Aleph number

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

In the branch of mathematics known as set theory, 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 (aleph-null, also aleph-naught 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. While some alephs are larger than others, ∞ is just ∞.

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[edit] Aleph-null

Aleph-null (\aleph_0) is by definition the cardinality of the set of all natural numbers, and (assuming, as usual, the axiom of choice) is the smallest of all infinite cardinalities. 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, and the set of all finite subsets of any countably infinite set.

[edit] Aleph-one

\aleph_1 is the cardinality of the set of all countable ordinal numbers, called ω1 or 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 Ω: any countable subset of Ω has an upper bound in Ω. (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: any finite set of natural numbers has a maximum which is also a natural number; that is, finite unions of finite sets are finite.

Ω 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 Ω.

[edit] The continuum hypothesis

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. That it is consistent with ZFC was demonstrated by Kurt Gödel in 1940; that it is independent of ZFC was demonstrated by Paul Cohen in 1963.

[edit] Aleph-ω

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

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

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.

[edit] Aleph-α for general α

To define \aleph_\alpha for arbitrary ordinal number α, we must define the successor cardinal operation, which assigns to any cardinal number ρ the next bigger well-ordered cardinal ρ + . (If the axiom of choice holds, this is the next bigger 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 ωα. Its cardinality is written \aleph_\alpha. See initial ordinal.

[edit] Fixed points of aleph

For any ordinal α we have

\alpha\leq\aleph_\alpha.

In many cases \aleph_{\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 aleph function. The first such is the limit of the sequence

\aleph_0, \aleph_{\aleph_0}, \aleph_{\aleph_{\aleph_0}},\ldots

Any inaccessible cardinal is a fixed point of the aleph function as well.

[edit] Aleph number in popular culture

  • The theme of the infinite runs throughout the work of Jorge Luis Borges, whose short story "The Aleph" ("El Aleph") deals with a point in space that contains all other points, seen from all possible angles, at all possible times.
  • Scarlett Thomas's book "PopCo", features both a discussion of aleph-null and several events of importance that involve the concept.
  • Aleph One is the name of the open-source project for Bungie Studios' Marathon series of computer games. The last game of the series is entitled Marathon Infinity, so Aleph was chosen as the name because it was "going beyond Infinity".

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