Aleph number


In mathematics, and in particular set theory, the aleph numbers are a sequence of numbers used to represent the cardinality of infinite sets that can be well-ordered. They are named after the symbol used to denote them, the Hebrew letter aleph .
The cardinality of the natural numbers is , the next larger cardinality is aleph-one, then and so on. Continuing in this manner, it is possible to define a 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 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, or an extreme point of the extended real number line.

Aleph-naught

is the cardinality of the set of all natural numbers, and is an infinite cardinal. The set of all finite ordinals, called ' or ', 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
These infinite ordinals:,,,, and Epsilon numbers | are among the countably infinite sets. For example, the sequence of all positive odd integers followed by all positive even integers
is an 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, called or sometimes. This is itself an ordinal number larger than all countable ones, so it is an uncountable set. Therefore, is distinct from. 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. Using the axiom of choice we 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.
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 -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, 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.
Every uncountable coanalytic subset of a Polish space has
cardinality or.

Continuum hypothesis

The cardinality of the set of real numbers is. 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, CH, is equivalent to the identity
The CH states that there is no set whose cardinality is strictly between that of the integers 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 by the method of forcing.

Aleph-omega

Aleph-omega is
where the smallest infinite ordinal is denoted ω. That is, the cardinal number is the least upper bound of
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, and moreover it is possible to assume is as large as we like. We are only forced to avoid setting it to certain special cardinals with cofinality, meaning there is an unbounded function from to it.

Aleph-\alpha for general \alpha

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:
and for λ, an infinite limit ordinal,
The α-th infinite initial ordinal is written. Its cardinality is written.
In ZFC, the aleph function is a bijection from the ordinals to the infinite cardinals.

Fixed points of omega

For any ordinal α we have
In many cases 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 omega function, because of the fixed-point lemma for normal functions. The first such is the limit of the sequence
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.
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, 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 to be the set of sets with the same cardinality as S of minimum possible rank. This has the property that card = card if and only if S and T have the same cardinality.

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