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		Tarski Truth Theorem – ultrafilter Tarski Truth Theorem   Let T be a mathematical theory. Denote formulas of T by j, y, etc., and let Tj be the statement that “the sentence j is true in T ” Choose a canonical numbering of the formulas of T (e.g., Gödel numbering) that assigns to each sentence of T a unique positive integer, and denote the positive integer associated with a sentence j by ‘j’. Then there is no formula y in T such that y(‘j’)Tj. In other words, “truth” in a theory T is not definable in T. tesselation   A tiling of the plane, i.e. the use of plane figures to completely cover the plane without overlaps or gaps. A regular tesselation uses only a finite number of distinct shapes. Most regular tesselations are periodic, but some are aperiodic. Cf. polygon, Penrose tiles. Related MiniText: Mathematical Art of M.C. Escher tetrahedron   A polyhedron having four faces. The faces of a regular tetrahedron are congruent, equilateral triangles. Cf. Platonic solid. tiling   See tesselation. topology   Generally, topology is the study of those properties of a space which are invariant under continuous deformations, i.e., deformations which do not create “tears” or “holes.” More specifically, given a set X, a topology on X is a collection of subsets of X, called the open sets of X, such that the empty set and X itself are included in the collection, and such that the collection is closed under the formation of finite intersections and arbitrary (i.e., not necessarily finite or countable) unions. A set X with a topology defined upon it is called a topological space. Cf. homeomorphism. totally ordered set   A set with a total order defined on it. total order   An order relation “<” on a set S is a total order exactly if, for any two elements x and y of S, either x < y, x = y, or y < x, but no two of the three. Sometimes also called a linear order. Every totally ordered set is order-isomorphic to an ordinal, and this ordinal is called its order type (or sometimes just “type.”) Cf. partial order transcendental function   A function which is not an algebraic function, i.e., a function whose action on its argument(s) cannot be represented by the arithmetic and algebraic operations: addition and subtraction, multiplication and division, raising to a power, or extraction of roots. The exponential function, the logarithmic function, and the trigonometric functions are examples of transcendental functions. transcendental number   A number which is not an algebraic number, i.e., that is not the root of any polynomial with rational coefficients. It is known that e and p (pi) are transcendental. Since the algebraic numbers are countable and the set of all real numbers is uncountable, this means that the set of transcendental numbers is uncountably large as well. transitive closure   If x is any set, the transitive closure of x is the set containing all of the elements of x, and all of the elements of the elements of x, and all of the elements of the elements of the elements of x, and so on. Formally, the transitive closure Tc(x) of x is defined by the recursive formulas where w is the set of finite ordinals. transitive relation   A relation “ ~ ” on a set X is called transitive if it is the case that for every x, y, and z in X, if x ~ y and y ~ z , then x ~ z. For example, the relation “ < ” (less-than) on the set of natural numbers is transitive. Cf. reflexive relation, symmetric relation. transitive set   A set is called transitive if it is equal to its transitive closure. That is, x is a transitive set if whenever y is an element of x and z is an element of y, then z is an element of x. tree   Graph Theory: A connected graph that does not contain any cycles. So named because they vaguely resemble trees in appearance. Cf. forest. Set Theory: A partial order T in which, for each x in T, the set {yT: y < x} is well-ordered. The height of x, denoted ht(x, T ), is the order type of the set {y T: y < x}. For each ordinal a, the a-th level of T, denoted Leva(T ), is the set {x T: ht(x, T ) = a}. The height ht(T ) of T is the least a such that Leva(T ) = . A sub-tree of T is a subset S T with the induced order such that whenever x S and y < x, then y S. triangle   Geometry: A closed plane figure with three straight sides meeting at three vertices. If one side of a triangle is chosen as the base, then the height of the triangle is the perpendicular distance to the base from the vertex opposite the base. Triangles are classified by their angle measures: Acute – all angles less than 90°. Obtuse – one angle greater than 90°. Right – one angle exactly 90°. Scalene – all angles and sides unequal. Isosceles – two angles equal (equivalently, two sides equal). Equilateral – all angles equal (equivalently, all sides equal). On a right triangle, the sides adjacent to the right angle are called the legs, and the side opposite is called the hypotenuse. Cf. Pythagorean Theorem. Graph Theory: A cycle with three vertices. trigonometric function   The “trig” functions are transcendental functions defined on angles. They include the sine, cosine, tangent, secant, cosecant, and cotangent functions. See the related article for a complete description. Related article: Trig Functions and Identities trigonometry   The mathematical subject concerned with trigonometric functions, which are defined on angles. They include the sine, cosine, tangent, secant, cosecant, and cotangent functions. Related article: Trig Functions and Identities trivial   The smallest, simplest, and usually least interesting example of any object or construction. Every field has a specific definition of what is considered the trivial object of study in that field. The following entries provide examples. Logic: A conclusion is trivial if it is so obvious that no proof or demonstration is required to establish its truth. trivial automorphism   The identity function is a trivial automorphism of any set. tuple   An ordered list of elements from a set, usually represented as (a1, a2, a3, ... ,an). (Sometimes angle brackets are used in place of parentheses.) A tuple with only two elements is called an ordered pair, and tuples with 3, 4, and 5 elements are called ordered triples, quadruples, and quintuples, respectively. In general, a tuple with n elements is called an n-tuple. A relation on a family of sets is represented by a set of tuples. Cf. flat pair. Turing machine   An abstract machine consisting of a collection of ordered quadruples, corresponding to i) the contents of the current memory location; ii) the current state of the machine; iii) the action to be performed (erase or write to the current memory location and move to a new memory location); and iv) the new state of the machine. All modern computers are universal Turing machines. ultrafilter   A filter F on a set X is called an ultrafilter if for every subset Y of X, either Y is in F or the complement of Y is in F.
		Tarski Truth Theorem – ultrafilter

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