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natural number – paradox
natural number
An element of the set N = {1, 2, 3, ...} consisting of all the “counting numbers.” When the number 0 is included, this set is sometimes called the whole numbers. In set theory, the natural numbers (incuding 0) are identified with the set w of finite ordinals. The natural numbers are a well-founded linear order with no largest member, and are countably infinite.
 Cf. Peano axioms, rational number, real number.
Related MiniText: Number -- What Is How Many?
negation
If j is a statement, sentence, or formula of logic, then the negation of j, denoted by  j, is that formula which is true whenever j is false, and false whenever j is true.
negative
The negative of a number or quantity x is the number, denoted -x, which when added to x yields 0. That is, the negative of a number is its additive inverse.
non-denumerable
Uncountable.
normal
A line intersecting a curve (or surface) perpendicular to the tangent line (or tangent plane) at the point of intersection. The normal to a surface expressed as a function of several variables xi is given by the gradient.
nowhere dense
Given a space X and a subset A of X, we say that A is nowhere dense if every open set of X contains an open subset that is disjoint from A. This is equivalent to saying that the complement of A is dense, or that A has empty interior.
number
There is no precise mathematical definition of the word “number.” There are however precise definitions of the terms “natural number,” “rational number,” “real number,” “complex number,” and other less commonly used kinds of number. When a mathematician speaks about numbers she usually has one of these cases in mind and she should, at the outset, make it clear to which type of number she is referring. The naive, inborn concept of number that is shared to some degree by all humans is a matter for philosophical rather than strictly mathematical inquiry, and it may be noted that there has historically been strong opposition to the introduction of new generalizations of established concepts of number.
numeral
Graphical symbol representing a number.
obtuse
An angle is called obtuse if it is greater than a right angle, that is, if its measure is greater than 90° (p/2 radians). A triangle is called obtuse if one of its angles is obtuse.
Cf. acute.
octahedron
A polyhedron having eight faces.
The faces of a regular octahedron are congruent, equilateral triangles.
Cf. Platonic solid, polyhedron.
open interval
An interval of the real number line (or any other totally ordered set) which does not include its endpoints. An interval containing only one of its endpoints is called half-open.
Cf. closed interval.
ordered field
See field.
ordered pair
An ordered tuple (a,b), the first element of which is called the abscissa, and the second element the ordinate, and for which (a,b) = (b,a) if and only if a = b. Functions, graphs of functions, and binary relations are represented as sets of ordered pairs. In standard set theory, the ordered pair (a,b) is defined to be the set { {a}, {a,b} }.
Cf. flat pair.
ordered set
A set with an order relation defined on it.
Cf. partial order, total order.
order of operations
As a matter of convention, in any given expression involving arithmetic and/or algebraic operations, operations within parentheses (or other grouping symbols) are evaluated first. Within this constraint, exponentiation precedes multiplication and division, and the latter precede addition and subtraction. Within these constraints, operations are evaluated from left to right.
order-preserving function
A function f is called order-preserving if it preserves the order of its domain elements, that is, if whenever x and y are elements of its domain such that x  y then f(x) f(y). Also called isotone or inctreasing. If f reverses the order of its domain elements, then it is called antitone or decreasing. In either case f is called monotone or monotonic. If whenever x < y we have f(x) < f(y), then f is called strictly increasing (resp. decreasing).
order relation
A relation R on a set S is an order relation exactly if it is reflexive, transitive and antisymmetric. Order relations are usually denoted by “ < ” or “ ”.
Cf. partial order, total order
order type
See total order.
ordinal
The class of ordinals is defined by:- 0 is an ordinal;
- if a is an ordinal, then a + 1 = a union {a}) is an ordinal;
- if A is a collection of ordinals, then union(A) is an ordinal;
- nothing else is an ordinal.
The class of ordinals is transitive, and is a well-founded, linear ordering. An ordinal of the form a + 1 is called a successor ordinal, and is otherwise called a limit ordinal.
Cf. Von Neumann Heirarchy.
ordinate
The second element of an ordered pair.
Cf. abscissa.
paradox
A seemingly 'necessary' contradiction or absurdity. A paradox arising logically out of formal axioms is called an antinomy. True paradoxes may be broadly classified as paradoxes of logic, paradoxes of infinity, paradoxes of knowledge, paradoxes of language, and paradoxes of self-reference. See:Antistrephon Paradox
Banach-Tarski Paradox
Berry Paradox
Boundary Paradox
Finitude Paradox
First Boring Number Paradox
Grelling's Paradox
Grue-Bleen Paradox
Liar Paradox
Quine's Paradox
Prisoner's Dilemma
Russell Paradox
Santa Sentence Paradox
Sid's Paradox
Sorites Paradox
Unexpected Hanging Paradox
Zeno's Paradox of the Arrow
Zenos Paradox of the Moving Rows
Zeno's Paradox of the Tortoise and Achilles
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