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.

[a] They were introduced by the mathematician Georg Cantor[1] and are named after the symbol he used to denote them, the Hebrew letter aleph (ℵ).

[2][b] The smallest cardinality of an infinite set is that of the natural numbers, denoted by

(read aleph-nought, aleph-zero, or aleph-null); the next larger cardinality of a well-ordered set is

Continuing in this manner, it is possible to define an infinite cardinal number

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.

) 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.

is the lowercase Greek letter omega), also 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.

is the cardinality of the set of all countable ordinal numbers.

is itself an ordinal number larger than all countable ones, so it is an uncountable set.

implies (in ZF, Zermelo–Fraenkel set theory without the axiom of choice) that no cardinal number is between

If the axiom of choice is used, it can be further proved that the class of cardinal numbers is totally ordered, and thus

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, 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

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.

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.

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

is the limit of a countable-length sequence of smaller cardinals.

(if the axiom of choice holds, this is the (unique) next larger cardinal).

is not a function, but a function-like class, as it is not a set (due to the Burali-Forti paradox).

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

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.

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.