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Aleph number

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Infinite cardinal number

"ℵ" redirects here. For the letter, seeAleph. For other uses, seeAleph (disambiguation) andAlef (disambiguation).

Inmathematics, particularly inset theory, thealeph numbers are asequence of numbers used to represent thecardinality (or size) ofinfinite sets.^[a] They were introduced by the mathematicianGeorg Cantor^[1] and are named after the symbol he used to denote them, the Hebrew letteraleph (ℵ).^[2]^[b]

The smallest cardinality of an infinite set is that of thenatural numbers, denoted by $\aleph _{0}$ (readaleph-nought,aleph-zero, oraleph-null); the next larger cardinality of awell-ordered set is $\aleph _{1},$ then $\aleph _{2},$ then $\aleph _{3},$ and so on. Continuing in this manner, it is possible to define an infinitecardinal number $\aleph _{\alpha }$ for everyordinal number $\alpha ,$ as described below.

The concept and notation are due toGeorg Cantor,^[5]who defined the notion of cardinality and realized thatinfinite sets can have different cardinalities.

The aleph numbers differ from theinfinity ( $\infty$ ) commonly found in algebra and calculus, in that the alephs measure the sizes of sets, while infinity is commonly defined either as an extremelimit of thereal number line (applied to afunction orsequence that "diverges to infinity" or "increases without bound"), or as an extreme point of theextended real number line.

Aleph-zero

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$\aleph _{0}$ (aleph-nought,aleph-zero, oraleph-null) is the cardinality of the set of all natural numbers, and is aninfinite cardinal. The set of all finiteordinals, called $\omega$ or $\omega _{0}$ (where $\omega$ is the lowercase Greek letteromega), also has cardinality $\aleph _{0}$ . A set has cardinality $\aleph _{0}$ if and only if it iscountably infinite, that is, there is abijection (one-to-one correspondence) between it and the natural numbers. Examples of such sets are:

the set ofnatural numbers, irrespective of including or excluding zero,
the set of allintegers,
any infinite subset of the integers, such as the set of allsquare numbers or the set of allprime numbers,
the set of allrational numbers,
the set of allconstructible numbers (in the geometric sense),
the set of allalgebraic numbers,
the set of allcomputable numbers,
the set of allcomputable functions,
the set of all binarystrings of finite length, and
the set of all finitesubsets of any given countably infinite set.

Among the countably infinite sets are certain infinite ordinals,^[c] including for example $\omega$ , $\omega +1$ , $\omega \cdot 2$ , $\omega ^{2}$ , $\omega ^{\omega }$ , and $\varepsilon _{0}$ .^[6] For example, the sequence (withorder type $\omega \cdot 2$ ) of all positive odd integers followed by all positive even integers $\{1,3,5,7,9,\cdots ;2,4,6,8,10,\cdots \}$ is a well-ordering of the set (with cardinality $\aleph _{0}$ ) of positive integers.

If theaxiom of countable choice (a weaker version of theaxiom of choice) holds, then $\aleph _{0}$ is smaller than any other infinite cardinal.

Aleph-one

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"Aleph One" redirects here. For other uses, seeAleph One (disambiguation).

$\aleph _{1}$ is the cardinality of the set of all countableordinal numbers.^[7] This set is denoted by $\omega _{1}$ (or sometimes Ω). The set $\omega _{1}$ is itself an ordinal number larger than all countable ones, so it is anuncountable set. Therefore, $\aleph _{1}$ is the smallest cardinality that is larger than $\aleph _{0},$ the smallest infinite cardinality.

The definition of $\aleph _{1}$ implies (in ZF,Zermelo–Fraenkel set theorywithout the axiom of choice) that no cardinal number is between $\aleph _{0}$ and $\aleph _{1}.$ If theaxiom of choice is used, it can be further proved that the class of cardinal numbers istotally ordered, and thus $\aleph _{1}$ is the second-smallest infinite cardinal number. One can show one of the most useful properties of the set⁠ $\omega _{1}$ ⁠: Any countable subset of $\omega _{1}$ has an upper bound in $\omega _{1}$ (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 $\aleph _{0}$ : Every finite set of natural numbers has a maximum, which is also a natural number, andfinite unions of finite sets are finite.

An example application of the ordinal $\omega _{1}$ is "closing" with respect to countable-arity operations; e.g., trying to explicitly describe theσ-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 spaces,groups, 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, viatransfinite induction, a set by "throwing in" all possiblecountable unions and complements, and taking the union of all that over all of $\omega _{1}.$

Continuum hypothesis

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Main article:Continuum hypothesis

Aleph-omega

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Aleph-omega is $\aleph _{\omega }=\aleph _{n}^{n\in \omega }=\aleph _{n}^{n\in \{0,1,2,\cdots \}}$ where the smallest infinite ordinal is denoted as $\omega$ . That is, the cardinal number $\aleph _{\omega }$ is theleast upper bound of $\aleph _{n}^{n\in \{0,1,2,\cdots \}}$ .

Notably, $\aleph _{\omega }$ is the first uncountable cardinal number that can be demonstrated within Zermelo–Fraenkel set theorynot to be equal to the cardinality of the set of allreal numbers $2^{\aleph _{0}}$ : For any natural number $n\geq 1$ , we can consistently assume that $2^{\aleph _{0}}=\aleph _{n}$ , and moreover it is possible to assume that $2^{\aleph _{0}}$ is at least as large as any cardinal number we like. The main restriction ZFC puts on the value of $2^{\aleph _{0}}$ is that it cannot equal certain special cardinals withcofinality $\aleph _{0}$ . An uncountably infinite cardinal $\kappa$ having cofinality $\aleph _{0}$ means that there is a (countable-length) sequence $\kappa _{0}\leq \kappa _{1}\leq \kappa _{2}\leq \cdots$ of cardinals $\kappa _{i}<\kappa$ whose limit (i.e. its least upper bound) is $\kappa$ (seeEaston's theorem). As per the definition above, $\aleph _{\omega }$ is the limit of a countable-length sequence of smaller cardinals.

Aleph-α for generalα

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To define $\aleph _{\alpha }$ for arbitrary ordinal number $\alpha$ , we must define thesuccessor cardinal operation, which assigns to any cardinal number $\rho$ the next largerwell-ordered cardinal $\rho ^{+}$ (if theaxiom of choice holds, this is the (unique) next larger cardinal).

We can then define the aleph numbers as follows:

\aleph _{0}=\omega

\aleph _{\alpha +1}=(\aleph _{\alpha })^{+}

\aleph _{\lambda }=\bigcup \{\aleph _{\alpha }|\alpha <\lambda \}

for

\lambda

an infinitelimit ordinal,

The $\alpha$ -th infiniteinitial ordinal is written $\omega _{\alpha }$ . Its cardinality is written $\aleph _{\alpha }$ .

Informally, thealeph function $\aleph :{\text{On}}\rightarrow {\text{Cd}}$ is a bijection from the ordinals to the infinite cardinals.Formally, inZFC, $\aleph$ isnot a function, but a function-like class, as it is not a set (due to theBurali-Forti paradox).

Fixed points of omega

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For any ordinal $\alpha$ we have $\alpha \leq \omega _{\alpha }$ .

In many cases $\omega _{\alpha }$ is strictly greater thanα. For example, it is true for any successorordinal: $\alpha +1\leq \omega _{\alpha }<\omega _{\alpha +1}$ holds. There are, however, some limit ordinals that arefixed points of the omega function, because of thefixed-point lemma for normal functions. The first such is the limit of the sequence

\omega ,\omega _{\omega },\omega _{\omega _{\omega }},\cdots

which is sometimes denoted ${\textstyle \omega _{\omega _{\ddots }}}$ .

Anyweakly inaccessible cardinal is also a fixed point of the aleph function.^[11] This can be shown in ZFC as follows. Suppose $\kappa =\aleph _{\lambda }$ is a weakly inaccessible cardinal. If $\lambda$ were asuccessor ordinal, then $\aleph _{\lambda }$ would be asuccessor cardinal and hence not weakly inaccessible. If $\lambda$ were alimit ordinal less than $\kappa$ then itscofinality (and thus the cofinality of $\aleph _{\lambda }$ ) would be less than $\kappa$ and so $\kappa$ would not be regular and thus not weakly inaccessible. Thus $\lambda \geq \kappa$ and consequently $\lambda =\kappa$ , which makes it a fixed point.

Role of axiom of choice

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The cardinality of any infiniteordinal number is an aleph number. Every aleph is the cardinality of some ordinal. The least of these is itsinitial ordinal. Any set whose cardinality is an aleph isequinumerous with an ordinal and is thuswell-orderable.

Eachfinite set is well-orderable, but does not have an aleph as its cardinality.

Over ZF, the assumption that the cardinality of eachinfinite set is an aleph number is equivalent to the existence of a well-ordering of every set, which in turn is equivalent to theaxiom 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 ofScott'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 ${\text{card}}(S)$ to be the set of sets with the same cardinality as $S {\displaystyle S}$ of minimum possible rank. This has the property that ${\text{card}}(S)={\text{card}}(T)$ if and only if $S {\displaystyle S}$ and $T {\displaystyle T}$ have the same cardinality. (The set ${\text{card}}(S)$ does not have the same cardinality of $S {\displaystyle S}$ in general, but all its elements do.)

Notes

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^Given theaxiom 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 bewell-ordered.
^In older mathematics books, the letter aleph is often printed upside down by accident—for example, in Sierpiński (1958)^[3]^: 402 the letter aleph appears both the right way up and upside down, partly because amonotype matrix for aleph was mistakenly constructed the wrong way up.^[4]
^This is using the convention that an ordinal is identified with the set of all ordinals less than itself (the so-calledvon Neumann ordinals).

References

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^"Aleph".Encyclopedia of Mathematics.
^Weisstein, Eric W."Aleph".mathworld.wolfram.com. Retrieved2020-08-12.
^Sierpiński, Wacław (1958).Cardinal and Ordinal Numbers. Polska Akademia Nauk Monografie Matematyczne. Vol. 34. Warsaw, PL: Państwowe Wydawnictwo Naukowe.MR 0095787.
^Swanson, Ellen; O'Sean, Arlene Ann; Schleyer, Antoinette Tingley (2000) [1979].Mathematics into type: Copy editing and proofreading of mathematics for editorial assistants and authors (updated ed.). Providence, RI:American Mathematical Society. p. 16.ISBN 0-8218-0053-1.MR 0553111.
^Miller, Jeff."Earliest uses of symbols of set theory and logic".jeff560.tripod.com. Retrieved2016-05-05; who quotesDauben, Joseph Warren (1990).Georg Cantor: His mathematics and philosophy of the infinite. Princeton University Press.ISBN 9780691024479.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 ...
^Jech, Thomas (2003).Set Theory. Springer Monographs in Mathematics. Berlin, New York:Springer-Verlag.
^"Power of the continuum | mathematics | Britannica".www.britannica.com. Retrieved2025-02-06.
^^a ^bSzudzik, Mattew (31 July 2018)."Continuum Hypothesis".Wolfram Mathworld. Wolfram Web Resources. Retrieved15 August 2018.
^Weisstein, Eric W."Continuum Hypothesis".mathworld.wolfram.com. Retrieved2020-08-12.
^Chow, Timothy Y. (2007). "A beginner's guide to forcing".arXiv:0712.1320 [math.LO].
^Harris, Kenneth A. (April 6, 2009)."Lecture 31"(PDF). Department of Mathematics.kaharris.org. Intro to Set Theory.University of Michigan. Math 582. Archived fromthe original(PDF) on March 4, 2016. RetrievedSeptember 1, 2012.

External links

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"Aleph-zero",Encyclopedia of Mathematics,EMS Press, 2001 [1994]
Weisstein, Eric W."Aleph-0".MathWorld.

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