Aleph number

In mathematics, particularly in set theory, the aleph numbers are a sequence of numbers used to represent the cardinality (or size) of infinite sets that can be well-ordered. They were introduced by the mathematician Georg Cantor and are named after the symbol he used to denote them, the Hebrew letter aleph (formula_1).
(Though in older mathematics books, the letter aleph is often printed upside down by accident, partly because a monotype matrix for aleph was mistakenly constructed the wrong way up).
The cardinality of the natural numbers is formula_2 (read "aleph-nought" or "aleph-zero"; the term "aleph-null" is also sometimes used), the next larger cardinality of a well-orderable set is aleph-one formula_3, then formula_4 and so on. Continuing in this manner, it is possible to define a cardinal number formula_5 for every ordinal number formula_6, 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 (formula_7) 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 (applied to a function or sequence that "diverges to infinity" or "increases without bound"), or as an extreme point of the extended real number line.
Aleph-nought.
formula_2 (aleph-nought, also aleph-zero or aleph-null) is the cardinality of the set of all natural numbers, and is an infinite cardinal. The set of all finite ordinals, called formula_9 or formula_10 (where formula_9 is the lowercase Greek letter omega), has cardinality formula_2. A set has cardinality formula_2 if and only if it is countably infinite, that is, there is a bijection (one-to-one correspondence) between it and the natural numbers. Examples of such sets are
These infinite ordinals: formula_9, formula_15, formula_16, formula_17, formula_18 and formula_19 are among the countably infinite sets. For example, the sequence (with ordinality ω·2) of all positive odd integers followed by all positive even integers
is an ordering of the set (with cardinality formula_2) of positive integers.
If the axiom of countable choice (a weaker version of the axiom of choice) holds, then formula_2 is smaller than any other infinite cardinal.
Aleph-one.
formula_3 is the cardinality of the set of all countable ordinal numbers, called formula_24 or sometimes formula_25. This formula_24 is itself an ordinal number larger than all countable ones, so it is an uncountable set. Therefore, formula_3 is distinct from formula_2. The definition of formula_3 implies (in ZF, Zermelo–Fraenkel set theory "without" the axiom of choice) that no cardinal number is between formula_2 and formula_3. If the axiom of choice is used, it can be further proved that the class of cardinal numbers is totally ordered, and thus formula_3 is the second-smallest infinite cardinal number. Using the axiom of choice, one can show one of the most useful properties of the set formula_24: any countable subset of formula_24 has an upper bound in formula_35 (This follows from the fact that the union of a countable number of countable sets is itself countable—one of the most common applications of the axiom of choice.) This fact is analogous to the situation in formula_2: every finite set of natural numbers has a maximum which is also a natural number, and finite unions of finite sets are finite.
formula_24 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 formula_38-algebra generated by an arbitrary collection of subsets (see e.g. Borel hierarchy). 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 formula_24.
Every uncountable coanalytic subset of a Polish space formula_40 has cardinality formula_3 or formula_42.
Continuum hypothesis.
The cardinality of the set of real numbers (cardinality of the continuum) is formula_42. It cannot be determined from ZFC (Zermelo–Fraenkel set theory with the axiom of choice) 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 (provided that ZFC is consistent). 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 (then novel) method of forcing.
Aleph-omega.
Aleph-omega is
where the smallest infinite ordinal is denoted ω. That is, the cardinal number formula_45 is the least upper bound of
formula_45 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 formula_47, and moreover it is possible to assume formula_42 is as large as we like. We are only forced to avoid setting it to certain special cardinals with cofinality formula_2, meaning there is an unbounded function from formula_2 to it (see Easton's theorem).
Aleph-formula_6 for general formula_6.
To define formula_5 for arbitrary ordinal number formula_6, we must define the successor cardinal operation, which assigns to any cardinal number formula_55 the next larger well-ordered cardinal formula_56 (if the axiom of choice holds, this is the next larger 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 formula_57. Its cardinality is written formula_5.
In ZFC, the aleph function formula_1 is a bijection from the ordinals to the infinite cardinals.
Fixed points of omega.
For any ordinal α we have
In many cases formula_60 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 formula_61 is a weakly inaccessible cardinal. If formula_62 were a successor ordinal, then formula_63 would be a successor cardinal and hence not weakly inaccessible. If formula_62 were a limit ordinal less than formula_65, then its cofinality (and thus the cofinality of formula_63) would be less than formula_67 and so formula_67 would not be regular and thus not weakly inaccessible. Thus formula_69 and consequently formula_70 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 (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.)