Aleph number Totally Explained

Everything about Aleph Number totally explained

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's 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 ∞.

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's 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.

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's 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; for example, 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.

The continuum hypothesis

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

2^,ldots

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

Aleph number in popular culture

The theme of the infinite runs throughout the work of Jorge Luis Borges, whose short story "The Aleph" ("") deals with a point in space that contains all other points, seen from all possible angles, at all possible times.

In the Futurama episode "Raging Bender", the movie theater's name is Loew's

aleph_0

-plex.

The science fiction novel White Light by Rudy Rucker uses an imaginary universe to elucidate the set theory concept of aleph numbers.

The science fiction novel The Infinitive of Go by John Brunner concerns a teleportation device based on transfinite mathematics which gives access to a multiverse of parallel realities whose cardinality is "at least aleph-four".

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".Further Information

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