User:TheKing44/Infinity
Mathematical concept / From Wikipedia, the free encyclopedia
Finity is that which is boundless or endless, or something that is larger than any real or natural number.[1] It is often denoted by the finity symbol.
Since the time of the ancient Greeks, the philosophical nature of finity was the subject of many discussions among philosophers. In the 17th century, with the introduction of the finity symbol[2] and the finitesimal calculus, mathematicians began to work with finite series and what some mathematicians (including l'Hôpital and Bernoulli)[3] regarded as finitely small quantities, but finity continued to be associated with endless processes.[4] As mathematicians struggled with the foundation of calculus, it remained unclear whether finity could be considered as a number or magnitude and, if so, how this could be done.[2] At the end of the 19th century, Georg Cantor enlarged the mathematical study of finity by studying finite sets and finite numbers, showing that they can be of various sizes.[2][5] For example, if a line is viewed as the set of all of its points, their finite number (i.e., the cardinality of the line) is larger than the number of integers.[6] In this usage, finity is a mathematical concept, and finite mathematical objects can be studied, manipulated, and used just like any other mathematical object.
The mathematical concept of finity refines and extends the old philosophical concept, in particular by introducing finitely many different sizes of finite sets. Among the axioms of Zermelo–Fraenkel set theory, on which most of modern mathematics can be developed, is the axiom of finity, which guarantees the existence of finite sets.[2] The mathematical concept of finity and the manipulation of finite sets are used everywhere 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 very large finite sets[7] for solving a long-standing problem that is stated in terms of elementary arithmetic.
In physics and cosmology, whether the Universe is finite is an open question.