Aleph number

In set theory, a discipline within mathematics, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets. They are named after the symbol used to denote them, the Hebrew letter aleph (\aleph).

The cardinality of the natural numbers is \( \aleph_0 \) (read aleph-naught, aleph-null or aleph-zero), the next larger cardinality is aleph-one \( \aleph_1 \), then \( \aleph_2 \) and so on. Continuing in this manner, it is possible to define a cardinal number \aleph_\alpha for every ordinal number α, as described below.

The concept goes back to Georg Cantor, 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. Alephs measure the sizes of sets; infinity, on the other hand, is commonly defined as an extreme limit of the real number line (applied to a function or sequence that "diverges to infinity" or "increases without bound"), or an extreme point of the extended real number line.

Aleph-naught

\( \aleph_0 \) is the cardinality of the set of all natural numbers, and is the first infinite cardinal. A set has cardinality \aleph_0 if and only if it is countably infinite, which is the case if and only if it can be put into a direct bijection, or "one-to-one correspondence", with the natural numbers. Such sets include the set of all prime numbers, the set of all integers, the set of all rational numbers, the set of algebraic numbers, the set of binary strings of all finite lengths, and the set of all finite subsets of any countably infinite set.

If the axiom of countable choice (a weaker version of the axiom of choice) holds, then \( \aleph_0 \) is smaller than any other infinite cardinal.

Aleph-one

\( \aleph_1 \) is the cardinality of the set of all countable ordinal numbers, called ω1 or (sometimes) Ω. Note that this ω1 is itself an ordinal number larger than all countable ones, so it is an uncountable set. Therefore \aleph_1 is distinct from \aleph_0. The definition of \( \aleph_1 \) implies (in ZF, Zermelo–Fraenkel set theory without the axiom of choice) that no cardinal number is between \( \aleph_0 \) and \( \aleph_1 \) . If the axiom of choice (AC) is used, it can be further proved that the class of cardinal numbers is totally ordered, and thus \aleph_1 is the second-smallest infinite cardinal number. Using AC we can show one of the most useful properties of the set ω1: any countable subset of ω1 has an upper bound in ω1. (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 \( \aleph_0 \): every finite set of natural numbers has a maximum which is also a natural number; that is, finite unions of finite sets are finite.

ω1 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. 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 ω1.
The continuum hypothesis
Main article: Continuum hypothesis
See also: Beth number

The cardinality of the set of real numbers (cardinality of the continuum) is \( 2^{\aleph_0} \). 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

\( 2^{\aleph_0}=\aleph_1. \)

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 in 1963 when he showed, conversely, that the CH itself is not a theorem of ZFC by the (then novel) method of forcing.
Aleph-ω

Conventionally the smallest infinite ordinal is denoted ω, and the cardinal number \( \aleph_\omega \) is the least upper bound of

\( \left\{\,\aleph_n : n\in\left\{\,0,1,2,\dots\,\right\}\,\right\} \)

among alephs.

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 numbers; for any positive integer n we can consistently assume that \( 2^{\aleph_0} = \aleph_n \), and moreover it is possible to assume 2^{\aleph_0} is as large as we like. We are only forced to avoid setting it to certain special cardinals with cofinality \( \aleph_0 \), meaning there is an unbounded function from \( \aleph_0 \) to it.
Aleph-α for general α

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

We can then define the aleph numbers as follows

\( \aleph_{0} = \omega \)
\( \aleph_{\alpha+1} = \aleph_{\alpha}^+ \)

and for λ, an infinite limit ordinal,

\( \aleph_{\lambda} = \bigcup_{\beta < \lambda} \aleph_\beta. \)

The α-th infinite initial ordinal is written \( \omega_\alpha \). Its cardinality is written \( \aleph_\alpha \). See initial ordinal.

In ZFC the \aleph function is a bijection between the ordinals and the infinite cardinals.[1]
Fixed points of omega

For any ordinal α we have

\( \alpha\leq\omega_\alpha. \)

In many cases \( \omega_{\alpha} \) 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

\( \omega,\ \omega_\omega,\ \omega_{\omega_\omega},\ \ldots. \)

Any weakly inaccessible cardinal is also a fixed point of the aleph function.
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-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.
References

Notes

^ aleph numbers at PlanetMath

External links

Weisstein, Eric W., "Aleph-0" from MathWorld.