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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."''Cambridge Dictionary of Philosophy'', Second Edition, p. 429
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."''Cambridge Dictionary of Philosophy'', Second Edition, p. 429
The word '''infinity''' comes from the [[Latin]] ''infinitas'' or "unboundedness." It refers to several distinct concepts (usually linked to the idea of "without end") which arise in [[philosophy]], [[mathematics]], and [[theology]].
In [[mathematics]], "infinity" is often used in contexts where it is treated as if it were a number (i.e., it counts or measures things: "an infinite number of terms") but it is a different type of "number" than the [[real numbers]]. Infinity is related to [[limit (mathematics)|limit]]s, [[aleph number]]s, [[class (set theory)|class]]es in [[set theory]], [[Dedekind-infinite set]]s, [[large cardinal]]s,<ref>Large cardinals are quantitative infinities defining the number of things in a [[Set|collection]], which are so large that they cannot be proven to exist in the ordinary mathematics of [[ZFC|Zermelo-Fraenkel plus Choice]] (ZFC).</ref> [[Russell's paradox]], [[non-standard arithmetic]], [[hyperreal number]]s, [[projective geometry]], [[Affinely extended real number system|extended real number]]s and the [[absolute Infinite]].
=== Infinity symbol ===
{{Unreferencedsection|date=June 2007}}
[[Image:John Wallis.jpg|thumb|200px|right|John Wallis introduced the infinity symbol to mathematical literature.]]
The precise origin of the infinity symbol '''∞''' is unclear. One possibility is suggested by the name it is sometimes called — the [[lemniscate]], from the Latin ''lemniscus'', meaning "ribbon." One can imagine walking forever along a simple loop formed from a ribbon.
A popular explanation is that the infinity symbol is derived from the shape of a [[Möbius strip]]. Again, one can imagine walking along its surface forever. However, this explanation is improbable, since the symbol had been in use to represent infinity for over two hundred years before [[August Ferdinand Möbius]] and [[Johann Benedict Listing]] discovered the Möbius strip in [[1858]].
It is also possible that it is inspired by older [[religious]]/[[alchemical]] [[symbolism]]. For instance, it has been found in [[Tibet]]an [[rock carvings]], and the [[ouroboros]], or infinity snake, is often depicted in this shape. In the [[Rider-Waite tarot deck]], the lemniscate represents the [[balance]] of [[forces]] and is often associated with the [[The Magician (Tarot card)|magician card]].
[[John Wallis]] is usually credited with introducing ∞ as a symbol for infinity in [[1655]] in
his ''De sectionibus conicis''. One conjecture about why he chose this symbol is that he derived it from a [[Roman numeral]] for 1000 that was in turn derived from the [[Etruscan numerals|Etruscan numeral]] for 1000, which looked somewhat like <font face="Arial Unicode MS, Lucida Sans Unicode">CIƆ</font> and was sometimes used to mean "many." Another conjecture is that he derived it from the Greek letter ω ([[omega]]), the last letter in the [[Greek alphabet]].<ref>[http://www.roma.unisa.edu.au/07305/symbols.htm#Infinity The History of Mathematical Symbols], By Douglas Weaver, Mathematics Coordinator, Taperoo High School with the assistance of Anthony D. Smith, Computing Studies teacher, Taperoo High School.</ref>
Another possibility is that the symbol was chosen because it was easy to rotate an "8" character by 90° when [[typesetting]] was done by hand. The symbol is sometimes called a "lazy eight", evoking the image of an "8" lying on its side.
Another popular belief is that the infinity symbol is a clear depiction of the hour glass turned 90°. Obviously, this action would cause the hour glass to take infinite time to empty thus presenting a tangible example of infinity. The invention of the hourglass predates the existence of the infinite symbol allowing this theory to be plausible. <!-- Is this true? -->
The infinity symbol is represented in [[Unicode]] by the character ∞ (U+221E).
== History ==
=== Early Indian views of infinity ===
The [[Isha Upanishad]] of the [[Yajurveda]] (c. 4th to 3rd century BC) states that "if you remove a part from infinity or add a part to infinity, still what remains is infinity".
:'''{{Unicode|Pūrṇam adaḥ pūrṇam idam}}''' (That is full, this is full)
:'''{{Unicode|pūrṇāt pūrṇam udacyate}}''' (From the full, the full is subtracted)
:'''{{Unicode|pūrṇasya pūrṇam ādāya}}''' (When the full is taken from the full)
:'''{{Unicode|pūrṇam evāvasiṣyate'''}} (The full still will remain.) - [[Isha Upanishad]]
The Indian [[Indian mathematics|mathematical]] text ''Surya Prajnapti'' (c. [[400 BC]]) classifies all numbers into three sets: enumerable, innumerable, and infinite. Each of these was further subdivided into three orders:
* Enumerable: lowest, intermediate and highest
* Innumerable: nearly innumerable, truly innumerable and innumerably innumerable
* Infinite: nearly infinite, truly infinite, infinitely infinite
The [[Jainism|Jains]] were the first to discard the idea that all infinites were the same or equal. They recognized different types of infinities: infinite in length (one [[dimension]]), infinite in area (two dimensions), infinite in volume (three dimensions), and infinite perpetually (infinite number of dimensions).
According to Singh (1987), Joseph (2000) and Agrawal (2000), the highest enumerable number ''N'' of the Jains corresponds to the modern concept of [[Aleph number|aleph-null]] <math>\aleph_0</math> (the [[cardinal number]] of the infinite set of integers 1, 2, ...), the smallest cardinal [[transfinite number]]. The Jains also defined a whole system of infinite cardinal numbers, of which the highest enumerable number ''N'' is the smallest.
In the Jaina work on the [[Set theory|theory of sets]], two basic types of infinite numbers are distinguished. On both physical and [[Ontology|ontological]] grounds, a distinction was made between {{IAST|''asaṃkhyāta''}} ("countless, innumerable") and ''ananta'' ("endless, unlimited"), between rigidly bounded and loosely bounded infinities.
[[Category: General Reference]]
[[Category: General Reference]]
