Von Neumann–Bernays–Gödel set theory
In the foundations of mathematics, von Neumann–Bernays–Gödel set theory is an axiomatic set theory, a conservative extension of Zermelo–Fraenkel set theory. NBG introduces the notion of class, a collection of sets defined by a formula whose quantifiers range only over sets. NBG can define classes that are larger than sets, such as the class of all sets and the class of all ordinals. Morse–Kelley set theory allows classes to be defined by formulas whose quantifiers range over classes. NBG is finitely axiomatizable, while MK are not. A key theorem of NBG is the class existence theorem, which states that for every formula whose quantifiers range only over sets, there is a class consisting of the sets satisfying the formula; this class is built by mirroring the step-by-step construction of the formula with classes. Since all set-theoretic formulas are constructed from two kinds of atomic formulas and finitely many logical symbols, only finitely many axioms are needed to build the classes satisfying them.
This is. Classes are used for other constructions, for handling the set-theoretic paradoxes, for stating the axiom of global choice, stronger than ZFC's axiom of choice. John von Neumann introduced classes into set theory in 1925; the primitive notions of his theory were argument. Using these notions, he set. Paul Bernays set as primitive notions. Kurt Gödel simplified Bernays' theory for his relative consistency proof of the axiom of choice and the generalized continuum hypothesis. Classes have several uses in NBG: They produce a finite axiomatization of set theory, they are used to state a "very strong form of the axiom of choice"—namely, the axiom of global choice: There exists a global choice function G defined on the class of all nonempty sets such that G ∈ x for every nonempty set x. This is stronger than ZFC's axiom of choice: For every set s of nonempty sets, there exists a choice function f defined on s such that f ∈ x for all x ∈ s; the set-theoretic paradoxes are handled by recognizing.
For example, assume that the class O r d of all ordinals is a set. O r d is a transitive set well-ordered by ∈. So, by definition, O r d is an ordinal. Hence, O r d ∈ O r d, which contradicts ∈ being a well-ordering of O r d. Therefore, O r d is not a set; because a class, not a set is called a proper class, O r d is a proper class. Proper classes are useful in constructions. In his proof of the relative consistency of the axiom of global choice and the generalized continuum hypothesis, Gödel used proper classes to build the constructible universe, he constructed a function on the class of all ordinals that, for each ordinal, builds a constructible set by applying a set-building operation to constructed sets. The constructible universe is the image of this function. Once classes are added to the language, it is simple to axiomatize a set theory with classes, similar to ZFC. First, the axiom schema of class comprehension is added; this axiom schema states: For every formula ϕ that quantifies only over sets, there exists a class A consisting of the n -tuples satisfying the formula—that is, ∀ x 1 ⋯ ∀ x n.
The axiom schema of replacement is replaced by a single axiom that uses a class. ZFC's axiom of extensionality is modified to handle classes: If two classes have the same elements they are identical; the other axioms of ZFC are not modified. This theory is not finitely axiomatized. ZFC's replacement schema has been replaced by a single axiom, but the axiom schema of class comprehension has been introduced. To produce a theory with finitely many axioms, the axiom schema of class comprehension is first replaced with finitely many class existence axioms; these axioms are used to prove the class existence theorem which implies every instance of the axiom schema. The proof of this theorem requires only seven class existence axioms, which are used to convert the construction of a formula into the construction of a class satisfying the formula. NBG sets. Intuitively, every set is a class. There are two ways to axioma
In mathematics, a group is a set equipped with a binary operation which combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study. Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie groups are the symmetry groups used in the Standard Model of particle physics; the concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s.
After contributions from other fields such as number theory and geometry, the group notion was generalized and established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. In addition to their abstract properties, group theorists study the different ways in which a group can be expressed concretely, both from a point of view of representation theory and of computational group theory. A theory has been developed for finite groups, which culminated with the classification of finite simple groups, completed in 2004. Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become an active area in group theory; the modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4.
The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagrange, gave a criterion for the solvability of a particular polynomial equation in terms of the symmetry group of its roots. The elements of such a Galois group correspond to certain permutations of the roots. At first, Galois' ideas were rejected by his contemporaries, published only posthumously. More general permutation groups were investigated in particular by Augustin Louis Cauchy. Arthur Cayley's On the theory of groups, as depending on the symbolic equation θn = 1 gives the first abstract definition of a finite group. Geometry was a second field in which groups were used systematically symmetry groups as part of Felix Klein's 1872 Erlangen program. After novel geometries such as hyperbolic and projective geometry had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas, Sophus Lie founded the study of Lie groups in 1884; the third field contributing to group theory was number theory.
Certain abelian group structures had been used implicitly in Carl Friedrich Gauss' number-theoretical work Disquisitiones Arithmeticae, more explicitly by Leopold Kronecker. In 1847, Ernst Kummer made early attempts to prove Fermat's Last Theorem by developing groups describing factorization into prime numbers; the convergence of these various sources into a uniform theory of groups started with Camille Jordan's Traité des substitutions et des équations algébriques. Walther von Dyck introduced the idea of specifying a group by means of generators and relations, was the first to give an axiomatic definition of an "abstract group", in the terminology of the time; as of the 20th century, groups gained wide recognition by the pioneering work of Ferdinand Georg Frobenius and William Burnside, who worked on representation theory of finite groups, Richard Brauer's modular representation theory and Issai Schur's papers. The theory of Lie groups, more locally compact groups was studied by Hermann Weyl, Élie Cartan and many others.
Its algebraic counterpart, the theory of algebraic groups, was first shaped by Claude Chevalley and by the work of Armand Borel and Jacques Tits. The University of Chicago's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein, John G. Thompson and Walter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, led to the classification of finite simple groups, with the final step taken by Aschbacher and Smith in 2004; this project exceeded previous mathematical endeavours by its sheer size, in both length of proof and number of researchers. Research is ongoing to simplify the proof of this classification; these days, group theory is still a active mathematical branch, impacting many other fields. One of the most familiar groups is the set of integers Z which consists of the numbers... − 4, − 3, − − 1, 0, 1, 2, 3, 4... together with addition. The following properties of integer addition serve as a model for the group axioms given in the definition below.
For any two integers a and b, the sum a + b is an integer. That is, addition of integers always yields an integer; this property is known as closure under addition. For all integers a, b and c, + c = a +. Expressed in words
In mathematics, a proof is an inferential argument for a mathematical statement. In the argument, other established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms, along with accepted rules of inference. Axioms may be treated as conditions. Proofs are examples of exhaustive deductive reasoning or inductive reasoning and are distinguished from empirical arguments or non-exhaustive inductive reasoning. A proof must demonstrate that a statement is always true, rather than enumerate many confirmatory cases. An unproved proposition, believed to be true is known as a conjecture. Proofs employ logic but include some amount of natural language which admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory; the distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, so-called folk mathematics.
The philosophy of mathematics is concerned with the role of language and logic in proofs, mathematics as a language. The word "proof" comes from the Latin probare meaning "to test". Related modern words are the English "probe", "probation", "probability", the Spanish probar, Italian provare, the German probieren; the early use of "probity" was in the presentation of legal evidence. A person of authority, such as a nobleman, was said to have probity, whereby the evidence was by his relative authority, which outweighed empirical testimony. Plausibility arguments using heuristic devices such as pictures and analogies preceded strict mathematical proof, it is that the idea of demonstrating a conclusion first arose in connection with geometry, which meant the same as "land measurement". The development of mathematical proof is the product of ancient Greek mathematics, one of the greatest achievements thereof. Thales and Hippocrates of Chios proved some theorems in geometry. Eudoxus and Theaetetus formulated did not prove them.
Aristotle said definitions should describe the concept being defined in terms of other concepts known. Mathematical proofs were revolutionized by Euclid, who introduced the axiomatic method still in use today, starting with undefined terms and axioms, used these to prove theorems using deductive logic, his book, the Elements, was read by anyone, considered educated in the West until the middle of the 20th century. In addition to theorems of geometry, such as the Pythagorean theorem, the Elements covers number theory, including a proof that the square root of two is irrational and that there are infinitely many prime numbers. Further advances took place in medieval Islamic mathematics. While earlier Greek proofs were geometric demonstrations, the development of arithmetic and algebra by Islamic mathematicians allowed more general proofs that no longer depended on geometry. In the 10th century CE, the Iraqi mathematician Al-Hashimi provided general proofs for numbers as he considered multiplication, etc. for "lines."
He used this method to provide a proof of the existence of irrational numbers. An inductive proof for arithmetic sequences was introduced in the Al-Fakhri by Al-Karaji, who used it to prove the binomial theorem and properties of Pascal's triangle. Alhazen developed the method of proof by contradiction, as the first attempt at proving the Euclidean parallel postulate. Modern proof theory treats proofs as inductively defined data structures. There is no longer an assumption; as practiced, a proof is expressed in natural language and is a rigorous argument intended to convince the audience of the truth of a statement. The standard of rigor has varied throughout history. A proof can be presented differently depending on the intended audience. In order to gain acceptance, a proof has to meet communal statements of rigor; the concept of a proof is formalized in the field of mathematical logic. A formal proof is written in a formal language instead of a natural language. A formal proof is defined as sequence of formulas in a formal language, in which each formula is a logical consequence of preceding formulas.
Having a definition of formal proof makes the concept of proof amenable to study. Indeed, the field of proof theory studies formal proofs and their properties, for example, the property that a statement has a formal proof. An application of proof theory is to show; the definition of a formal proof is intended to capture the concept of proofs as written in the practice of mathematics. The soundness of this definition amounts to the belief that a published proof can, in principle, be converted into a formal proof. However, outside the field of automated proof assistants, this is done in practice. A classic question in philosophy a
Naive set theory
Naïve set theory is any of several theories of sets used in the discussion of the foundations of mathematics. Unlike axiomatic set theories, which are defined using formal logic, naïve set theory is defined informally, in natural language, it describes the aspects of mathematical sets familiar in discrete mathematics, suffices for the everyday use of set theory concepts in contemporary mathematics. Sets are of great importance in mathematics. Naïve set theory suffices for many purposes, while serving as a stepping-stone towards more formal treatments. A naïve theory in the sense of "naïve set theory" is a non-formalized theory, that is, a theory that uses a natural language to describe sets and operations on sets; the words and, or, if... not, for some, for every are treated as in ordinary mathematics. As a matter of convenience, use of naïve set theory and its formalism prevails in higher mathematics – including in more formal settings of set theory itself; the first development of set theory was a naïve set theory.
It was created at the end of the 19th century by Georg Cantor as part of his study of infinite sets and developed by Gottlob Frege in his Begriffsschrift. Naïve set theory may refer to several distinct notions, it may refer to Informal presentation of an axiomatic set theory, e.g. as in Naïve Set Theory by Paul Halmos. Early or versions of Georg Cantor's theory and other informal systems. Decidedly inconsistent theories, such as a theory of Gottlob Frege that yielded Russell's paradox, theories of Giuseppe Peano and Richard Dedekind; the assumption that any property may be used to form a set, without restriction, leads to paradoxes. One common example is Russell's paradox: there is no set consisting of "all sets that do not contain themselves", thus consistent systems of naïve set theory must include some limitations on the principles which can be used to form sets. Some believe that Georg Cantor's set theory was not implicated in the set-theoretic paradoxes. One difficulty in determining this with certainty is that Cantor did not provide an axiomatization of his system.
By 1899, Cantor was aware of some of the paradoxes following from unrestricted interpretation of his theory, for instance Cantor's paradox and the Burali-Forti paradox, did not believe that they discredited his theory. Cantor's paradox can be derived from the above assumption—that any property P may be used to form a set—using for P "x is a cardinal number". Frege explicitly axiomatized a theory in which a formalized version of naïve set theory can be interpreted, it is this formal theory which Bertrand Russell addressed when he presented his paradox, not a theory Cantor, who, as mentioned, was aware of several paradoxes had in mind. Axiomatic set theory was developed in response to these early attempts to understand sets, with the goal of determining what operations were allowed and when. A naïve set theory is not inconsistent, if it specifies the sets allowed to be considered; this can be done by the means of definitions. It is possible to state all the axioms explicitly, as in the case of Halmos' Naïve Set Theory, an informal presentation of the usual axiomatic Zermelo–Fraenkel set theory.
It is "naïve" in that the language and notations are those of ordinary informal mathematics, in that it doesn't deal with consistency or completeness of the axiom system. An axiomatic set theory is not consistent: not free of paradoxes, it follows from Gödel's incompleteness theorems that a sufficiently complicated first order logic system cannot be proved consistent from within the theory itself – if it is consistent. However, the common axiomatic systems are believed to be consistent. Based on Gödel's theorem, it is just not known – and never can be – if there are no paradoxes at all in these theories or in any first-order set theory; the term naïve set theory is still today used in some literature to refer to the set theories studied by Frege and Cantor, rather than to the informal counterparts of modern axiomatic set theory. The choice between an axiomatic approach and other approaches is a matter of convenience. In everyday mathematics the best choice may be informal use of axiomatic set theory.
References to particular axioms then occur only when demanded by tradition, e.g. the axiom of choice is mentioned when used. Formal proofs occur only when warranted by exceptional circumstances; this informal usage of axiomatic set theory can have the appearance of naïve set theory as outlined below. It is easier to read and write and is less error-prone than a formal approach. In naïve set theory, a set is described as a well-defined collection of objects; these objects are called the members of the set. Objects can be anything: numbers, other sets, etc. For instance, 4 is a member of the set of all integers; the set of numbers is infinitely large. The definition of sets goes back to Georg Cantor, he wrote 1915 in his article Beiträge zur Begründung der transfiniten Mengenlehre: “Unter einer'Menge' verstehen wir jede Zusammenfassung M von bestimmten wohlunterschi
Set theory is a branch of mathematical logic that studies sets, which informally are collections of objects. Although any type of object can be collected into a set, set theory is applied most to objects that are relevant to mathematics; the language of set theory can be used to define nearly all mathematical objects. The modern study of set theory was initiated by Richard Dedekind in the 1870s. After the discovery of paradoxes in naive set theory, such as Russell's paradox, numerous axiom systems were proposed in the early twentieth century, of which the Zermelo–Fraenkel axioms, with or without the axiom of choice, are the best-known. Set theory is employed as a foundational system for mathematics in the form of Zermelo–Fraenkel set theory with the axiom of choice. Beyond its foundational role, set theory is a branch of mathematics in its own right, with an active research community. Contemporary research into set theory includes a diverse collection of topics, ranging from the structure of the real number line to the study of the consistency of large cardinals.
Mathematical topics emerge and evolve through interactions among many researchers. Set theory, was founded by a single paper in 1874 by Georg Cantor: "On a Property of the Collection of All Real Algebraic Numbers". Since the 5th century BC, beginning with Greek mathematician Zeno of Elea in the West and early Indian mathematicians in the East, mathematicians had struggled with the concept of infinity. Notable is the work of Bernard Bolzano in the first half of the 19th century. Modern understanding of infinity began in 1870–1874 and was motivated by Cantor's work in real analysis. An 1872 meeting between Cantor and Richard Dedekind influenced Cantor's thinking and culminated in Cantor's 1874 paper. Cantor's work polarized the mathematicians of his day. While Karl Weierstrass and Dedekind supported Cantor, Leopold Kronecker, now seen as a founder of mathematical constructivism, did not. Cantorian set theory became widespread, due to the utility of Cantorian concepts, such as one-to-one correspondence among sets, his proof that there are more real numbers than integers, the "infinity of infinities" resulting from the power set operation.
This utility of set theory led to the article "Mengenlehre" contributed in 1898 by Arthur Schoenflies to Klein's encyclopedia. The next wave of excitement in set theory came around 1900, when it was discovered that some interpretations of Cantorian set theory gave rise to several contradictions, called antinomies or paradoxes. Bertrand Russell and Ernst Zermelo independently found the simplest and best known paradox, now called Russell's paradox: consider "the set of all sets that are not members of themselves", which leads to a contradiction since it must be a member of itself and not a member of itself. In 1899 Cantor had himself posed the question "What is the cardinal number of the set of all sets?", obtained a related paradox. Russell used his paradox as a theme in his 1903 review of continental mathematics in his The Principles of Mathematics. In 1906 English readers gained the book Theory of Sets of Points by husband and wife William Henry Young and Grace Chisholm Young, published by Cambridge University Press.
The momentum of set theory was such. The work of Zermelo in 1908 and the work of Abraham Fraenkel and Thoralf Skolem in 1922 resulted in the set of axioms ZFC, which became the most used set of axioms for set theory; the work of analysts such as Henri Lebesgue demonstrated the great mathematical utility of set theory, which has since become woven into the fabric of modern mathematics. Set theory is used as a foundational system, although in some areas—such as algebraic geometry and algebraic topology—category theory is thought to be a preferred foundation. Set theory begins with a fundamental binary relation between an object o and a set A. If o is a member of A, the notation o. Since sets are objects, the membership relation can relate sets as well. A derived binary relation between two sets is the subset relation called set inclusion. If all the members of set A are members of set B A is a subset of B, denoted A ⊆ B. For example, is a subset of, so is but is not; as insinuated from this definition, a set is a subset of itself.
For cases where this possibility is unsuitable or would make sense to be rejected, the term proper subset is defined. A is called a proper subset of B if and only if A is a subset of B, but A is not equal to B. Note that 1, 2, 3 are members of the set but are not subsets of it. Just as arithmetic features binary operations on numbers, set theory features binary operations on sets. The: Union of the sets A and B, denoted A ∪ B, is the set of all objects that are a member of A, or B, or both; the union of and is the set. Intersection of the sets A and B, denoted A ∩ B, is the set of all objects that are members of both A and B; the intersection of and is the set. Set difference of U and A, denoted U \ A, is the set of all members of U that are not members of A; the set difference \ is, conversely, the set difference \ is. When A is a subset of U, the set difference U \ A is called the complement of A in U. In this case, if the choice of U is clear from the context, the notation Ac is sometimes used instead of U \ A if U is a universal set as in the study of Venn diagrams.
Symmetric difference of sets A and B, denoted A △ B or A ⊖ B, is
Classical logic is the intensively studied and most used class of logics. Classical logic has had much influence on analytic philosophy, the type of philosophy most found in the English-speaking world; each logical system in this class shares characteristic properties: Law of excluded middle and double negation elimination Law of noncontradiction, the principle of explosion Monotonicity of entailment and idempotency of entailment Commutativity of conjunction De Morgan duality: every logical operator is dual to anotherWhile not entailed by the preceding conditions, contemporary discussions of classical logic only include propositional and first-order logics. In other words, the overwhelming majority of time spent studying classical logic has been spent studying propositional and first-order logic, as opposed to the other forms of classical logic. Most semantics of classical logic are bivalent, meaning all of the possible denotations of propositions can be categorised as either true or false.
Classical logic is a 20th century innovation. The name does not refer to classical antiquity. In fact, classical logic was the reconciliation of Aristotle's logic, which dominated most of the last 2000 years, with the propositional Stoic logic; the two were sometimes seen as irreconcilable. Leibniz's calculus ratiocinator can be seen as foreshadowing classical logic. Bernard Bolzano has the understanding of existential import found in classical logic and not in Aristotle. Though he never questioned Aristotle, George Boole's algebraic reformulation of logic, so called Boolean logic, was a predecessor of modern mathematical logic and classical logic. William Stanley Jevons and John Venn, who had the modern understanding of existential import, expanded Boole's system; the original first-order, classical logic is found in Gottlob Frege's Begriffsschrift. It has a wider application than Aristotle's logic, is capable of expressing Aristotle's logic as a special case, it explains the quantifiers in terms of mathematical functions.
It was the first logic capable of dealing with the problem of multiple generality, for which Aristotle's system was impotent. Frege, considered the founder of analytic philosophy, invented it so as to show all of mathematics was derivable from logic, make arithmetic rigorous as David Hilbert had done for geometry, the doctrine known as logicism in the foundations of mathematics; the notation Frege used never much caught on. Hugh MacColl published a variant of propositional logic two years prior; the writings of Augustus De Morgan and Charles Sanders Peirce pioneered classical logic with the logic of relations. Peirce influenced Ernst Schröder. Classical logic reached fruition in Bertrand Russell and A. N. Whitehead's Principia Mathematica, Ludwig Wittgenstein's Tractatus Logico Philosophicus. Russell and Whitehead were influenced by Peano and Frege, sought to show mathematics was derived from logic. Wittgenstein was influenced by Frege and Russell, considered the Tractatus to have solved all problems of philosophy.
Willard Van Orman Quine insisted on classical, first-order logic as the true logic, saying higher-order logic was "set theory in disguise". Jan Łukasiewicz pioneered non-classical logic; the results of Kurt Goedel and Alfred Tarski undermined the logicist project. With the advent of algebraic logic it became apparent that classical propositional calculus admits other semantics. In Boolean-valued semantics, the truth values are the elements of an arbitrary Boolean algebra. Intermediate elements of the algebra correspond to truth values other than "true" and "false"; the principle of bivalence holds only when the Boolean algebra is taken to be the two-element algebra, which has no intermediate elements. Warren Goldfard, "Deductive Logic", 1st edition, 2003, ISBN 0-87220-660-2
In mathematics, an element, or member, of a set is any one of the distinct objects that make up that set. Writing A = means that the elements of the set A are the numbers 1, 2, 3 and 4. Sets of elements of A, for example, are subsets of A. Sets can themselves be elements. For example, consider the set B =; the elements of B are not 1, 2, 3, 4. Rather, there are only three elements of B, namely the numbers 1 and 2, the set; the elements of a set can be anything. For example, C =, is the set whose elements are the colors red and blue; the relation "is an element of" called set membership, is denoted by the symbol " ∈ ". Writing x ∈ A means that "x is an element of A". Equivalent expressions are "x is a member of A", "x belongs to A", "x is in A" and "x lies in A"; the expressions "A includes x" and "A contains x" are used to mean set membership, however some authors use them to mean instead "x is a subset of A". Logician George Boolos urged that "contains" be used for membership only and "includes" for the subset relation only.
For the relation ∈, the converse relation ∈T may be written A ∋ x, meaning "A contains x". The negation of set membership is denoted by the symbol "∉". Writing x ∉ A means that "x is not an element of A"; the symbol ∈ was first used by Giuseppe Peano 1889 in his work Arithmetices principia, nova methodo exposita. Here he wrote on page X: Signum ∈ significat est. Ita a ∈ b legitur a est quoddam b. So a ∈ b is read; every relation R: U → V is subject to two involutions: complementation R → R ¯ and conversion RT: V → U. The relation ∈ has for its domain a universal set U, has the power set P for its codomain or range; the complementary relation ∈ ¯ = ∉ expresses the opposite of ∈. An element x ∈ U may have x ∉ A, in which case x ∈ U \ A, the complement of A in U; the converse relation ∈ T = ∋ swaps the domain and range with ∈. For any A in P, A ∋ x is true when x ∈ A; the number of elements in a particular set is a property known as cardinality. In the above examples the cardinality of the set A is 4, while the cardinality of either of the sets B and C is 3.
An infinite set is a set with an infinite number of elements, while a finite set is a set with a finite number of elements. The above examples are examples of finite sets. An example of an infinite set is the set of positive integers. Using the sets defined above, namely A =, B = and C =: 2 ∈ A ∈ B 3,4 ∉ B is a member of B Yellow ∉ C The cardinality of D = is finite and equal to 5; the cardinality of P = is infinite. Halmos, Paul R. Naive Set Theory, Undergraduate Texts in Mathematics, NY: Springer-Verlag, ISBN 0-387-90092-6 - "Naive" means that it is not axiomatized, not that it is silly or easy. Jech, Thomas, "Set Theory", Stanford Encyclopedia of Philosophy Suppes, Axiomatic Set Theory, NY: Dover Publications, Inc. ISBN 0-486-61630-4 - Both the notion of set, membership or element-hood, the axiom of extension, the axiom of separation, the union axiom are needed for a more thorough understanding of "set element". Weisstein, Eric W. "Element". MathWorld