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Aleph number

In the branch of mathematics known as set theory, aleph usually refers to a series of numbers used to represent the cardinality of infinite sets. The name is that of the symbol used to denote these numbers, the Hebrew letter aleph.

For example,

(said aleph-null or aleph-naught) is the cardinality of the set of rational numbers or integers. A set of this cardinality is said to be countably infinite.
(said aleph-one) is the cardinality of the set of all countable ordinal numbers.
is cardinality of the set of all ordinal numbers of cardinality no greater than aleph-one.

For any ordinal number κ there is a corresponding cardinal number

Conventionally the smallest infinite ordinal is denoted ω, and the cardinal number

is the smallest upper bound of

This last is one of relatively few cardinal numbers that can be demonstrated within Zermelo-Fraenkel set theory not to be equal to the cardinality of the set of all real numbers.

Aleph-null

Aleph-null is a transfinite number as defined by Cantor when he proved that infinite sets can have different cardinalities or sizes. Aleph-null is by definition the cardinality of the set of all natural numbers, and is the smallest of all infinite cardinalities. Any set of cardinality Aleph-null can be put into a direct one-to-one correspondence (see bijection) with the integers, and thus is a countably infinite set. Such sets include the set of all prime numbers, the set of all squares of integers, the set of all positive integers, and the set of all integer multiples of a given non-zero real number n.

Aleph-one

Aleph-one is the cardinality of the set of all countably infinite ordinal numbers. It can be demonstrated within the Zermelo-Fraenkel axioms (without the axiom of choice) that no cardinal number is between aleph-null and aleph-one. If the axiom of choice is used, it can be further proved that the class of cardinal numbers is totally ordered, and thus aleph-one is the second-smallest infinite cardinal number.

The continuum hypothesis

In Zermelo-Fraenkel set theory with the axiom of choice, the celebrated continuum hypothesis is equivalent to the identity

This proposition is independent of "ZFC", i.e., of Zermelo-Fraenkel set theory conjoined with the axiom of choice, i.e., it can be neither proved nor disproved within the context of that axiom system. That it is consisent with ZFC was demonstrated by Kurt Gödel in 1940; that it is independent of ZFC was demonstrated by Paul Cohen in 1963.