Aleph number

In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor[1] and are named after the symbol he used to denote them, the Semitic letter aleph ().[2][a]

Aleph-nought, aleph-zero, or aleph-null, the smallest infinite cardinal number

The cardinality of the natural numbers is (read aleph-nought or aleph-zero; the term aleph-null is also sometimes used), the next larger cardinality of a well-orderable set is aleph-one then and so on. Continuing in this manner, it is possible to define a cardinal number for every ordinal number as described below.

The concept and notation are due to Georg Cantor,[5] who defined the notion of cardinality and realized that infinite sets can have different cardinalities.

The aleph numbers differ from the infinity () 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 of the real number line (applied to a function or sequence that "diverges to infinity" or "increases without bound"), or as an extreme point of the extended real number line.


  (aleph-nought, also aleph-zero or aleph-null) is the cardinality of the set of all natural numbers, and is an infinite cardinal. The set of all finite ordinals, called   or   (where   is the lowercase Greek letter omega), has cardinality  . A set has cardinality   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

These infinite ordinals:           and   are among the countably infinite sets.[6] For example, the sequence (with ordinality  ) of all positive odd integers followed by all positive even integers


is an ordering of the set (with cardinality  ) of positive integers.

If the axiom of countable choice (a weaker version of the axiom of choice) holds, then   is smaller than any other infinite cardinal.


  is the cardinality of the set of all countable ordinal numbers, called   or sometimes  . This   is itself an ordinal number larger than all countable ones, so it is an uncountable set. Therefore,   is distinct from  . The definition of   implies (in ZF, Zermelo–Fraenkel set theory without the axiom of choice) that no cardinal number is between   and  . If the axiom of choice is used, it can be further proved that the class of cardinal numbers is totally ordered, and thus   is the second-smallest infinite cardinal number. Using the axiom of choice, one can show one of the most useful properties of the set  : any countable subset of   has an upper bound in  . (This follows from the fact that the union of a countable number of countable sets is itself countable – one of the most common applications of the axiom of choice.) This fact is analogous to the situation in   : every finite set of natural numbers has a maximum which is also a natural number, and finite unions of finite sets are finite.

 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 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 spaces, groups, 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  .

Continuum hypothesisEdit

The cardinality of the set of real numbers (cardinality of the continuum) is   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


The CH states that there is no set whose cardinality is strictly between that of the integers and the real numbers.[8] 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 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.[7]


Aleph-omega is


where the smallest infinite ordinal is denoted ω. That is, the cardinal number   is the least upper bound of


  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 numbers; for any positive integer n we can consistently assume that   and moreover it is possible to assume   is as large as we like. We are only forced to avoid setting it to certain special cardinals with cofinality   meaning there is an unbounded function from   to it (see Easton's theorem).

Aleph-α for general αEdit

To define   for arbitrary ordinal number   we must define the successor cardinal operation, which assigns to any cardinal number   the next larger well-ordered cardinal   (if the axiom of choice holds, this is the next larger cardinal).

We can then define the aleph numbers as follows:


and for λ, an infinite limit ordinal,


The α-th infinite initial ordinal is written  . Its cardinality is written  

Informally, the aleph function   is a bijection from the ordinals to the infinite cardinals. Formally, in ZFC,   is not a function, since it is not a set (due to the Burali-Forti paradox).

Fixed points of omegaEdit

For any ordinal α we have


In many cases   is strictly greater than α. For example, for any successor ordinal α this holds. There are, however, some limit ordinals which are fixed points of the omega function, because of the fixed-point lemma for normal functions. The first such is the limit of the sequence


Any weakly inaccessible cardinal is also a fixed point of the aleph function.[9] This can be shown in ZFC as follows. Suppose   is a weakly inaccessible cardinal. If   were a successor ordinal, then   would be a successor cardinal and hence not weakly inaccessible. If   were a limit ordinal less than   then its cofinality (and thus the cofinality of  ) would be less than   and so   would not be regular and thus not weakly inaccessible. Thus   and consequently   which makes it a fixed point.

Role of axiom of choiceEdit

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.

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

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.

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 card(S) to be the set of sets with the same cardinality as S of minimum possible rank. This has the property that card(S) = card(T) if and only if S and T have the same cardinality. (The set card(S) does not have the same cardinality of S in general, but all its elements do.)

See alsoEdit


  1. ^ 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 a monotype matrix for aleph was mistakenly constructed the wrong way up.[4]


  1. ^ "Aleph". Encyclopedia of Mathematics.
  2. ^ Weisstein, Eric W. "Aleph". Retrieved 2020-08-12.
  3. ^ Sierpiński, Wacław (1958). Cardinal and Ordinal Numbers. Polska Akademia Nauk Monografie Matematyczne. Vol. 34. Warsaw, PL: Państwowe Wydawnictwo Naukowe. MR 0095787.
  4. ^ Swanson, Ellen; O'Sean, Arlene Ann; Schleyer, Antoinette Tingley (1999) [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.
  5. ^ Miller, Jeff. "Earliest uses of symbols of set theory and logic". Retrieved 2016-05-05; who quotes Dauben, Joseph Warren (1990). Georg Cantor: His mathematics and philosophy of the infinite. 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 ...
  6. ^ Jech, Thomas (2003). Set Theory. Springer Monographs in Mathematics. Berlin, New York: Springer-Verlag.
  7. ^ a b Szudzik, Mattew (31 July 2018). "Continuum Hypothesis". Wolfram Mathworld. Wolfram Web Resources. Retrieved 15 August 2018.
  8. ^ Weisstein, Eric W. "Continuum Hypothesis". Retrieved 2020-08-12.
  9. ^ Harris, Kenneth A. (April 6, 2009). "Lecture 31" (PDF). Department of Mathematics. Intro to Set Theory. University of Michigan. Math 582. Archived from the original (PDF) on March 4, 2016. Retrieved September 1, 2012.

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