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

The smallest cardinality of an infinite set is that of the natural numbers (read aleph-nought, aleph-zero, or aleph-null); the next larger cardinality of a well-ordered and so on. Continuing in this manner, it is possible to define an infinite cardinal number for every ordinal number as described below.

The concept and notation are due 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, 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-zero

(aleph-nought, 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 is the lowercase Greek letter omega 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:

Among the countably infinite sets are certain infinite ordinals,, and \varepsilon_0. For example, the sequence (with order type) of positive integers.

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

Aleph-one

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 the smallest infinite cardinality.

implies (in ZF, Zermelo–Fraenkel set theory without If the axiom of choice is used, it can be further proved that the class of cardinal numbers is totally ordered is the second-smallest infinite cardinal number. One can show one of the most useful properties of the set : Every finite set of natural numbers has a maximum, which is also a natural number, and finite unions of finite sets are finite.

is "closing" with respect to countable-arity 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, groupss, 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

Continuum hypothesis

The cardinality of the set of real numbers (cardinality of the continuum) is 2. 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

  1. .

The CH states that there is no set whose cardinality is strictly between that of the natural numbers and the real numbers. 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, using the (then-novel) method of forcing.

Aleph-omega

is the least upper bound.

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 that it cannot equal certain special cardinals with cofinality (see Easton's theorem is the limit of a countable-length sequence of smaller cardinals.

Aleph-α for general α

, we must define the successor cardinal operation the next larger well-ordered (if the axiom of choice holds, this is the (unique) next larger cardinal).

We can then define the aleph numbers as follows:

  1. an infinite limit ordinal,

-th infinite initial ordinal.

Informally, the aleph function is a bijection from the ordinals to the infinite cardinals that can be well ordered. Formally, in ZFC is not a function, but a function-like class, as it is not a set (due to the Burali-Forti paradox).

Fixed points of omega

.

is strictly greater than α. For example, it is true for any successor ordinal holds. There are, however, some limit ordinals that are fixed pointss 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. were a successor ordinal would be a successor cardinal were a limit ordinal then its cofinality, which makes it a fixed point.

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.

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 in general, but all its elements do.)

See also

Notes

References