Wednesday, January 26, 2022

What are the Aleph numbers?

Aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered.

The cardinality of the natural numbers is ℵ0.

Examples of such sets are:
  • the set of all integers,
  • any infinite subset of the integers, such as the set of all square numbers or the set of all prime numbers,
  • the set of all rational numbers,
  • the set of all constructible numbers (in the geometric sense),
  • the set of all algebraic numbers,
  • the set of all computable numbers,
  • the set of all binary strings of finite length, and
  • the set of all finite subsets of any given countably infinite set.

The next larger cardinality of a well-orderable set is aleph-one ℵ1, then ℵ2. Continuing in this manner, it is possible to define a cardinal number ℵα

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.

The assumption that the cardinality of each infinite set is an aleph number is equivalent over ZF 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. For example, one can define card(S) to be the set of sets with the same cardinality as S of minimum possible rank.

This has the property that card(S) = card(T) if and only if S and T have the same cardinality. (The set card(S) does not have the same cardinality of S in general, but all its elements do.)

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