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'''Infinity''' means beyond enumeration.<ref>This improved definition comports best with the use of the term in mathematics, in everyday life, and across cultures and languages.</ref>  Appreciation of infinity is remarkably useful mentally, emotionally, and on a personal level.  "Paradise" is infinite happiness; "[[Hell]]" is infinite distress; in electronics, a "open circuit" has infinite resistance. Anxiety or fear about infinity is a psychiatric condition known as "apeirophobia", which exists because infinity is real.

'''Infinity''' means beyond enumeration.<ref>This improved definition comports best with the use of the term in mathematics, in everyday life, and across cultures and languages.</ref>  Appreciation of infinity -- particularly infinite time -- is remarkably useful mentally and emotionally.  "Paradise" is infinite [[happiness]] and [[joy]] eternally; "[[Hell]]" is infinite distress; in electronics, an "open circuit" has infinite resistance.<ref>Anxiety or fear about infinity is a psychiatric condition known as "apeirophobia", which exists because infinity is real.</ref> '''''Recognition of infinite time can overcome almost any anxiety'''''. Financial-related stress can be reduced by recognizing that money is infinite.  A human being has infinite potential.

The food supply is essentially infinite in cultures where obesity is a problem; wealth is essentially infinite for those who cannot spend it all in their lifetime; and time is infinite for those who waste it.  "More" in those situations is meaningless; less is nothing to have anxiety about.  Some scoff at primitive cultures that use infinity rather than large numbers, but that is actually productive in viewing quantities larger than meaningful.  Also, infinity reinforces the primacy of the future (which is infinite) over the past (merely finite).  Given that "time is money," "infinite time" ([[eternal life]]) is "infinite money" and the concept is helpful in dealing with financial anxiety.

[[File:Abbey Church Saint Louis.jpg|right|200px|thumb|Abbey Church: infinite access]]

 

The food supply is essentially infinite in cultures where obesity is a problem; wealth is essentially infinite for those who cannot spend it all in their lifetime; and time is infinite for those who waste it.  "More" in those situations is meaningless; less is nothing to have anxiety about.  Some scoff at primitive cultures that use infinity rather than large numbers, but that is a productive way to view quantities larger than meaningful.  Also, infinity reinforces the primacy of the future (which is infinite) over the past (merely finite).  Given that "time is money," "infinite time" ([[eternal life]]) is "infinite money" which helps reduce financial anxiety.

[[File:Abbey Church Saint Louis.jpg|right|200px|thumb|church's infinite accessibility and outreach]]

Understanding infinity has advanced many fields, including:

Understanding infinity has advanced many fields, including:

*[[addiction]], which is an preference or obsession of infinite intensity;

*[[addiction]], which is a preference or obsession of infinite intensity;

*[[architecture]], as in the [[Abbey Church]] in [[St. Louis]] that embraces infinite equal access and reach;

*[[architecture]], as in the [[Abbey Church]] in [[St. Louis]] that embraces infinite accessibility and outreach;

*[[mathematics]] through [[calculus]], [[set theory]] and [[singularity|singularities]];  

*[[mathematics]] through [[calculus]], [[set theory]] and [[singularity|singularities]];  

*[[quantum mechanics]] through infinite [[uncertainty]];  

*[[quantum mechanics]] through infinite [[uncertainty]];  

*[[economics]] through the infinite opportunity discovered by the [[Coase theorem]];  

*[[economics]] through the infinite opportunity discovered by the [[Coase theorem]];  

*[[physics]] through the [[Big Bang]] (infinite [[energy]] density) and the speculative [[black holes]] (infinite mass density);  

*[[physics]] through the [[Big Bang]] (infinite [[energy]] density) and the speculative [[black holes]] (infinite mass density);  

*[[politics]] through dictatorships (infinite authority) and [[separation of powers]] to avoid concentration of infinite power;

*[[politics]] through dictatorships (infinite authority) and [[separation of powers]] to avoid concentration of infinite power;

*[[religion]] through the concepts like [[eternity]] (infinite time), [[Heaven]]/[[Hell]] (infinite joy/suffering), [[faith]] (infinite belief), [[Parable of the Vineyard Workers]] and the [[Prodigal Son]] (infinite wealth), the infinity of [[God]],<ref>https://www.newadvent.org/cathen/08004a.htm</ref> and the number of [[angel]]s (infinite);<ref>[[Epistle_to_the_Hebrews_(Translated)#12:22|Hebrew 12:22]] ("infinite").</ref>

*[[religion]] through the concepts like [[eternity]] (infinite time), [[Heaven]]/[[Hell]] (infinite joy/suffering), [[faith]] (infinite good from the force of will), [[Parable of the Vineyard Workers]] and the [[Prodigal Son]] (infinite wealth), [[Widow's Mite]] (infinite [[charity]]), the infinity of [[God]],<ref>https://www.newadvent.org/cathen/08004a.htm</ref> and the number of [[angel]]s (infinite);<ref>[[Epistle_to_the_Hebrews_(Translated)#12:22|Hebrew 12:22]] ("infinite").</ref>

*[[statistics]] through the [[central limit theorem]], which states that the distribution of the average for a series of nearly any type of distribution approaches the [[normal distribution]];

*[[statistics]] through the [[central limit theorem]], which states that the distribution of the average for a series of nearly any type of distribution approaches the [[normal distribution]];

*[[literature]], as in ''An Occurrence at Owl Creek Bridge'' (1890) and many great works inspired by it, which dramatically use a nearly infinite expansion of an infinitesimal time; writings by [[Nathaniel Hawthorne]] ("Nothing for the present hour is worthy of man's infinite capacity save to imbibe the warm smile of heaven and sympathize with the reviving earth." - ''Buds and Bird Voices''); and plays by [[Shakespeare]], who used infinity copiously (see below);

*[[literature]], as in ''An Occurrence at Owl Creek Bridge'' (1890) and many great works inspired by it, which dramatically use a nearly infinite expansion of an infinitesimal time; writings by [[Nathaniel Hawthorne]] ("Nothing for the present hour is worthy of man's infinite capacity save to imbibe the warm smile of heaven and sympathize with the reviving earth." - ''Buds and Bird Voices''); plays by [[Shakespeare]], who used infinity copiously (see below); and 13 times in ''[[Moby-Dick]]'';

*[[biology]], as observed by [[Louis Pasteur]]: “The role of the infinitely small in nature is infinitely great”;<ref>https://www.goodreads.com/author/quotes/692216.Louis_Pasteur</ref>

*[[biology]], as observed by [[Louis Pasteur]]: “The role of the infinitely small in nature is infinitely great”;<ref>https://www.goodreads.com/author/quotes/692216.Louis_Pasteur</ref>

*[[sociology]]: traditional [[marriage]] is infinitely more stable than other types of relationships;

*[[sociology]]: traditional [[marriage]] is infinitely more stable than other types of relationships;

*[[rock music]]: paradise (in love songs, such as "Two Tickets to Paradise") represents infinite happiness;  

*[[rock music]]: paradise (in love songs, such as "Two Tickets to Paradise") represents infinite happiness;  

*paintings: [[Salvador Dali]]'s work (including his infinite time in ''[[The Persistence of Memory]]''), and other [[Surrealism]] such as ''Mama, Papa Is Wounded!'' (1927);<ref>https://www.history.com/topics/art-history/surrealism-history</ref>

*paintings: [[Salvador Dali]]'s work (including his infinite time in ''[[The Persistence of Memory]]''), and other [[Surrealism]] such as ''Mama, Papa Is Wounded!'' (1927);<ref>https://www.history.com/topics/art-history/surrealism-history</ref>

*movie titles: as in ''From Here to Eternity'' (1953), ''Avengers: Infinity War'' (2018), ''Infinity'' (1996), and the TV show ''Forever'' (2014-15);

*movies: ''From Here to Eternity'' (1953), ''Avengers: Infinity War'' (2018), ''Infinity'' (1996), and the TV show ''Forever'' (2014-15); famous lines from movies include "To Infinity and Beyond!" (Buzz Lightyear in the ''Toy Story'' films).

*understanding other terms: infinity helps define other terms concisely, such as "open circuit" (infinite resistance);

*understanding other terms: infinity helps define other terms concisely, such as "open circuit" (infinite resistance);

*[[Resurrection]]: personal awareness of infinite time implies, as a matter of logic, personal realization of infinite time;

*[[Resurrection]]: personal awareness of infinite time implies, as a matter of logic, actual realization of infinite time through resurrection;

*[[Abraham Lincoln]]: "Surely God would not have created such a being as man, with an ability to grasp the infinite, to exist only for a day! No, no, man was made for immortality";<ref>https://www.brainyquote.com/quotes/abraham_lincoln_121271</ref> and

*[[Abraham Lincoln]]: "Surely God would not have created such a being as man, with an ability to grasp the infinite, to exist only for a day! No, no, man was made for immortality";<ref>https://www.brainyquote.com/quotes/abraham_lincoln_121271</ref>

*in [[infinity in engineering|engineering]]: in a DC circuit, a capacitor has infinite resistance, while [[Earth]] has infinite capacitance.

In addition, recognition of infinity — such as infinite time or infinite strength — is helpful on a personal level in dealing with adversity.  Infinite time means infinite money, since time is convertible into money.  [[Jesus]] often emphasized the concept of infinity in his [[miracle]]s and [[parable]]s (see below).  He rejected the mistaken view of the ancient Greeks that infinity is non-existent and incoherent.<ref>{{cquote|The Ancient Greeks generally conceived of the infinite as formless, characterless, indefinite, indeterminate, chaotic, and unintelligible. The term had negative connotations and was especially vague, having no clear criteria for distinguishing the finite from the infinite.}} [http://www.iep.utm.edu/infinite/]

*in [[infinity in engineering|engineering]]: in a DC circuit, a capacitor has infinite resistance, while [[Earth]] has infinite capacitance;

*in carpentry and [[materials science]]: wood of infinite density is needed to build a spiral staircase with multiple twists and no supporting column, which can hold a dozen people at once - see [[Loretto spiral staircase]].

 

In addition, recognition of infinity — such as infinite time or infinite strength — is helpful on a personal level in dealing with adversity.  Infinite time means infinite money because time is convertible into money.  [[Jesus]] often emphasized the concept of infinity in his [[miracle]]s and [[parable]]s (see below).  He rejected the mistaken view of the ancient Greeks that infinity is non-existent and incoherent.<ref>{{cquote|The Ancient Greeks generally conceived of the infinite as formless, characterless, indefinite, indeterminate, chaotic, and unintelligible. The term had negative connotations and was especially vague, having no clear criteria for distinguishing the finite from the infinite.}} [http://www.iep.utm.edu/infinite/]

The ancient Greeks were flat-out wrong.</ref>  Today acceptance of infinity is universal, despite the adamant denial by the ancient Greeks and others throughout history.  The devoutly [[Christian]] mathematician [[Georg Cantor]] even proved that some infinities are larger than others.

The ancient Greeks were flat-out wrong.</ref>  Today acceptance of infinity is universal, despite the adamant denial by the ancient Greeks and others throughout history.  The devoutly [[Christian]] mathematician [[Georg Cantor]] even proved that some infinities are larger than others.

The existence of an infinite [[God]] implies the necessary existence of the opposite: the void described in [[Genesis]].  Negative infinity, as implied by the existence of positive infinity, is the numeric equivalent of the [[Hell]] mentioned frequently in the [[New Testament]].  The [[logic of infinity]] permeates the [[Gospels]].

The existence of an infinite [[God]] implies the necessary existence of the opposite: the void described in [[Genesis]].  Negative infinity, as implied by the existence of positive infinity, is the numeric equivalent of the [[Hell]] mentioned frequently in the [[New Testament]].  The [[logic of infinity]] permeates the [[Gospels]].

== Quantity in general ==

== Infinity and biblical theology ==

 

 

 

 

 

 

 

 

The infinite is thought to be a type of quantity. But what is a quantity?  The idea of quantity may not be definable, as Aristotle regarded it as a category, that is, something that applies to everything, but we may be able to note the conditions under which it exists.

'''4.''' Because God is omnipotent and omniscient, he knew which possible world was the best and was able to create it, and, because he is omnibenevolent, he chose to create that world;

But such conditions made distinct, may be assimilable only by the most compendious of mental capacities, and those with active intellects without the capacity to receive it may not benefit from the distinctions, due to difficulties in assimilation. We first learn by ignoring all but a single aspect of a problem, for which a solution is then found. So if in assimilating such attempts at solution through conceptualization, more important ones suggest themselves to us, it is best to ignore other aspects of the proposed solution until those new problems have been resolved, resolutions which might very well serve as grounds for impeaching the proposed solution prior to any further assimilation.

'''5.''' Therefore, the existing world, the one that God created, is the best of all possible worlds.<ref>[https://www.britannica.com/topic/best-of-all-possible-worlds Best of all possible worlds], Encyclopedia Britannica</ref>}}

== Teachings about infinity by Jesus ==

=== Teachings about infinity by Jesus ===

Infinity is depicted by several miracles, including the [[Multiplication of the loaves]] and the [[Miraculous catch of fish]].  Infinity is implicit in several parables, including the [[Prodigal Son]], and the parable of the vineyard workers.<ref>Matthew 20:1-16.</ref>  In addition, one's soul is of infinite value to him and thus cannot be exceeded in value by anything else, as explained by Matthew 16:26.

Infinity is depicted by several miracles, including the [[Multiplication of the loaves]] and the [[Miraculous catch of fish]].  Infinity is implicit in several parables, including the [[Prodigal Son]], and the parable of the vineyard workers.<ref>Matthew 20:1-16.</ref>  In addition, one's soul is of infinite value to him and thus cannot be exceeded in value by anything else, as explained by Matthew 16:26.

Infinity is also portrayed by the Parables of the Hidden Treasure and the Pearl:<ref>Matthew 13:44-46 [[NIV]] version.</ref>

Infinity is also portrayed in the Parables of the Hidden Treasure and the Pearl:<ref>Matthew 13:44-46 [[NIV]] version.</ref>

:“The kingdom of heaven is like treasure hidden in a field. When a man found it, he hid it again, and then in his joy went and sold all he had and bought that field.

:“The kingdom of heaven is like treasure hidden in a field. When a man found it, he hid it again, and then in his joy went and sold all he had and bought that field.

:“Again, the kingdom of heaven is like a merchant looking for fine pearls. When he found one of great value, he went away and sold everything he had and bought it.”

:“Again, the kingdom of heaven is like a merchant looking for fine pearls. When he found one of great value, he went away and sold everything he had and bought it.”

[[Eternity]] is infinite time, a concept that was also first introduced and emphasized by Jesus.  The [[Old Testament]] has few references to "eternity" and "eternal", while the [[New Testament]] repeatedly teaches it.  For example, Jesus taught about infinity in connection with [[Judgment Day]] when "these will go off to eternal punishment, but the righteous to eternal life."<ref>Matt 45:26 ([[NAB]]).</ref>

[[Eternity]] is infinite time, a concept that was also first introduced and emphasized by Jesus.  The [[Old Testament]] has few references to "eternity" and "eternal", while the [[New Testament]] repeatedly teaches it.  For example, Jesus taught about infinity in connection with [[Judgment Day]] when "these will go off to eternal punishment, but the righteous to eternal life."<ref>Matt 45:26 ([[NAB]]).</ref>

== Mistranslated as "unsearchable" ==

=== Mistranslated as "unsearchable" ===

Many translations of the [[Bible]] mistakenly use the term "unsearchable" when "infinite" is the intended meaning. Ancient Greek lacked a word for "infinity", and instead used "apeiron" for indefinite or unlimited.<ref>Some say that obscure Greek term "aphorismenon" was closer to the modern term of infinity.</ref>

Many translations of the [[Bible]] mistakenly use the term "unsearchable" when "infinite" is the intended meaning. Ancient Greek lacked a word for "infinity", and instead used "apeiron" for indefinite or unlimited.<ref>Some say that the obscure Greek term "aphorismenon" was closer to the modern term of infinity.</ref>

For example, the otherwise superb [[New American Standard Bible]] mistranslates with "unsearchable" rather than "infinite" in six places:

For example, the otherwise superb [[New American Standard Bible]] mistranslates with "unsearchable" rather than "infinite" in six places:

* Ephesians 3:8 - To me, though I am the very least of all the saints, this grace was given, to preach to the Gentiles the unsearchable riches of Christ

* Ephesians 3:8 - To me, though I am the very least of all the saints, this grace was given, to preach to the Gentiles the unsearchable riches of Christ

The [[New American Standard Bible]] uses "unfathomable" rather than "unsearchable" there.  But why not use the more precise word, "infinite"?

The [[New American Standard Bible]] uses "unfathomable" rather than "unsearchable" there.  But why not use the more precise word, "infinite"?

Bible Hub has a superb analysis of the use of infinite in Bible translations.<ref>https://biblehub.com/topical/i/infinite.htm</ref>

Bible Hub has a superb analysis of the use of infinite in Bible translations.<ref>https://biblehub.com/topical/i/infinite.htm</ref>

== Anselm and Shakespeare ==

== Anselm and Shakespeare ==

[[Anselm]] focused on infinity to prove the divinity of [[Christ]].  [[Chaucer]] referenced "infinity" in 1374, according to the Oxford English Dictionary.  But [[Shakespeare]] used infinity more than anyone up until his time, in such classics as [[Hamlet]] ("A king of infinite space") and [[Romeo and Juliet]] (“My bounty is as boundless as the sea, My love as deep; the more I give to thee, The more I have, for both are infinite.”).  This passage from Hamlet is particularly memorable in referring to the infinite:<ref>Hamlet, 2.2.314</ref>

[[Anselm]] focused on infinity to prove the divinity of [[Christ]].  [[Chaucer]] referenced "infinity" in 1374, according to the Oxford English Dictionary.  But [[Shakespeare]] used infinity more than anyone up until his time, in such classics as [[Hamlet]] ("A king of infinite space") and [[Romeo and Juliet]] (“My bounty is as boundless as the sea, My love as deep; the more I give to thee, The more I have, for both are infinite.”).  This passage from Hamlet is particularly memorable in referring to the infinite:<ref>Hamlet, 2.2.314</ref>

{{cquote|What a piece of work is a man! how noble in reason! how infinite in faculty! in form and moving how express and admirable! in action how like an angel! in apprehension how like a god! the beauty of the world! the paragon of animals!}}

{{cquote|What a piece of work is a man! how noble in reason! how infinite in faculty! in form and moving how express and admirable! in action how like an angel! in apprehension how like a god! the beauty of the world! the paragon of animals!}}

Famous quotations by Shakespeare includes a remarkable 12 separate uses of infinity.<ref>https://keytopoetry.com/dev/william-shakespeare/quotations/</ref>

Famous quotations by Shakespeare include a remarkable 12 separate uses of infinity.<ref>https://keytopoetry.com/dev/william-shakespeare/quotations/</ref> Shakespeare's phrase "infinite ... liar" is superior to modern expressions like "complete liar" and "pathological liar," neither of which are as strong and descriptive.

Shakespeare in particular warns of the vicious circle of unlimited desire, will and power in his ''Troilus and Cressida'' (1600), speaking of moderation and degree:

Shakespeare in particular warns of the vicious circle of unlimited desire, will and power in his ''Troilus and Cressida'' (1600), speaking of moderation and degree:

:''Cressida'':  Will you walk in, my lord? (Act III, Scene II)

:''Cressida'':  Will you walk in, my lord? (Act III, Scene II)

== Similarities to zero ==

== Similarities to zero ==

Infinity has surprising similarities to [[zero]].  Neither can be diminished, and the multiplication or division of either cannot change it.  While infinity and zero are opposites, neither is stronger or superior to the other: multiplication of infinity by zero yields an indeterminate result.  As a pair they illustrate the logical truth that "the last shall be first, and the first shall be last" as taught by Jesus at the end of the parable of the vineyard workers, a parable that illustrates that infinity cannot be diminished.

Infinity has surprising similarities to [[zero]].  Neither can be diminished, and the multiplication or division of either cannot change it.  While infinity and zero are opposites, neither is stronger or superior to the other: multiplication of infinity by zero yields an indeterminate result.  As a pair, they illustrate the logical truth that "the last shall be first, and the first shall be last" as taught by Jesus at the end of the parable of the vineyard workers, a parable that illustrates that infinity cannot be diminished.

[[Leonhard Euler]], the most prolific mathematician in history, famously said about infinity and zero:<ref>https://www.brainyquote.com/quotes/leonhard_euler_184025</ref>

[[Leonhard Euler]], the most prolific mathematician in history, famously said about infinity and zero:<ref>https://www.brainyquote.com/quotes/leonhard_euler_184025</ref>

==Infinitesimal==

==Infinitesimal==

Inherent in the concept of infinity is the concept of [[infinitesimal]]:  infinity small. [[Archimedes]] used small numbers and areas, as in calculating the area under a parabola, but the [[Ancient Greeks]] rejected the concept of infinity.  Beginning with the 17th century, mathematicians like [[Fermat]], [[Euler]], [[Newton]], and [[Leibniz]] used the concept of infinitesimal in integrals and differentials. [[Bernhard Riemann]] used the concept of limits to develop a rigorous definition of the integral, now known as the [[Riemann Integral]].

Inherent in the concept of infinity is the concept of [[infinitesimal]]:  infinitely small. [[Archimedes]] used small numbers and areas, as in calculating the area under a parabola, but the [[Ancient Greeks]] rejected the concept of infinity.  Beginning with the 17th century, mathematicians like [[Fermat]], [[Euler]], [[Newton]], and [[Leibniz]] used the concept of infinitesimal in integrals and differentials. [[Bernhard Riemann]] used the concept of limits to develop a rigorous definition of the integral, now known as the [[Riemann Integral]].

== Infinity and economics ==

== Infinity and economics ==

== Infinity and sports ==

== Infinity and sports ==

In [[baseball]], a pitcher has an infinite Earned Run Average (ERA) when he allows earned runs without obtaining any "outs" by the other side, because the ERA is a pitcher's average earned runs allowed per nine innings pitched (i.e., 27 outs obtained).  In 2019, Nationals relief pitcher Trevor Rosenthal had an infinite ERA until he finally obtained an out.<ref>https://www.freep.com/story/sports/mlb/2019/04/11/trevor-rosenthal-nationals-reliever-infinity/3433399002/</ref>

In [[baseball]], a pitcher has an infinite Earned Run Average (ERA) when he allows earned runs without obtaining any "outs" by the opposing team, because the ERA is a pitcher's average earned runs allowed per nine innings pitched (i.e., 27 outs obtained).  In 2019, Nationals relief pitcher Trevor Rosenthal had an infinite ERA until he finally obtained an out.<ref>https://www.freep.com/story/sports/mlb/2019/04/11/trevor-rosenthal-nationals-reliever-infinity/3433399002/</ref>

Baseball has a potentially infinite duration, with no time limits.  For this and other reasons the game has been described as timeless.

Baseball has no clock and no time limits.  For this and other reasons, this classic game has been described as timeless (which is the same as infinite time).

== Ongoing disputes about infinity ==

== Ongoing disputes about infinity ==

The simplest place to see this seeming paradox is in the fact that all integers are finite, but that there are an infinite number of them.

The simplest place to see this seeming paradox is in the fact that all integers are finite, but that there are an infinite number of them.

*Why are all integers finite?  Because, if infinity were an integer, what would <math>\infty+1\,</math> be?

*Why are all integers finite?  Because, if infinity were an integer, what would <math>\infty+1\,</math> be?

*Why is the ''set'' of all integers an infinite set?  Because they go on forever. If the set were finite, there would be a biggest integer. But that can't be, because we can always add one to any integer. This kind of thinking is often schoolchildren's first introduction to logical reasoning.<ref>Merrian-Webster defines infinity as an "unlimited extent of time, space, or quantity ... an indefinitely great number or amount." [https://www.merriam-webster.com/dictionary/infinity Merriam-Webster dictionary]  Put another way, infinity is something which "goes on without end."</ref>

*Why is the ''set'' of all integers an infinite set?  Because they go on forever. If the set were finite, there would be a largest integer. But that can't be, because we can always add one to any integer. This kind of thinking is often schoolchildren's first introduction to logical reasoning.<ref>[[Merriam-Webster]] defines infinity as an "unlimited extent of time, space, or quantity ... an indefinitely great number or amount." [https://www.merriam-webster.com/dictionary/infinity Merriam-Webster dictionary]  Put another way, infinity is something which "goes on without end."</ref>

While infinity is not a number, it appears in many other contexts.  As we have seen, the size of the set of integers is infinite. We say that the [[cardinality]] of the integers (this set is often denoted <math>\mathbb{Z}</math>) is infinite.  The rational numbers (<math>\mathbb{Q}</math>) and the real numbers (<math>\mathbb{R}</math>) also have infinite cardinality.  An interesting result from set theory (see below) is that <math>\mathbb{Z}</math> and <math>\mathbb{Q}</math> have the same cardinality, while <math>\mathbb{R}</math> has larger cardinality than the other two.

While infinity is not a number, it appears in many other contexts.  As we have seen, the size of the set of integers is infinite. We say that the [[cardinality]] of the integers (this set is often denoted <math>\mathbb{Z}</math>) is infinite.  The rational numbers (<math>\mathbb{Q}</math>) and the real numbers (<math>\mathbb{R}</math>) also have infinite cardinality.  An interesting result from set theory (see below) is that <math>\mathbb{Z}</math> and <math>\mathbb{Q}</math> have the same cardinality, while <math>\mathbb{R}</math> has larger cardinality than the other two.

The cardinality of the natural numbers is known as aleph-null. There are as many integers as natural numbers; and as many rational numbers as natural numbers. But it can be shown, using Cantor's diagonalization argument, that the reals are strictly larger than the natural numbers - you cannot put the natural numbers into one to one correspondence with the reals. The cardinality of the reals is the cardinality of the power set of the natural numbers, which is known as the power of the continuum (c) or as beth-one. (Aleph-null is the same as beth-null; whether aleph-one is the same as beth-one is a mathematical question, which essentially has no answer - and we can even show it has no answer.)

The cardinality of the natural numbers is known as aleph-null. There are as many integers as natural numbers; and as many rational numbers as natural numbers. But it can be shown, using Cantor's diagonalization argument, that the reals are strictly larger than the natural numbers - you cannot put the natural numbers into one to one correspondence with the reals. The cardinality of the reals is the cardinality of the power set of the natural numbers, which is known as the power of the continuum (c) or as beth-one. (Aleph-null is the same as beth-null; whether aleph-one is the same as beth-one is a mathematical question, which essentially has no answer - and we can even show it has no answer.)

The '''generalized continuum hypothesis''' (GCH) states that <math>\beth_{a} = \aleph_{a}</math> for all ordinals <math>a</math>. So, the continuum hypothesis is a special case of the generalised continuum hypothesis for ordinal <math>0</math>. Like CH, GCH is independent of the axioms of ZFC.

The '''generalized continuum hypothesis''' (GCH) states that <math>\beth_{a} = \aleph_{a}</math> for all ordinals <math>a</math>. So, the continuum hypothesis is a special case of the generalized continuum hypothesis for ordinal <math>0</math>. Like CH, GCH is independent of the axioms of ZFC.

A set having cardinality equal to or less than <math>\aleph_{0}</math> is called countable; a set of greater cardinality is called uncountable.

A set having cardinality equal to or less than <math>\aleph_{0}</math> is called countable; a set of greater cardinality is called uncountable.

Other than transfinite cardinals and ordinals, there are other types of infinite numbers:

Other than transfinite cardinals and ordinals, there are other types of infinite numbers:

*the '''affinely extended reals''': these are elements of the set <math>\mathbb{R} \cup \{ +\infty, -\infty \}</math>. In other words, the real numbers extended by the addition of positive and negative infinity. <math>+\infty</math> is greater than every real number, and <math>-\infty</math> is less than every real number.

*the '''affinely extended reals''': these are elements of the set <math>\mathbb{R} \cup \{ +\infty, -\infty \}</math>. In other words, the real numbers extended by the addition of positive and negative infinity. <math>+\infty</math> is greater than every real number, and <math>-\infty</math> is less than every real number.

*the '''projectively extended reals''': these are elements of the set <math>\mathbb{R} \cup \{ \infty \}</math>. In other words, the real numbers extended by the addition of a single signless infinity. The ordering relation is not defined for the projectively extended reals - but informally, <math>\infty</math> can be said to be both greater than ''and'' less than every real number.

*the '''projectively extended reals''': these are elements of the set <math>\mathbb{R} \cup \{ \infty \}</math>. In other words, the real numbers are extended by the addition of a single signless infinity. The ordering relation is not defined for the projectively extended reals - but informally, <math>\infty</math> can be said to be both greater than ''and'' less than every real number.

*the '''hyperreals''': denoted ^*<math>\mathbb{R}</math>, these extend the real numbers with both infinite and infinitesimal quantities. They are used in non-standard analysis. Other extensions of the real numbers with infinite and infinitesimal quantities include the '''surreal numbers''', the '''superreal numbers''', and the '''Levi-Civita field'''.

*the '''hyperreals''': denoted ^*<math>\mathbb{R}</math>, these extend the real numbers with both infinite and infinitesimal quantities. They are used in non-standard analysis. Other extensions of the real numbers with infinite and infinitesimal quantities include the '''surreal numbers''', the '''superreal numbers''', and the '''Levi-Civita field'''.

The conclusion to be drawn from all of this, is that although in popular usage we speak about ''infinity'' as if it were a single thing, there are actually numerous different infinities, all belonging to different systems. If we want to talk about ''infinity'', we ought to be careful to specify which infinity we mean.

The conclusion to be drawn is that although in popular usage we speak about ''infinity'' as if it were a single thing, there are actually numerous different infinities, all belonging to different systems. If we want to talk about ''infinity'', we ought to be careful to specify which infinity we mean.

==Archimedean property==

==Archimedean property==

   

   

== Philosophy of mathematics ==

== Philosophy of mathematics ==

Different positions in the [[philosophy of mathematics]] have different attitudes towards infinity. At one extreme, Platonists and formalists generally have no philosophical objection to any form of infinity than can be defined. At the other extreme, finitists and ultrafinitists deny the existence of infinity; to them, only finite quantities, and finite objects, actually exist. In the middle, many constructivists and intuitionists adopt a position of countablism - an acceptance of the existence of countable sets, but denying that uncountable sets exist.

Different positions in the [[philosophy of mathematics]] have different attitudes towards infinity. At one extreme, Platonists and formalists generally have no philosophical objection to any form of infinity that can be defined. At the other extreme, finitists and ultrafinitists deny the existence of infinity; to them, only finite quantities, and finite objects, actually exist. In the middle, many constructivists and intuitionists adopt a position of countablism - an acceptance of the existence of countable sets, but denying that uncountable sets exist.

 

{{cquote|Not only do we lack evidence for the infinite but we don’t need the infinite to do physics. ...

== See also==

== See also==

*[[Infinity denial]]

*[[Infinity denial]]