Draft:Finiteness

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{{AFC submission|d|reason|Please see WP:NOTDICT and WP:ESSAY. This is just a definition and list of examples. We need sources that discuss finiteness as a unitary, overriding concept (beyond just a definition).|u=Kevincook13|ns=118|decliner=WeirdNAnnoyed|declinets=20250314111035|small=yes|ts=20250207164627}}

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{{AFC submission|d|dict|u=Kevincook13|ns=118|decliner=Samoht27|declinets=20250205175827|small=yes|ts=20250205165940}}

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{{AFC comment|1=You have not made any changes addressing the concerns of previous reviews. Please refrain from resubmitting until you have done so. – AllCatsAreGrey ^(talk) 19:43, 13 April 2025 (UTC)}}

{{AFC comment|1=Totally inappropriate for an encyclopaedia, you are wasting everybody's time including your own. Theroadislong (talk) 16:03, 21 March 2025 (UTC)}}

{{AFC comment|1=Partly this is a dictionary definition - which is not the purpose of Wikipedia. Also just a random set of dubious points without sources. KylieTastic (talk) 12:24, 6 February 2025 (UTC)}}

{{AFC comment|1=This is a dictionary definition of finite -Samoht27 (talk) 17:58, 5 February 2025 (UTC)}}

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Finiteness is the state of being limited or ended. Humans are considered to be in this state because of their limited life span.{{cite book | vauthors = Carey JR | date = 2003 | title = Longevity. The biology and Demography of Life Span | publisher = Princeton University Press | doi = 10.2307/j.ctv18zhf9v | isbn = 0-691-08848-9 |oclc=1231563351 |jstor=j.ctv18zhf9v}} Natural numbers are considered to be in this state because counting to a natural number comes to an end, such as the number of months in a year.

Whether or not something comes to an end is not always self-evident. In writing, a full stop unambiguously denotes the end, or completion, of a sentence. An ellipsis denotes a lack of completion.{{cite web | url=https://www.ox.ac.uk/sites/files/oxford/University%20of%20Oxford%20Style%20Guide%20%28updated%20Hilary%20term%202016%29.pdf | title = University of Oxford Style Guide: Hilary term 2016 | location = Oxford |publisher = University of Oxford | date= 2016 | page=15 |access-date= 18 May 2017}} A controversial use of ellipses is to simultaneously intend both completion and non-completion, as in 0.999... = 1.{{cite book |last=Byers |first=William |title=How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics |year=2007 |publisher=Princeton University Press |isbn=978-0-691-12738-5 |url=https://archive.org/details/howmathematician00byer}}

Coming to an end, such as a person dying, or a sentence being completed, is one of the two aspects of the state of finiteness. The other aspect is to be limited. Limitations, bounds, and constraints play a significant role in science and everyday life, such as error bounds and seat belts.

Because so much of the human experience is in the state of finiteness, it is normal to assume the state as a given. For example, although the number one is a finite number we normally do not distinguish it as such.

Although normally assumed, finiteness is sometimes specified. Such specification is common when referring to sums in the state of finiteness, to distinguish them from those which are not. The term finite is used to specify finiteness of a sum, but so also is the term partial.

A famous finite sum is the Riemann sum, named after Bernhard Riemann who used it to rigorously define the integral.

In 1847, George Boole used lower case to denote finiteness in his essay on the mathematical analysis of logic.George Boole, [https://books.google.com/books?id=zv4YAQAAIAAJ&pg=PP9 The Mathematical Analysis of Logic, Being an Essay towards a Calculus of Deductive Reasoning] {{Webarchive|url=https://web.archive.org/web/20160511151325/https://books.google.com/books?id=zv4YAQAAIAAJ&pg=PP9 |date=11 May 2016 }} (London, England: Macmillan, Barclay, & Macmillan, 1847). Boole called these lowercase letters elective symbols. A capital letter was used to represent a class and individuals belonging to the class. For example, X represents the class animals. Sometimes, the capital letter was written in the plural, Xs, referring specifically to individuals belonging to the class. Both X and Xs refer to members of the class separately, rather than collectively. The lowercase x refers collectively to selected individuals belonging to the class. Such an elective symbol denotes an elected, selected, collected whole.

Today, the term finite set is used to refer to what Boole described as a collection of objects, and denoted with a lowercase letter.

Draft:Finiteness

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