Infinity
Infinity is something which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol .
For the symbol, see Infinity symbol. For other uses of "Infinity" and "Infinite", see Infinity (disambiguation). Not to be confused with Infiniti.
Since the time of the ancient Greeks, the philosophical nature of infinity has been the subject of many discussions among philosophers. In the 17th century, with the introduction of the infinity symbol[1] and the infinitesimal calculus, mathematicians began to work with infinite series and what some mathematicians (including l'Hôpital and Bernoulli)[2] regarded as infinitely small quantities, but infinity continued to be associated with endless processes. As mathematicians struggled with the foundation of calculus, it remained unclear whether infinity could be considered as a number or magnitude and, if so, how this could be done.[1] At the end of the 19th century, Georg Cantor enlarged the mathematical study of infinity by studying infinite sets and infinite numbers, showing that they can be of various sizes.[1][3] For example, if a line is viewed as the set of all of its points, their infinite number (i.e., the cardinality of the line) is larger than the number of integers.[4] In this usage, infinity is a mathematical concept, and infinite mathematical objects can be studied, manipulated, and used just like any other mathematical object.
The mathematical concept of infinity refines and extends the old philosophical concept, in particular by introducing infinitely many different sizes of infinite sets. Among the axioms of Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of infinity, which guarantees the existence of infinite sets.[1] The mathematical concept of infinity and the manipulation of infinite sets are widely used in mathematics, even in areas such as combinatorics that may seem to have nothing to do with them. For example, Wiles's proof of Fermat's Last Theorem implicitly relies on the existence of Grothendieck universes, very large infinite sets,[5] for solving a long-standing problem that is stated in terms of elementary arithmetic.
In physics and cosmology, whether the universe is spatially infinite is an open question.
means that does not bound a finite area from to
means that the area under is infinite.
means that the total area under is finite, and is equal to
Logic[edit]
In logic, an infinite regress argument is "a distinctively philosophical kind of argument purporting to show that a thesis is defective because it generates an infinite series when either (form A) no such series exists or (form B) were it to exist, the thesis would lack the role (e.g., of justification) that it is supposed to play."[55]
Computing[edit]
The IEEE floating-point standard (IEEE 754) specifies a positive and a negative infinity value (and also indefinite values). These are defined as the result of arithmetic overflow, division by zero, and other exceptional operations.[56]
Some programming languages, such as Java[57] and J,[58] allow the programmer an explicit access to the positive and negative infinity values as language constants. These can be used as greatest and least elements, as they compare (respectively) greater than or less than all other values. They have uses as sentinel values in algorithms involving sorting, searching, or windowing.
In languages that do not have greatest and least elements, but do allow overloading of relational operators, it is possible for a programmer to create the greatest and least elements. In languages that do not provide explicit access to such values from the initial state of the program, but do implement the floating-point data type, the infinity values may still be accessible and usable as the result of certain operations.
In programming, an infinite loop is a loop whose exit condition is never satisfied, thus executing indefinitely.
Arts, games, and cognitive sciences[edit]
Perspective artwork uses the concept of vanishing points, roughly corresponding to mathematical points at infinity, located at an infinite distance from the observer. This allows artists to create paintings that realistically render space, distances, and forms.[59] Artist M.C. Escher is specifically known for employing the concept of infinity in his work in this and other ways.
Variations of chess played on an unbounded board are called infinite chess.[60][61]
Cognitive scientist George Lakoff considers the concept of infinity in mathematics and the sciences as a metaphor. This perspective is based on the basic metaphor of infinity (BMI), defined as the ever-increasing sequence <1,2,3,...>.[62]