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 uncountable – ZFC uncountable   A set is uncountable (uncountably infinite) if it is infinite but not countable, i.e., no complete one-to-one match-up of the set with the set of natural numbers (finite ordinals) can be performed. Georg Cantor proved that the set of real numbers is uncountably infinite. (This is sometimes called the “non-denumerability of the continuum”). Related MiniText: Infinity -- You Can't Get There From Here... uncountably infinite   See uncountable. union   The union of two sets A and B is the set consisting of those elements which are in A or in B or in both, and is denoted byIf the union is taken over a family of sets {Ai}i = 1, 2, ..., n, then it is the set consisting of those elements that are in at least one of the sets in the collection.Sometimes the word “union” is used to indicate the sumset of a set A, and is then loosely described as “union A.” In this latter case the union is understood to be the union over the elements of A, and is denoted byWhen considering an algebra of sets, the union of two or more sets is sometimes called the join of the sets.Cf. intersection. unit interval   The interval on the real number line from 0 to 1, inclusive. unit square   The set of points of the Cartesian plane with domain and range values in the unit interval, that is the square region with vertices (0, 0), (0, 1), (1, 0), and (1, 1), including its boundary. universal quantifier   universal Turing machine   A Turing machine capable of performing, given the appropriate program, the actions of any other Turing machine. unordered pair   See flat pair. upper bound   An upper bound of a set with an order relation (such as “ < ”) defined on it is an element which is greater than or equal to every element in the set.Cf. least upper bound. Von Neumann Heirarchy    ARTICLE   A construction in set theory giving rise to the class of ordinals. One begins with the empty set, and successor sets are obtained by forming the union of the current set and the set containing the current set; at limit stages, take unions. See the article for a detailed exposition. weakly connected digraph   A directed graph every two vertices of which are connected by a semipath. Also called weak for short. A directed graph is weak if and only if it has a spanning semiwalk. Cf. unilaterally connected digraph, strongly connected digraph, connected graph. well-founded   In set theory, a collection is well-founded if every subcollection has a least member under the membership relation. For example, the set of natural numbers is well-founded. In ZFC, the foundation axiom asserts this property of all sets. A set which is not well-founded is sometimes called a hyperset. well-ordered   A set S with a linear order is called well-ordered if every non-empty subset T of S has a least element under the ordering relation.Cf. well-ordering principle. well-ordering principle   The assertion that every set can be well-ordered. Equivalent to the Axiom of Choice. whole number   An element of the set consisting of the number 0 together with the counting numbers, 1, 2, 3, etc.; i.e., the set N of natural numbers with 0 included. Sometimes the term “whole” is meant to refer to negative integers also; the intended meaning should be clear from the context. Zeno’s Paradox of the Arrow   One of the famous paradoxes of Zeno of Elea. Consider an arrow in flight. At each moment of time the arrow may be considered to occupy a specific region in space. This poses the following problem: if the arrow is in a particular point of space in any given moment, then how is it moving? And if it is not moving, how does it get from point to point in the passage of time? Zermelo Fraenkel set theory   The standard set theory in which most mathematics is formalized. Its axioms include the pairing axiom, the power set axiom, the axiom of infinity, the axiom of extensionality, the axiom of replacement, the separation axiom, the union axiom, and the foundation axiom. Abbreviated ZF. When the axiom of choice is assumed, this theory is abbreviated ZFC. ZF   ZFC uncountable – ZFC
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