Aleph number


In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality 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.
The smallest cardinality of an infinite set is that of the natural numbers, denoted by ; the next larger cardinality of a well-ordered set is then then 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 Georg Cantor's first [set theory article|infinite sets can have different cardinalities].
The aleph numbers differ from the Extended [real number line|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, or as an extreme point of the extended real number line.

Aleph-zero

is the cardinality of the set of all natural numbers, and is an infinite cardinal. The set of all finite ordinals, called or , also has cardinality. A set has cardinality if and only if it is countably infinite, that is, there is a bijection between it and the natural numbers. Examples of such sets are:
Among the countably infinite sets are certain infinite ordinals, including for example,,,,, and Epsilon numbers |. For example, the sequence of all positive odd integers followed by all positive even integers is a well-ordering of the set of positive integers.
If the axiom of countable choice holds, then is smaller than any other infinite cardinal.

Aleph-one

is the cardinality of the set of all countable ordinal numbers. This set is denoted by . The set is itself an ordinal number larger than all countable ones, so it is an uncountable set. Therefore, is the smallest cardinality that is larger than the smallest infinite cardinality.
The definition of implies that no cardinal number is between and If the axiom of choice is used, it can be further proved that the class of cardinal numbers is totally ordered, and thus is the second-smallest infinite cardinal number. One can show one of the most useful properties of the set : Any countable subset of has an upper bound in . This fact is analogous to the situation in : Every finite set of natural numbers has a maximum, which is also a natural number, and finite unions of finite sets are finite.
An example application of the ordinal is "closing" with respect to countable-arity 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 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 unions and complements, and taking the union of all that over all of

Continuum hypothesis

The cardinality of the set of real numbers is 2. It cannot be determined from ZFC where this number fits exactly in the aleph number hierarchy, but it follows from ZFC that the continuum hypothesis is equivalent to the identity
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. 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 method of forcing.

Aleph-omega

Aleph-omega is where the smallest infinite ordinal is denoted as. That is, the cardinal number is the least upper bound of.
Notably, 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 natural number, we can consistently assume that, and moreover it is possible to assume that is at least as large as any cardinal number we like. The main restriction ZFC puts on the value of is that it cannot equal certain special cardinals with cofinality. An uncountably infinite cardinal having cofinality means that there is a sequence of cardinals whose limit is . As per the definition above, is the limit of a countable-length sequence of smaller cardinals.

Aleph-''α'' for general ''α''

To define for arbitrary ordinal number, we must define the successor cardinal operation, which assigns to any cardinal number the next larger well-ordered cardinal .
We can then define the aleph numbers as follows:
The -th infinite initial ordinal is written. Its cardinality is written.
Informally, the aleph function is a bijection from the ordinals to the infinite cardinals.
Formally, in ZFC, is not a function, but a function-like class, as it is not a set.

Fixed points of omega

For any ordinal we have.
In many cases is strictly greater than α. For example, it is true for any successor ordinal: holds. There are, however, some limit ordinals that are fixed points of the omega function, because of the fixed-point lemma for normal functions. The first such is the limit of the sequence
which is sometimes denoted.
Any weakly inaccessible cardinal is also a fixed point of the aleph function. This can be shown in ZFC as follows. Suppose is a weakly inaccessible cardinal. If were a successor ordinal, then would be a successor cardinal and hence not weakly inaccessible. If were a limit ordinal less than then its cofinality would be less than and so would not be regular and thus not weakly inaccessible. Thus and consequently, 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, 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 to be the set of sets with the same cardinality as of minimum possible rank. This has the property that if and only if and have the same cardinality.