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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]].

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,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).[[Russell's paradox]], [[non-standard arithmetic]], [[hyperreal number]]s, [[projective geometry]], [[Affinely extended real number system|extended real number]]s and the [[absolute Infinite]].

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,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).[[Russell's paradox]], [[non-standard arithmetic]], [[hyperreal number]]s, [[projective geometry]], [[Affinely extended real number system|extended real number]]s and the [[absolute Infinite]].

=== Logic ===

=== Logic ===

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]].

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 ===

=== Infinity symbol ===

[[John Wallis]] is usually credited with introducing ∞ as a symbol for infinity in [[1655]] in

[[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>

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>[https://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 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.

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".

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)

:'''Pūrṇam adaḥ pūrṇam idam''' (That is full, this is full)

:'''{{Unicode|pūrṇāt pūrṇam udacyate}}''' (From the full, the full is subtracted)

:'''pūrṇāt pūrṇam udacyate''' (From the full, the full is subtracted)

:'''{{Unicode|pūrṇasya pūrṇam ādāya}}''' (When the full is taken from the full)

:'''pūrṇasya pūrṇam ādāya''' (When the full is taken from the full)

:'''{{Unicode|pūrṇam evāvasiṣyate'''}} (The full still will remain.) - [[Isha Upanishad]]

:'''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:

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:

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).

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.

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]] (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.

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 ''asaṃkhyāta'' ("countless, innumerable") and ''ananta'' ("endless, unlimited"), between rigidly bounded and loosely bounded infinities.

[[Category: General Reference]]

[[Category: General Reference]]