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generalized continuum hypothesis – infinity
generalized continuum hypothesis
The claim that there is no set of intermediate cardinality between any given set and its power set. The GCH is independent of (i.e., cannot be proved true or false by) the known axioms of set theory.
 Cf. continuum hypothesis.
Related article: Gödel's Theorems
Related MiniText: Infinity -- You Can't Get There From Here...
g.l.b.
Abbreviation for greatest lower bound.
graph
General: A picture or diagram representing one or more relationships among certain objects or quantities.
Functions: The collection of ordered tuples (x,y), consisting of domain elements x (possibly also tuples) and corresponding range elements y. Although informally we often mean the picture when we speak of the graph of a function, the graph should always be formally thought of as a set of tuples.
Graph Theory: A collection of nodes (also called vertices), together with a set of edges joining the nodes. Formally, a graph G is a set of vertices V, a set of edges E, and a symmetric relation between them specifying incidence. Every edge must be incident on exactly two (not necessarily distinct) vertices, called its ends. Two vertices incident on the same edge are called adjacent. Graphs are often represented pictorially, with the vertices being points (or small circles or squares or some such), and the edges being curves (often line segments), with the endpoints of each such curve being the ends of that edge in the graph.
Cf. Konigsberg Bridge Problem.
greatest common divisor
The greatest common divisor of two natural numbers a and b is the largest natural number c such that c divides a and c divides b.
greatest lower bound
A lower bound which is greater than or equal to every other lower bound.
Cf. Least upper bound.
Greek
ARTICLE
Letters of the Greek alphabet are commonly used in mathematics. See the article for a full description.
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.
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
See identity function.
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.
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...
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