Infinity

Infinity (symbol: ) is a concept describing something without any bound or larger than any natural number. Philosophers have speculated about the nature of the infinite, for example Zeno of Elea, who proposed many paradoxes involving infinity, and Eudoxus of Cnidus, who used the idea of infinitely small quantities in his method of exhaustion. Modern mathematics uses the general concept of infinity in the solution of many practical and theoretical problems, such as in calculus and set theory, and the idea is also used in physics and the other sciences.

In mathematics, "infinity" is often treated as a number (i.e., it counts or measures things: "an infinite number of terms") but it is not the same sort of number as either a natural or a real number.

Georg Cantor formalized many ideas related to infinity and infinite sets during the late 19th and early 20th centuries. In the theory he developed, there are infinite sets of different sizes (called cardinalities).[1] For example, the set of integers is countably infinite, while the infinite set of real numbers is uncountable.[2]

History

Ancient cultures had various ideas about the nature of infinity. The ancient Indians and Greeks did not define infinity in precise formalism as does modern mathematics, and instead approached infinity as a philosophical concept.

Early Greek

The earliest recorded idea of infinity comes from Anaximander, a pre-Socratic Greek philosopher who lived in Miletus. He used the word apeiron which means infinite or limitless.[3] However, the earliest attestable accounts of mathematical infinity come from Zeno of Elea (born c. 490 BCE), a pre-Socratic Greek philosopher of southern Italy and member of the Eleatic School founded by Parmenides. Aristotle called him the inventor of the dialectic.[4][5] He is best known for his paradoxes,[4] described by Bertrand Russell as "immeasurably subtle and profound".[6]

In accordance with the traditional view of Aristotle, the Hellenistic Greeks generally preferred to distinguish the potential infinity from the actual infinity; for example, instead of saying that there are an infinity of primes, Euclid prefers instead to say that there are more prime numbers than contained in any given collection of prime numbers.[7]

However, recent readings of the Archimedes Palimpsest have found that Archimedes had an understanding about actual infinite quantities. According to Nonlinear Dynamic Systems and Controls, Archimedes has been found to be "the first to rigorously address the science of infinity with infinitely large sets using precise mathematical proofs."[8]

Early Indian

The Jain mathematical text Surya Prajnapti (c. 4th–3rd century BCE) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:[9]

  • Enumerable: lowest, intermediate, and highest
  • Innumerable: nearly innumerable, truly innumerable, and innumerably innumerable
  • Infinite: nearly infinite, truly infinite, infinitely infinite

In this work, two basic types of infinite numbers are distinguished. On both physical and ontological grounds, a distinction was made between asaṃkhyāta ("countless, innumerable") and ananta ("endless, unlimited"), between rigidly bounded and loosely bounded infinities.[10]

17th century

European mathematicians started using infinite numbers and expressions in a systematic fashion in the 17th century. In 1655 John Wallis first used the notation for such a number in his De sectionibus conicis and exploited it in area calculations by dividing the region into infinitesimal strips of width on the order of [11] But in Arithmetica infinitorum (1655 also) he indicates infinite series, infinite products and infinite continued fractions by writing down a few terms or factors and then appending "&c." For example, "1, 6, 12, 18, 24, &c."[12]

In 1699 Isaac Newton wrote about equations with an infinite number of terms in his work De analysi per aequationes numero terminorum infinitas.[13]

Other Languages
Alemannisch: Unendlichkeit
አማርኛ: አዕላፍ
العربية: لانهاية
aragonés: Infinito
অসমীয়া: অসীম
asturianu: Infinitu
azərbaycanca: Sonsuzluq
বাংলা: অসীম
Bân-lâm-gú: Bû-hān
башҡортса: Сикһеҙлек
беларуская: Бесканечнасць
беларуская (тарашкевіца)‎: Бясконцасьць
български: Безкрайност
bosanski: Beskonačnost
català: Infinit
Чӑвашла: Вĕçсĕрлĕх
čeština: Nekonečno
corsu: Infinitu
Cymraeg: Anfeidredd
Deutsch: Unendlichkeit
eesti: Lõpmatus
Ελληνικά: Άπειρο
español: Infinito
Esperanto: Senfineco
euskara: Infinitu
فارسی: بی‌نهایت
français: Infini
Gaeilge: Éigríoch
galego: Infinito
贛語: 無限
ગુજરાતી: અનંત
한국어: 무한
हिन्दी: अनंत
hrvatski: Beskonačnost
Ilokano: Awan inggana
Bahasa Indonesia: Tak hingga
íslenska: Óendanleiki
עברית: אינסוף
ಕನ್ನಡ: ಅನಂತ
ქართული: უსასრულობა
қазақша: Шексіздік
kurdî: Bêdawî
Latina: Infinitas
latviešu: Bezgalība
lietuvių: Begalybė
la .lojban.: li ci'i
magyar: Végtelen
македонски: Бесконечност
Malagasy: Tsiefa
മലയാളം: അനന്തത
मराठी: अनंत
Bahasa Melayu: Ketakterhinggaan
монгол: Хязгааргүй
မြန်မာဘာသာ: အနန္တ
Nederlands: Oneindigheid
日本語: 無限
norsk: Uendelig
norsk nynorsk: Uendeleg
occitan: Infinit
oʻzbekcha/ўзбекча: Cheksizlik
ਪੰਜਾਬੀ: ਅਨੰਤ
پنجابی: انانتی
Patois: Infiniti
português: Infinito
română: Infinit
русиньскый: Бесконечность
Scots: Infinity
සිංහල: අනන්තය
Simple English: Infinity
slovenčina: Nekonečno
slovenščina: Neskončnost
کوردی: بێکۆتایی
српски / srpski: Бесконачност
srpskohrvatski / српскохрватски: Beskonačnost
svenska: Oändlighet
தமிழ்: முடிவிலி
татарча/tatarça: Чиксезлек
తెలుగు: అనంతము
тоҷикӣ: Беинтиҳоӣ
Türkçe: Sonsuz
українська: Нескінченність
Tiếng Việt: Vô tận
文言: 無限
Winaray: Infinidad
粵語: 無限大
žemaitėška: Begalībė
中文: 无穷