BROWSE ALPHABETICALLY LEVEL:    Elementary    Advanced    Both INCLUDE TOPICS:    Basic Math    Algebra    Analysis    Biography    Calculus    Comp Sci    Discrete    Economics    Foundations    Geometry    Graph Thry    History    Number Thry    Phys Sci    Statistics    Topology    Trigonometry Hempel’s Ravens Paradox – integral Hempel’s Ravens Paradox    ARTICLE   A paradox of inductive logic described by the philosopher C.G. Hempel. See the article for a complete exposition. homomorphism   A function f from one algebra to another is called a homomorphism if it preserves operations on elements, that is, if for any a, b in the domain, f(ab) = f(a)f(b).Cf. group homomorphism, ring homomorphism. hyperbola   The locus of points in the plane, the difference of whose distances from two fixed points, called the foci, remains constant.Like the ellipse and parabola, the hyperbola is a conic section. Related article: Conics hyperset   A set which is not well-founded, i.e., which involves self-membership or, equivalently, an infinite descending membership chain. Example: the Quine atom x = {x}. hypotenuse   On a right triangle, the side opposite the right angle. i   See imaginary number. icosahedron   A polyhedron having twenty faces.The faces of a regular icosahedron are congruent, equilateral triangles.Cf. Platonic solid. identity function   A function that maps each domain element to itself. Also called the identity map. identity map   image   Given a function f with domain X, the image under f of a subset A of X, denoted f(A), is the subset of the range consisting of those elements to which elements of A are mapped by f. imaginary number   By definition, the square root of –1, i.e., i 2 = –1.Cf. complex number. inaccessible cardinal   A cardinal k is called weakly inaccessible if and only if it is a regular limit cardinal. A cardinal k is called strongly inaccessible if and only if k > w and for every cardinal l less than k, 2l is less than k. If GCH is assumed, then weakly inaccessibles are strongly inaccessible. Ordinary set theory (ZFC) cannot prove that weakly inaccessibles exist.Cf. König’s Lemma. independent axiom   In a formal mathematical theory, a formula or statement in the theory is said to be independent if it cannot be derived (proved, deduced) from the other axioms of the theory. indeterminate form   A limit of an expression is said to be indeterminate, or in indeterminate form, if when evaluated directly it resolves to one of the formsSuch limits may often be evaluated by manipulating them algebraically before applying the limit, or, in the case of the first two indeterminate forms shown, by applying L'Hospital's Rule. Related article: Limits inf   Abbreviation of infimum. infimum   The infimum of any subset of a linear order (linearly ordered set) is the greatest lower bound of the subset. In particular, the infimum of any set of numbers is the largest number in the set which is less than or equal to every other number in the set. In a complete linear order the infimum of any bounded set always exists.Cf. supremum, least upper bound axiom. infinity   Infinity is a concept understood in different ways depending upon the context in which the word is used. In particular, infinity is not a number in the ordinary sense. The so-called extended real numbers include either a positive or a negative infinity (but not both). When this is done, the algebraic forms "infinity plus negative infinity", "infinity times zero", and "infinity divided by infinity" are undefined. Infinite ordinals may be countable or uncountable. Whether actually infinite totalities may be admitted, used, or analyzed remains a contentious issue in the philosophy of mathematics. See the minitext for a thorough treatment. Related MiniText: Infinity -- You Can't Get There From Here... injection   An injective function, i.e., a function that is “one-to-one.” Equivalently, a function that maps exactly one element of its domain to each element of its range.Cf. surjection, bijection. injective   A function is injective, also called “one-to-one,” if to each element of the range at most one element of the domain is mapped by the function.Cf. surjective, bijective. integer   An element of the set Z consisting of the natural numbers, zero, and the additive inverses (negatives) of the natural numbers. I.e., Z = { ... -3, -2, -1, 0, 1, 2, 3, ... }. The use of Z to denote the set of integers stems from the German word zahlen, which means “to count.”Cf. natural number. Related MiniText: Number -- What Is How Many? integral   An antiderivative of a function. That is, if f(x) is a real-valued function, an antiderivative F(x) of f(x) has the property that the derivative of F with respect to x is f.The definite integral (called the Riemann integral) of a real-valued function f(x) from x = a to x = b is the limit of the Riemann sum:assuming this limit exists, where ci is in the i th subinterval of the partition of (a, b), and where a is the lower limit, and b is the upper limit, of the integral.Cf. fundamental theorem of calculus, Lebesgue integral. Hempel’s Ravens Paradox – integral
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