Infinite: Difference between revisions

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== Infinity ==

A separate concept to that of an infinite set is that of infinity itself, denoted \(\infty\). This generally refers to an object that is larger than all natural numbers. It has different precise meanings in different contexts, such as denoting ~~certain~~ [https://en.wikipedia.org/wiki/Limit_(mathematics)#Infinity_as_a_limit limits] or [https://en.wikipedia.org/wiki/Improper_integral improper integrals], or as an [https://en.wikipedia.org/wiki/Extended_real_number_line extended real ~~number~~]. However, there is no real number that serves the purpose of infinity, since the real numbers have the [https://en.wikipedia.org/wiki/Archimedean_property Archimedean property], meaning that for every real number

A separate concept to that of an infinite set is that of infinity itself, denoted \(\infty\). This generally refers to an object that is larger than all natural numbers. It has different precise meanings in different contexts. It can be used purely notationally, such as when denoting [https://en.wikipedia.org/wiki/Limit_(mathematics)#Infinity_as_a_limit limits to infinity], [https://en.wikipedia.org/wiki/Series_(mathematics) series], and [https://en.wikipedia.org/wiki/Improper_integral improper integrals], or as an object in a structure such as the [https://en.wikipedia.org/wiki/Extended_real_number_line extended real numbers]. However, there is no real number that serves the purpose of infinity, since the real numbers have the [https://en.wikipedia.org/wiki/Archimedean_property Archimedean property], meaning that for every real number

\(x\) there is a natural at least as large, such as \(\lceil x\rceil\). Infinity should also not be conflated with infinite ordinals or cardinals such as \(\omega\) and \(\aleph_0\).

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 == Infinity ==
-A separate concept to that of an infinite set is that of infinity itself, denoted \(\infty\). This generally refers to an object that is larger than all natural numbers. It has different precise meanings in different contexts, such as denoting certain [https://en.wikipedia.org/wiki/Limit_(mathematics)#Infinity_as_a_limit limits] or [https://en.wikipedia.org/wiki/Improper_integral improper integrals], or as an [https://en.wikipedia.org/wiki/Extended_real_number_line extended real number]. However, there is no real number that serves the purpose of infinity, since the real numbers have the [https://en.wikipedia.org/wiki/Archimedean_property Archimedean property], meaning that for every real number
+A separate concept to that of an infinite set is that of infinity itself, denoted \(\infty\). This generally refers to an object that is larger than all natural numbers. It has different precise meanings in different contexts. It can be used purely notationally, such as when denoting [https://en.wikipedia.org/wiki/Limit_(mathematics)#Infinity_as_a_limit limits to infinity], [https://en.wikipedia.org/wiki/Series_(mathematics) series], and [https://en.wikipedia.org/wiki/Improper_integral improper integrals], or as an object in a structure such as the [https://en.wikipedia.org/wiki/Extended_real_number_line extended real numbers]. However, there is no real number that serves the purpose of infinity, since the real numbers have the [https://en.wikipedia.org/wiki/Archimedean_property Archimedean property], meaning that for every real number
 \(x\) there is a natural at least as large, such as \(\lceil x\rceil\). Infinity should also not be conflated with infinite ordinals or cardinals such as \(\omega\) and \(\aleph_0\).

Infinite: Difference between revisions

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