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| {{redirect|Aleph One}}
| | The author is called Stacy Thurlow though he doesn't relish being called like that a majority of. To arrange flowers is what she does continuously. Meter reading is what she does in her day placement. Maine has always been her living place and her parents live closeby. Go to my website to know more: http://simpleit.edublogs.org/ |
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| [[Image:Aleph0.svg|thumb|right|150px|Aleph-null, the smallest infinite cardinal number]]
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| In [[set theory]], a discipline within mathematics, the '''aleph numbers''' are a sequence of numbers used to represent the [[cardinality]] (or size) of [[infinite set]]s. They are named after the symbol used to denote them, the [[Hebrew alphabet|Hebrew]] letter [[aleph]] (<math>\aleph</math>).
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| The cardinality of the [[natural number]]s is <math>\aleph_0</math> (read ''aleph-naught'', ''aleph-null'', or ''aleph-zero''), the next larger cardinality is aleph-one <math>\aleph_1</math>, then <math>\aleph_2</math> and so on. Continuing in this manner, it is possible to define a [[cardinal number]] <math>\aleph_\alpha</math> for every [[ordinal number]] α, as described below. | |
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| The concept goes back to [[Georg Cantor]], who defined the notion of cardinality and realized that infinite sets can have different cardinalities.
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| The aleph numbers differ from the [[Extended real number line|infinity]] (∞) commonly found in algebra and calculus. Alephs measure the sizes of sets; infinity, on the other hand, is commonly defined as an extreme [[limit (mathematics)|limit]] of the [[real number line]] (applied to a [[function (mathematics)|function]] or [[sequence (mathematics)|sequence]] that "[[divergent series|diverges]] to infinity" or "increases without bound"), or an extreme point of the [[extended real number line]].
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| ==Aleph-naught==
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| <math>\aleph_0</math> 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 '''ω''' or '''ω<sub>0</sub>''', 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. Such sets include the set of all [[prime number]]s, the set of all [[integer]]s, the set of all [[rational number]]s, the set of [[algebraic number]]s, the set of binary [[string (computer science)|string]]s of all finite lengths, and the set of all finite [[subset]]s of any given countably infinite set. These infinite ordinals: ω, ω+1, ω.2, ω<sup>2</sup>, ω<sup>ω</sup> and [[Ordinal number|ε<sub>0</sub>]] are among the countably infinite sets.<ref>{{Citation | last1=Jech | first1=Thomas | title=[[Set Theory]] | publisher= [[Springer-Verlag]]| location=Berlin, New York | series=Springer Monographs in Mathematics | year=2003}}</ref>
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| 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.
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| ==Aleph-one==
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| <math>\aleph_1</math> is the cardinality of the set of all countable [[ordinal number]]s, called '''ω<sub>1</sub>''' or (sometimes) '''Ω'''. This '''ω<sub>1</sub>''' is itself an ordinal number larger than all countable ones, so it is an [[uncountable set]]. Therefore <math>\aleph_1</math> is distinct from <math>\aleph_0</math>. 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]] (AC) 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. Using AC we can show one of the most useful properties of the set '''ω<sub>1</sub>''': any countable subset of '''ω<sub>1</sub>''' has an upper bound in '''ω<sub>1</sub>'''. (This follows from the fact that a countable union of countable sets is countable, one of the most common applications of AC.) 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 [[Union (set theory)#Finite unions|finite unions]] of finite sets are finite.
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| '''ω<sub>1</sub>''' is actually a useful concept, if somewhat exotic-sounding. An example application is "closing" with respect to countable operations; e.g., trying to explicitly describe the [[sigma-algebra|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, and the like. 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 '''ω<sub>1</sub>'''.
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| ==The continuum hypothesis==
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| {{main|Continuum hypothesis}}
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| {{see also|Beth number}}
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| The [[cardinality]] of the set of [[real number]]s ([[cardinality of the continuum]]) is <math>2^{\aleph_0}</math>. It is not clear where this number fits in the aleph number hierarchy. It follows from ZFC ([[Zermelo–Fraenkel set theory]] with the [[axiom of choice]]) that the celebrated continuum hypothesis, '''CH''', is equivalent to the identity
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| :<math>2^{\aleph_0}=\aleph_1.</math>
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| CH is independent of ZFC: it can be neither proven nor disproven within the context of that axiom system (provided that ZFC is consistent). That it 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 (mathematician)|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]].
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| ==Aleph-ω==
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| Conventionally the smallest infinite ordinal is denoted ω, and the cardinal number <math>\aleph_\omega</math> is the least upper bound of
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| :<math>\left\{\,\aleph_n : n\in\left\{\,0,1,2,\dots\,\right\}\,\right\}</math>
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| among alephs.
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| Aleph-ω 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; for any positive integer n we can consistently assume that <math>2^{\aleph_0} = \aleph_n</math>, and moreover it is possible to assume <math>2^{\aleph_0}</math> is as large as we like. We are only forced to avoid setting it to certain special cardinals with [[cofinality]] <math>\aleph_0</math>, meaning there is an unbounded function from <math>\aleph_0</math> to it (see [[Easton's theorem]]).
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| ==Aleph-α for general α==
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| 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 ρ the next larger [[well-order]]ed cardinal ρ{{sup|+}} (if the [[axiom of choice]] holds, this is the next larger cardinal).
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| We can then define the aleph numbers as follows:
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| :<math>\aleph_{0} = \omega</math>
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| :<math>\aleph_{\alpha+1} = \aleph_{\alpha}^+</math>
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| and for λ, an infinite limit ordinal,
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| :<math>\aleph_{\lambda} = \bigcup_{\beta < \lambda} \aleph_\beta.</math>
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| The α-th infinite initial ordinal is written <math>\omega_\alpha</math>. Its cardinality is written <math>\aleph_\alpha</math>. See [[initial ordinal]].
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| In ZFC the <math>\aleph</math> function is a bijection between the ordinals and the infinite cardinals.<ref>{{PlanetMath | urlname=AlephNumbers | title=aleph numbers | id=5710}}</ref>
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| ==Fixed points of omega==
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| For any ordinal α we have
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| :<math>\alpha\leq\omega_\alpha.</math> | |
| In many cases <math>\omega_{\alpha}</math> is strictly greater than α. For example, for any successor ordinal α this 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
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| :<math>\omega,\ \omega_\omega,\ \omega_{\omega_\omega},\ \ldots.</math>
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| Any [[inaccessible cardinal|weakly inaccessible cardinal]] is also a fixed point of the aleph function.<ref name="Harris 2009">{{cite web | url=http://kaharris.org/teaching/582/Lectures/lec31.pdf | title=Math 582 Intro to Set Theory, Lecture 31 | publisher=Department of Mathematics, University of Michigan | date=April 6, 2009 | accessdate=September 1, 2012 | author=Harris, Kenneth}}</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 \geq \kappa </math> and consequently <math>\lambda = \kappa </math> which makes it a fixed point.
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| ==Role of axiom of choice==
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| 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-orderable.
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| Each [[finite set]] is well-orderable, but does not have an aleph as its cardinality.
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| The assumption that the cardinality of each [[infinite set]] is an aleph number is equivalent over ZF 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.
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| 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.
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| ==See also==
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| * [[Regular cardinal]]
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| ==References==
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| ;Notes
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| {{reflist}}
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| ==External links==
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| *{{springer|title=Aleph-zero|id=p/a011280}}
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| * {{MathWorld | urlname=Aleph-0 | title=Aleph-0}}
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| <!-- [[he:אלף 0]] - this article is about aleph 0 specifically, not about aleph numbers. As of 2012-9-2- there is no page on hewiki that is a suitable interwiki target for this article, see talk page. -->
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| {{DEFAULTSORT:Aleph Number}}
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| [[Category:Cardinal numbers]]
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| [[Category:Infinity]]
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The author is called Stacy Thurlow though he doesn't relish being called like that a majority of. To arrange flowers is what she does continuously. Meter reading is what she does in her day placement. Maine has always been her living place and her parents live closeby. Go to my website to know more: http://simpleit.edublogs.org/