Editing Aleph number

{{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]].

==Aleph-zero<span class="anchor" id="Aleph-null"></span>==
<math>\aleph_0</math> ('''aleph-nought''', '''aleph-zero''', or '''aleph-null''') is the cardinality of the set of all natural numbers, and is an [[transfinite number|infinite cardinal]]. The set of all finite [[ordinal number|ordinals]], called <math>\omega</math> or <math>\omega_0</math> (where <math>\omega</math> is the lowercase Greek letter [[omega]]), also has cardinality <math>\aleph_0</math>. A set has cardinality <math>\aleph_0</math> if and only if it is [[countably infinite]], that is, there is a [[bijection]] (one-to-one correspondence) between it and the natural numbers. Examples of such sets are

* the set of [[natural numbers]], irrespective of including or excluding zero,
* the set of all [[integer]]s, 
* any infinite subset of the integers, such as the set of all [[square numbers]] or the set of all [[prime numbers]],
* the set of all [[rational number]]s, 
* the set of all [[constructible number]]s (in the geometric sense),
* the set of all [[algebraic number]]s, 
* the set of all [[computable number]]s, 
* the set of all [[computable function]]s, 
* the set of all binary [[string (computer science)|string]]s of finite length, and 
* the set of all finite [[subset]]s of any given countably infinite set.

Among the countably infinite sets are certain infinite ordinals,{{efn|This is using the convention that an ordinal is identified with the set of all ordinals less than itself (the so-called [[von Neumann ordinals]]).}} including for example <math>\omega</math>, <math>\omega+1</math>, <math>\omega \cdot 2</math>, <math>\omega^2</math>, <math>\omega^\omega</math>, and [[Epsilon numbers (mathematics)|<math>\varepsilon_0</math>]].<ref>{{cite book | last1=Jech | first1=Thomas | title=Set Theory | publisher= [[Springer-Verlag]]| location=Berlin, New York | series=Springer Monographs in Mathematics | year=2003}}</ref> For example, the sequence (with [[order type]] <math>\omega \cdot 2</math>) of all positive odd integers followed by all positive even integers <math>\{1, 3, 5, 7, 9, \cdots; 2, 4, 6, 8, 10, \cdots\}</math> is an ordering of the set (with cardinality <math>\aleph_0</math>) of positive integers.

If the [[axiom of countable choice]] (a weaker version of the [[axiom of choice]]) holds, then <math>\aleph_0</math> is smaller than any other infinite cardinal, and is therefore the (unique) least infinite ordinal.

==Aleph-one==
{{Redirect|Aleph One}}
<math>\aleph_1</math> is the cardinality of the set of all countable [[ordinal number]]s.<ref>{{Cite web |title=Power of the continuum {{!}} mathematics {{!}} Britannica |url=https://www.britannica.com/science/power-of-the-continuum |access-date=2025-02-06 |website=www.britannica.com |language=en}}</ref> This set is denoted by <math>\omega_1</math> (or sometimes Ω). The set <math>\omega_1</math> is itself an ordinal number larger than all countable ones, so it is an [[uncountable set]]. Therefore,  <math>\aleph_1</math> is the smallest cardinality that is larger than <math>\aleph_0,</math> the smallest infinite cardinality.

The definition of <math>\aleph_1</math> implies (in ZF, [[Zermelo–Fraenkel set theory]] ''without'' the axiom of choice) that no cardinal number is between <math>\aleph_0</math> and <math>\aleph_1.</math> If the [[axiom of choice]] is used, it can be further proved that the class of cardinal numbers is [[totally ordered]], and thus <math>\aleph_1</math> is the second-smallest infinite cardinal number. One can show one of the most useful properties of the set {{tmath|\omega_1}}: Any countable subset of <math>\omega_1</math> has an upper bound in <math>\omega_1</math> (this follows from the fact that the union of a countable number of countable sets is itself countable). This fact is analogous to the situation in <math>\aleph_0</math>: Every finite set of natural numbers has a maximum which is also a natural number, and [[finite unions]] of finite sets are finite.

An example application of the ordinal <math>\omega_1</math> is "closing" with respect to countable operations; e.g., trying to explicitly describe the [[σ-algebra]] generated by an arbitrary collection of subsets (see e.g. [[Borel hierarchy]]). This is harder than most explicit descriptions of "generation" in algebra ([[vector space]]s, [[group theory|group]]s, etc.) because in those cases we only have to close with respect to finite operations – sums, products, etc. The process involves defining, for each countable ordinal, via [[transfinite induction]], a set by "throwing in" all possible ''countable'' unions and complements, and taking the union of all that over all of <math>\omega_1.</math>

==Continuum hypothesis==
{{Main|Continuum hypothesis}}
{{See also|Beth number}}
The [[cardinality]] of the set of [[real number]]s ([[cardinality of the continuum]]) is 2<sup><math>\aleph_0</math></sup>. It cannot be determined from [[ZFC]] ([[Zermelo–Fraenkel set theory]] augmented with the [[axiom of choice]]) where this number fits exactly in the aleph number hierarchy, but it follows from ZFC that the continuum hypothesis (CH) is equivalent to the identity
:2<sup><math>\aleph_0</math></sup> = <math>\aleph_1</math>.<ref name=WolframCH>
{{cite web
 |url=http://mathworld.wolfram.com/ContinuumHypothesis.html
 |title=Continuum Hypothesis
 |last=Szudzik |first=Mattew
 |date=31 July 2018
 |website=Wolfram Mathworld
 |publisher=Wolfram Web Resources
 |access-date=15 August 2018
}}
</ref>

The CH states that there is no set whose cardinality is strictly between that of the natural numbers and the real numbers.<ref>
{{cite web
 |last=Weisstein |first=Eric W.
 |title=Continuum Hypothesis
 |url=https://mathworld.wolfram.com/ContinuumHypothesis.html
 |access-date=2020-08-12
 |website=mathworld.wolfram.com
 |language=en
}}
</ref> CH is independent of [[ZFC]]: It can be neither proven nor disproven within the context of that axiom system (provided that [[ZFC]] is [[consistency|consistent]]). That CH is consistent with [[ZFC]] was demonstrated by [[Kurt Gödel]] in 1940, when he showed that its negation is not a theorem of [[ZFC]]. That it is independent of [[ZFC]] was demonstrated by [[Paul Cohen]] in 1963, when he showed conversely that the CH itself is not a theorem of [[ZFC]] – by the (then-novel) method of [[Forcing (mathematics)|forcing]].<ref name=WolframCH/><ref>
{{cite arXiv
 |last=Chow |first=Timothy Y.
 |title=A beginner's guide to forcing
 |eprint=0712.1320
 |date=2007
|class=math.LO
 }}
</ref>

==Aleph-omega==
Aleph-omega is <math>\aleph_\omega = \sup\{\aleph_n| n \in \omega\} = \sup\{\aleph_n| n \in \{0, 1, 2,\cdots\}\}</math> where the smallest infinite ordinal is denoted as <math>\omega</math>. That is, the cardinal number <math>\aleph_\omega</math> is the [[least upper bound]] of <math>\sup\{\aleph_n| n \in \{0, 1, 2,\cdots\}\}</math>.

Notably, <math>\aleph_\omega</math> is the first uncountable cardinal number that can be demonstrated within Zermelo–Fraenkel set theory ''not'' to be equal to the cardinality of the set of all [[real number]]s <math>2^{\aleph_0}</math>: For any natural number <math> n \ge 1 </math>, we can consistently assume that <math>2^{\aleph_0} = \aleph_n</math>, and moreover it is possible to assume that <math>2^{\aleph_0}</math> is as least as large as any cardinal number we like. The main restriction ZFC puts on the value of <math>2^{\aleph_0}</math> is that it cannot equal certain special cardinals with [[cofinality]] <math>\aleph_0</math>. An uncountably infinite cardinal <math>\kappa</math> having cofinality <math>\aleph_0</math> means that there is a (countable-length) sequence <math>\kappa_0 \le \kappa_1 \le \kappa_2 \le \cdots</math> of cardinals <math>\kappa_i < \kappa</math> whose limit (i.e. its least upper bound) is <math>\kappa</math> (see [[Easton's theorem]]). As per the definition above, <math>\aleph_\omega</math> is the limit of a countable-length sequence of smaller cardinals.

==Aleph-''α'' for general ''α''==
To define <math>\aleph_\alpha</math> for arbitrary ordinal number <math>\alpha</math>, we must define the [[successor cardinal|successor cardinal operation]], which assigns to any cardinal number <math>\rho</math> the next larger [[well-order]]ed cardinal <math>\rho^{+}</math> (if the [[axiom of choice]] holds, this is the (unique) next larger cardinal).

We can then define the aleph numbers as follows:

:<math>\aleph_0 = \omega</math>
:<math>\aleph_{\alpha+1} = (\aleph_{\alpha})^{+}</math>
:<math>\aleph_{\lambda} = \bigcup\{\aleph_\alpha | \alpha < \lambda\}</math> for <math>\lambda</math> an infinite [[limit ordinal]],

The <math>\alpha</math>-th infinite [[initial ordinal]] is written <math>\omega_\alpha</math>. Its cardinality is written <math>\aleph_\alpha</math>.

Informally, the '''aleph function''' <math>\aleph : \text{On} \rightarrow \text{Cd}</math> is a bijection from the ordinals to the infinite cardinals.
Formally, in [[ZFC]], <math>\aleph</math> is ''not a function'', but a function-like class, as it is not a set (due to the [[Burali-Forti paradox]]).

==Fixed points of omega==
For any ordinal <math>\alpha</math> we have <math>\alpha \le \omega_\alpha</math>.

In many cases <math>\omega_\alpha</math> is strictly greater than ''α''. For example, it is true for any successor [[Ordinal number|ordinal]]: <math>\alpha + 1 \le \omega_{\alpha + 1}</math> holds. There are, however, some limit ordinals which are [[fixed point (mathematics)|fixed point]]s of the omega function, because of the [[fixed-point lemma for normal functions]]. The first such is the limit of the sequence

:<math> \omega, \omega_{\omega}, \omega_{\omega_{\omega}}, \cdots </math>

which is sometimes denoted <math display="inline">\omega_{\omega_{\ddots}}</math>.

Any [[inaccessible cardinal|weakly inaccessible cardinal]] is also a fixed point of the aleph function.<ref name=Harris-2009-04-06-Math-582>
{{cite web
 | author=Harris, Kenneth A.
 | date=6 April 2009
 | title=Lecture&nbsp;31
 | series=Intro to Set Theory
 | id=Math&nbsp;582 
 | department=Department of Mathematics
 | publisher=[[University of Michigan]]
 | website=kaharris.org
 | url=http://kaharris.org/teaching/582/Lectures/lec31.pdf
 | access-date=September 1, 2012
 | archive-url=https://web.archive.org/web/20160304121941/http://kaharris.org/teaching/582/Lectures/lec31.pdf
 | archive-date=March 4, 2016
 | url-status=dead
 | df=mdy-all
}}
</ref> This can be shown in ZFC as follows. Suppose <math>\kappa = \aleph_{\lambda}</math> is a weakly inaccessible cardinal. If <math>\lambda</math> were a [[successor ordinal]], then <math>\aleph_{\lambda}</math> would be a [[successor cardinal]] and hence not weakly inaccessible. If <math>\lambda</math> were a [[limit ordinal]] less than <math>\kappa</math> then its [[cofinality]] (and thus the cofinality of <math>\aleph_\lambda</math>) would be less than <math>\kappa</math> and so <math>\kappa</math> would not be regular and thus not weakly inaccessible. Thus <math>\lambda \ge \kappa</math> and consequently <math>\lambda = \kappa</math> which makes it a fixed point.

==Role of axiom of choice==
The cardinality of any infinite [[ordinal number]] is an aleph number. Every aleph is the cardinality of some ordinal. The least of these is its [[initial ordinal]]. Any set whose cardinality is an aleph is [[equinumerous]] with an ordinal and is thus [[well-order]]able.

Each [[finite set]] is well-orderable, but does not have an aleph as its cardinality.

Over ZF, the assumption that the cardinality of each [[infinite set]] is an aleph number is equivalent to the existence of a well-ordering of every set, which in turn is equivalent to the [[axiom of choice]]. ZFC set theory, which includes the axiom of choice, implies that every infinite set has an aleph number as its cardinality (i.e. is equinumerous with its initial ordinal), and thus the initial ordinals of the aleph numbers serve as a class of representatives for all possible infinite cardinal numbers.

When cardinality is studied in ZF without the axiom of choice, it is no longer possible to prove that each infinite set has some aleph number as its cardinality; the sets whose cardinality is an aleph number are exactly the infinite sets that can be well-ordered. The method of [[Scott's trick]] is sometimes used as an alternative way to construct representatives for cardinal numbers in the setting of ZF. For example, one can define <math>\text{card}(S)</math> to be the set of sets with the same cardinality as <math>S</math> of minimum possible rank. This has the property that <math>\text{card}(S) = \text{card}(T)</math> if and only if <math>S</math> and <math>T</math> have the same cardinality. (The set <math>\text{card}(S)</math> does not have the same cardinality of <math>S</math> in general, but all its elements do.)

==See also==
* [[Beth number]]
* [[Gimel function]]
* [[Regular cardinal]]
* [[Infinity]]
* [[Transfinite number]]
* [[Ordinal number]]

==Notes==
{{notelist}}

==References==
{{reflist|25em}}

==External links==
* {{springer|title=Aleph-zero|id=p/a011280|ref=none}}
* {{MathWorld | urlname=Aleph-0 | title=Aleph-0}}

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{{DEFAULTSORT:Aleph Number}}
[[Category:Cardinal numbers]]
[[Category:Hebrew alphabet]]
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