzig & Joseph Mazur, “Number: The Language of Science”, 1930)

"The principle underlying sampling is that a set of objects taken at random from a larger group tends to reproduce the characteristics of that larger group: this is called the Law of Statistical Regularity. There are exceptions to this rule, and a certain amount of judgment must be exercised, especially when there are a few abnormally large items in the larger group. With erratic data, the accuracy of sampling can often be tested by comparing several samples. On the whole, the larger the sample the more closely will it tend to resemble the population from which it is taken; too small a sample would not give reliable results." (Lewis R Connor, "Statistics in Theory and Practice", 1932)

"To say that mathematics in general has been reduced to logic hints at some new firming up of mathematics at its foundations. This is misleading. Set theory is less settled and more conjectural than the classical mathematical superstructure than can be founded upon it." (Willard van Orman Quine, "Elementary Logic", 1941)

"The emphasis on mathematical methods seems to be shifted more towards combinatorics and set theory - and away from the algorithm of differential equations which dominates mathematical physics." (Oskar Morgenstern, "Theory of Games and Economic Behavior", 1944)

"But, despite their remoteness from sense experience, we do have something like a perception of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don't see any reason why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception, which induces us to build up physical theories and to expect that future sense perception will agree with them and, moreover, to believe that a question not decidable now has meaning and may be decided in future." (Kurt Gödel, "What is Cantor’s Continuum problem?", American Mathematical Monthly 54, 1947)

"Topology begins where sets are implemented with some cohesive properties enabling one to define continuity." (Solomon Lefschetz, "Introduction to Topology", 1949)

"The notion of an abstract group arises by consideration of the formal properties of one-to-one transformations of a set onto itself. Similarly, the notion of a category is obtained from the formal properties of the class of all transformations y : X → Y of any one set into another, or of continuous transformations of one topological space into another, or of homomorphisms, of one group into another, and so on." (Saunders Mac Lane, "Duality for groups", Bulletin of the American Mathematical Society 56, 1950)

"The notion of an equilibrium point is the basic ingredient of our theory. This notion yields a generalization of the concept of the solution of a two-person zero-sum game. It turns out that the set of equilibrium points of a two-person zero-sum game is simply the set of all pairs of opposing 'good strategies'." (John F Nash, "Non-Cooperative Games", 1950)

"A classification is a scheme for breaking a category into a set of parts, called classes, according to some precisely defined differing characteristics possessed by all the elements of the category." (Alva M Tuttle, "Elementary Business and Economic Statistics", 1957)

"The mathematical theory of continuity is based, not on intuition, but on the logically developed theories of number and sets of points." (Carl B Boyer, "The History of the Calculus and Its Conceptual Development", 1959)

On Set Theory: On Sets (-1874)

"The existence of an actual infinite multitude is impossible. For any set of things one considers must be a specific set. And sets of things are specified by the number of things in them. Now no number is infinite, for number results from counting through a set of units. So no set of things can actually be inherently unlimited, nor can it happen to be unlimited." (St. Thomas Aquinas, "Summa Theologica", cca. 1266-1273)

"One very important genus of complex ideas that we encounter everywhere are those in which the idea of collection (Inbegriff ) appears. There are many types of the latter [...] I must first determine with more precision the concept I associate with the word collection. I use this word in the same sense as it is used in the common usage and thus understand by a collection of certain things exactly the same as what one would express by the words: a combination (Verbindung) or association (Vereinigung) of these things, a gathering (Zusammensein) of the latter, a whole (Ganzes) in which they occur as parts (Teile). Hence the mere idea of a collection does not allow us to determine in which order and sequence the things that are put together appear or, indeed, whether there is or can be such an order. [...] A collection, it seems to me, is nothing other than something complex (das Zusammengesetztheit hat)." (Bernard Bolzano, "Wissenschaftslehre" ["Theory of Science"], 1837)

"[a set is] an embodiment of the idea or concept which we conceive when we regard the arrangement of its parts as a matter of indifference." (Bernard Bolzano, 1847)

"Already within the domain of those things which do not have any pretension of reality, but only of possibility, there indisputably are sets that are infinite. The set of propositions and truths in themselves is infinite, as one can easily see." (Bernard Bolzano, "Paradoxien des Unedlichen" ["Paradoxes of the Infinite"], 1851)

"Even in the realm of things which do not claim actuality, and do not even claim possibility, there exist beyond dispute sets which are infinite. The set of all ‘absolute propositions and truths' is easily seen to be infinite." (Bernard Bolzano, "Paradoxien des Unedlichen" ["Paradoxes of the Infinite"], 1851)

"[...] from that circumstance alone we are not allowed to conclude that both sets, if they are infinite, are equal to each other with respect to the multiplicity of their parts (that is, if we abstract from all differences between them); [...] Equality of those multiplicities can only be inferred when some other reason is added, for instance that both sets have absolutely equal grounds of determination, i.e., that their mode of formation is absolutely equal." (Bernard Bolzano, "Paradoxien des Unedlichen", 1851)

"The existence of infinite sets, at least with non-actual members, is something which I now regard as sufficiently proved and defended; as also, that the set of all absolute truths is an infinite set." (Bernard Bolzano, "Paradoxien des Unedlichen" ["Paradoxes of the Infinite"], 1851)

On Set Theory: On Sets (1875-1899)

"If two well-defined manifolds M and N can be coordinated with each other univocally and completely, element by element (which, if possible in one way, can always happen in many others), we shall employ in the sequel the expression, that those manifolds have the same power or, also, that they are equivalent." (Georg Cantor, "Ein Beitrag zur Mannigfaltigkeitslehre", 1878)

"I say that a manifold (a collection, a set) of elements that belong to any conceptual sphere is well-defined, when on the basis of its definition and as a consequence of the logical principle of excluded middle it must be regarded as internally determined, both whether an object pertaining to the same conceptual sphere belongs or not as an element to the manifold, and whether two objects belonging to the set are equal to each other or not, despite formal differences in the ways of determination." (Georg Cantor, "Ober unendliche, lineare Punktmannichfaltigkeiten", 1879)

"Most of the difficulties of principle that are found in mathematics have their origins, it seems to me, in ignorance of the possibility of a purely arithmetical theory of magnitudes and manifolds." (Georg Cantor, "Ober unendliche, lineare Punktmannichfaltigkeiten", 1879)

"The concept of power, which includes as a special case the concept of whole number, that foundation of the theory of number, and which ought to be considered as the most general genuine origin of sets [Moment bei Mannigfaltigkeiten], is by no means restricted to linear point sets, but can be regarded as an attribute of any well-defined collection, whatever may be the character of its elements. [...] Set theory in the conception used here, if we only consider mathematics for now and forget other applications, includes the areas of arithmetic, function theory and geometry. It contains them in terms of the concept of power and brings them all together in a higher unity. Discontinuity and continuity are similarly considered from the same point of view and are thus measured with the same measure." (Georg Cantor, "Ober unendliche, lineare Punktmannichfaltigkeiten", 1879)

"The old and oft-repeated proposition 'Totum est majus sua parte' [the whole is larger than the part] may be applied without proof only in the case of entities that are based upon whole and part; then and only then is it an undeniable consequence of the concepts 'totum' and 'pars'. Unfortunately, however, this 'axiom' is used innumerably often without any basis and in neglect of the necessary distinction between 'reality' and 'quantity' , on the one hand, and 'number' and 'set', on the other, precisely in the sense in which it is generally false." (Georg Cantor, "Über unendliche, lineare Punktmannigfaltigkeiten", Mathematische Annalen 20, 1882)

"By a manifold or a set I understand in general every Many that can be thought of as One, i.e., every collection of determinate elements which can be bound up into a whole through a law, and with this I believe to define something that is akin to the Platonic εἷδος [form] or ἷδεα [idea]." (Georg Cantor, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", 1883) 

"If we now notice that all of the numbers previously obtained and their next successors fulfill a certain condition, [that the set of their predecessors is denumerable,] then this condition offers itself, if it is imposed as a requirement on all numbers to be formed next, as a new third principle [...] which I shall call principle of restriction or limitation and which, as I shall show, yields the result that the second number-class (II) defined with its assistance not only has a higher power than [the first number-class] (I), but precisely the next higher, that is, the second power." (Georg Cantor, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", 1883) 

"The foregoing account of my researches in the theory of manifolds has reached a point where further progress depends on extending the concept of true integral number beyond the previous boundaries; this extension lies in a direction which, to my knowledge, no one has yet attempted to explore. [...] My dependence on this extension of number concept is so great, that without it I should be unable to take freely the smallest step further in the theory of sets." (Georg Cantor, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", 1883) 

To the thought of considering the infinitely great not merely in the form of what grows without limits - and in the closely related form of the convergent infinite series first introduced in the seventeenth century-, but also fixing it mathematically by numbers in the determinate form of the completed-infinite, I have been logically compelled in the course of scientific exertions and attempts which have lasted many years, almost against my will, for it contradicts traditions which had become precious to me; and therefore I believe that no arguments can be made good against it which I would not know how to meet. (Georg Cantor, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", 1883)

"In calling arithmetic (algebra, analysis) just a part of logic, I declare already that I take the number-concept to be completely independent of the ideas or intuitions of space and time, that I see it as an immediate product of the pure laws of thought." (Richard Dedekind, "Was sind und was sollen die Zahlen?", 1888)

On Sets: Definitions

"[a set is] an embodiment of the idea or concept which we conceive when we regard the arrangement of its parts as a matter of indifference." (Bernard Bolzano, 1847)

"By a manifold or a set I understand in general every Many that can be thought of as One, i.e., every collection of determinate elements which can be bound up into a whole through a law, and with this I believe to define something that is akin to the Platonic εἷδος [form] or ἷδεα [idea]." (Georg Cantor, "Grundlagen einer allgemeinen Mannigfaltigkeitslehre", 1883)

"A set is formed by the grouping together of single objects into a whole. A set is a plurality thought of as a unit. If these or similar statements were set down as definitions, then it could be objected with good reason that they define idem per idemi or even obscurum per obscurius. However, we can consider them as expository, as references to a primitive concept, familiar to us all, whose resolution into more fundamental concepts would perhaps be neither competent nor necessary." (Felix Hausdorff, "Set Theory", 1962)

"Set theory is concerned with abstract objects and their relation to various collections which contain them. We do not define what a set is but accept it as a primitive notion. We gain an intuitive feeling for the meaning of sets and, consequently, an idea of their usage from merely listing some of the synonyms: class, collection, conglomeration, bunch, aggregate. Similarly, the notion of an object is primitive, with synonyms element and point. Finally, the relation between elements and sets, the idea of an element being in a set, is primitive." (Richard L Bishop & Samuel I Goldberg, "Tensor Analysis on Manifolds", 1968)

"Unfortunately, we are not in a position to give a rigorous definition of the fundamental concept of the theory : the concept of set. Of course, we could say that a set is a collection, a union, an ensemble, a family, a system, a class, etc. But this would not be a mathematical definition, but rather a misuse of the multitude of words available in the English language." (Naum Ya Vilenkin, "Stories about Sets", 1968)

"Formally, a Cantor set is defined as a set that is totally disconnected, closed, and perfect. A totally disconnected set is a set that contains no intervals and therefore has no interior points. A closed set is one that contains all its boundary elements. (A boundary element is an element that contains elements both inside and outside the set in arbitrarily small neighborhoods.) A perfect set is a nonempty set that is equal to the set of its accumulation points. All three conditions are met by our middle-third - erasing construction, the original Cantor set." (Manfred Schroeder, "Fractals, Chaos, Power Laws Minutes from an Infinite Paradise", 1990) 

"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)

"Unlike the classic set theory where a set is represented as an indicator function to specify if an object belongs or not to it, a fuzzy set is an extension of a classic set where a subset is represented as a membership function to characterize the degree that an object belongs to it. The indicator function of a classic set takes value of 1 or 0, whereas the membership function of a fuzzy set takes value between 1 and 0." (Jianchao Han & Nick Cerone, Principles and Perspectives of Granular Computing, 2009) 

"The invariance principle states that the result of counting a set does not depend on the order imposed on its elements during the counting process. Indeed, a mathematical set is just a collection without any implied ordering. A set is the collection of its elements - nothing more." (Alfred S Posamentier & Bernd Thaller, "Numbers: Their tales, types, and treasures", 2015)

"A set is a unity of which its elements are the constituents. It is a fundamental property of the mind to comprehend multitudes into unities. Sets are multitudes which are also unities. A multitude is the opposite of a unity. How can anything be both a multitude and a unity? Yet a set is just that. It is a seemingly contradictory fact that sets exist. It is surprising that the fact that multitudes are also unities leads to no contradictions: this is the main fact of mathematics. Thinking a plurality together seems like a triviality: and this appears to explain why we have no contradiction. But 'many things for one' is far from trivial." (Kurt Gödel)

"A set is a Many that allows itself to be thought of as a One." (Georg Cantor)

On Set Theory (2000-)

"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)

On Axioms: On the Axiom of Choice

"[...] hence upon the principle that even for an infinite totality of sets there are always mappings that associate with every set one of its elements, or, expressed formally, that the product of an infinite totality of sets, each containing at least one element, itself differs from zero. This logical principle cannot, to be sure, be reduced to a still simpler one, but it is applied without hesitation everywhere in mathematical deduction." (Ernst Zermelo, 1904)

"Many mathematicians continue to reject the axiom of choice. The growing realization that there are questions in mathematics that cannot be decided without this principle is likely to result in the gradual disappearance of the resistance to it." (Ernst Steinitz, "Algebraische Theorie der Körper" ["Algebraic theory of fields"], 1910) 

"Zermelo regards the axiom as an unquestionable truth. It must be confessed that, until he made it explicit, mathematicians had used it without a qualm; but it would seem that they had done so unconsciously. And the credit due to Zermelo for having made it explicit is entirely independent of the question whether it is true or false." (Bertrand Russel, "Introduction to Mathematical Philosophy", 1919)

"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)

"Today, most mathematicians have embraced the axiom of choice because of the order and simplicity it brings to mathematics in general. For example, the theorems that every vector space has a basis and every field has an algebraic closure hold only by virtue of the axiom of choice. Likewise, for the theorem that every sequentially continuous function is continuous. However, there are special places where the axiom of choice actually brings disorder. One is the theory of measure." (John Stillwell, "Roads to Infinity: The mathematics of truth and proof", 2010)

"Given any collection of infinite sets the Axiom of Choice tells us that there exists a set which has one element in common with each of the sets in the collection. Choice, which seems to be an intuitively sound principle, is equivalent to the much less plausible statement that every set has a well-ordering. Although many tried to prove Choice, they only seemed to be able to find equivalent statements which were just as difficult to prove." (Barnaby Sheppard, "The Logic of Infinity", 2014)

"Since the membership relation is well-founded, well-founded relations can be defined on any class, however, the existence of a well-ordering of every set cannot be proved without appealing to the Axiom of Choice. Indeed, the assumption that every set has a well-ordering is equivalent to the Axiom of Choice." (Barnaby Sheppard, "The Logic of Infinity", 2014)

"Objections to the Axiom of Choice, either the strong or the weak version, are typically either philosophical, based on the intuitive temporal implausibility of making an infinite number of choices, or on the non-constructive nature of the axiom, or are based on a peculiar identification of continuum-based models of physics with the physical objects being modelled; properties of the model which are implied by the Axiom of Choice are deemed to be counterintuitive because the physical objects they model don’t have these properties. Motivated by these objections, or just for curiosity, several alternatives to Choice have been explored." (Barnaby Sheppard, "The Logic of Infinity", 2014)

"The most obvious variations of the Axiom of Choice are those that restrict the cardinality of the sets in question. Other variations impose relational restrictions between the sets. When the early set theorists tried to prove the Axiom of Choice they invariably ended up showing it is equivalent to some other statement that they were unable to prove. This collection of equivalent statements has grown to an enormous size. One of its striking features is that some of the statements seem intuitively obvious while others are either wildly counterintuitive or evade any kind of evaluation." (Barnaby Sheppard, "The Logic of Infinity", 2014)

"The Axiom of Choice says that it is possible to make an infinite number of arbitrary choices. […] Mathematicians don’t exactly care whether or not the Axiom of Choice holds over all, but they do care whether you have to use it in any given situation or not." (Eugenia Cheng, "Beyond Infinity: An Expedition to the Outer Limits of Mathematics", 2017)