Editing Aleph number (section)

{{Short description|Infinite cardinal number}}
{{Redirect|ℵ|the letter|Aleph|other uses|Aleph (disambiguation)|and|Alef (disambiguation)}}
[[File:Aleph0.svg|thumb|right|150px|Aleph-nought, aleph-zero, or aleph-null, the smallest infinite cardinal number]]

In [[mathematics]], particularly in [[set theory]], the '''aleph numbers''' are a [[sequence]] of numbers used to represent the [[cardinality]] (or size) of [[infinite set]]s.{{efn|Given the [[axiom of choice]], every infinite set has a cardinality that is an aleph number.  In contexts where the axiom of choice is not available, the aleph numbers still constitute the cardinalities of those infinite sets that can be [[well-ordered]].}} They were introduced by the mathematician [[Georg Cantor]]<ref>{{cite encyclopedia |title=Aleph |encyclopedia=Encyclopedia of Mathematics  |url=https://encyclopediaofmath.org/wiki/Aleph}}</ref> and are named after the symbol he used to denote them, the Hebrew letter [[aleph]] (ℵ).<ref>{{Cite web |last=Weisstein |first=Eric W. |title=Aleph |website=mathworld.wolfram.com |language=en |url=https://mathworld.wolfram.com/Aleph.html |access-date=2020-08-12}}</ref>{{efn|
In older mathematics books, the letter aleph is often printed upside down by accident – for example, in Sierpiński (1958)<ref name=Sierpiński-1958>{{cite book |last=Sierpiński |first= Wacław |year=1958 |title=Cardinal and Ordinal Numbers |title-link=Cardinal and Ordinal Numbers |series=Polska Akademia Nauk Monografie Matematyczne |volume= 34 |publisher=Państwowe Wydawnictwo Naukowe |place=Warsaw, PL |mr=0095787}}
</ref>{{rp|page=402}} the letter aleph appears both the right way up and upside down – partly because a [[monotype]] matrix for aleph was mistakenly constructed the wrong way up.<ref>
{{cite book
 |last1=Swanson  |first1=Ellen
 |last2=O'Sean   |first2=Arlene Ann
 |last3=Schleyer |first3=Antoinette Tingley
 |year=2000 |orig-year=1979
 |edition=updated
 |title=Mathematics into type: Copy editing and proofreading of mathematics for editorial assistants and authors
 |publisher=[[American Mathematical Society]]
 |place=Providence, RI
 |isbn=0-8218-0053-1
 |mr=0553111
 |page=16
}}
</ref>
}}

The smallest cardinality of an infinite set is that of the [[natural number]]s, denoted by <math>\aleph_0</math> (read ''aleph-nought'', ''aleph-zero'', or ''aleph-null''); the next larger cardinality of a [[well-order|well-ordered]] set is <math>\aleph_1,</math> then <math>\aleph_2,</math> then <math>\aleph_3,</math> and so on. Continuing in this manner, it is possible to define an infinite [[cardinal number]] <math>\aleph_{\alpha}</math> for every [[ordinal number]] <math>\alpha,</math> as described below.

The concept and notation are due to [[Georg Cantor]],<ref>
{{cite web
 |first=Jeff |last=Miller
 |title=Earliest uses of symbols of set theory and logic
 |website=jeff560.tripod.com
 |url=http://jeff560.tripod.com/set.html
 |access-date=2016-05-05
 |postscript=;
}} who quotes
{{cite book
 |author=Dauben, Joseph Warren
 |date=1990
 |title=Georg Cantor: His mathematics and philosophy of the infinite
 |publisher=Princeton University Press
 |isbn=9780691024479
 |url-access=registration
 |url=https://archive.org/details/georgcantorhisma0000daub
 |quote=His new numbers deserved something unique. ... Not wishing to invent a new symbol himself, he chose the aleph, the first letter of the Hebrew alphabet ... the aleph could be taken to represent new beginnings&nbsp;...
}}
</ref>
who defined the notion of cardinality and realized that [[Georg Cantor's first set theory article|infinite sets can have different cardinalities]].

The aleph numbers differ from the [[Extended real number line|infinity]] (<math>\infty</math>) commonly found in algebra and calculus, in that the alephs measure the sizes of sets, while infinity is commonly defined either as an extreme [[limit (mathematics)|limit]] of the [[real number line]] (applied to a [[function (mathematics)|function]] or [[sequence]] that "[[divergent series|diverges]] to infinity" or "increases without bound"), or as an extreme point of the [[extended real number line]].