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[1] and are named after the symbol he used to denote them, the Hebrew letter aleph (ℵ).[2]
The cardinality of the natural numbers is ℵ0 (read aleph-nought or aleph-zero or aleph-null), the next larger cardinality of a well-ordered set is aleph-one ℵ1, then ℵ2 and so on. Continuing in this manner, it is possible to define a cardinal number ℵα for every ordinal number α, as described below.
The concept and notation are due to Georg Cantor,[3] 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, 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.
ℵ0 (aleph-zero, also aleph-nought 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 ω or ω0 (where ω is the lowercase Greek letter omega), has cardinality ℵ0. A set has cardinality ℵ0 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: ω, ω + 1, ω⋅2, ω2, ωω, and ε0 are among the countably infinite sets.[4] 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 ℵ0) of positive integers.
If the axiom of countable choice (a weaker version of the axiom of choice) holds, then ℵ0 is smaller than any other infinite cardinal, and is therefore the (unique) least infinite ordinal.
ℵ1 is, by definition, the cardinality of the set of all countable ordinal numbers. This set is denoted by ω1 (or sometimes Ω). The set ω1 is itself an ordinal number larger than all countable ones, so it is an uncountable set. Therefore, ℵ1 is distinct from ℵ0. The definition of ℵ1 implies (in ZF, Zermelo–Fraenkel set theory without the axiom of choice) that no cardinal number is between ℵ0 and ℵ1. If the axiom of choice is used, it can be further proved that the class of cardinal numbers is totally ordered, and thus ℵ1 is the second-smallest infinite cardinal number. One can show one of the most useful properties of the set ω1: Any countable subset of ω1 has an upper bound in ω1 (this follows from the fact that the union of a countable number of countable sets is itself countable). This fact is analogous to the situation in ℵ0: Every finite set of natural numbers has a maximum which is also a natural number, and finite unions of finite sets are finite.
The ordinal ω1 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 σ-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, 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 ω1.
See main article: Continuum hypothesis.
See also: Beth number. The cardinality of the set of real numbers (cardinality of the continuum) is 2ℵ0. It cannot be determined from ZFC (Zermelo–Fraenkel set theory augmented 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
2ℵ0 = ℵ1.[5]
The CH states that there is no set whose cardinality is strictly between that of the integers and the real numbers.[6] 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.[5] [7]
Aleph-omega is
ℵω = sup = sup
where the smallest infinite ordinal is denoted as ω. That is, the cardinal number ℵω is the least upper bound of
.
Notably, ℵω 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 2ℵ0: For any natural number n ≥ 1, we can consistently assume that 2ℵ0 = ℵn, and moreover it is possible to assume that 2ℵ0 is as least as large as any cardinal number we like. The main restriction ZFC puts on the value of 2ℵ0 is that it cannot equal certain special cardinals with cofinality ℵ0. An uncountably infinite cardinal κ having cofinality ℵ0 means that there is a (countable-length) sequence κ0 ≤ κ1 ≤ κ2 ≤ ... of cardinals κi < κ whose limit (i.e. its least upper bound) is κ (see Easton's theorem). As per the definition above, ℵω is the limit of a countable-length sequence of smaller cardinals.
To define ℵα for arbitrary ordinal number α, we must define the successor cardinal operation, which assigns to any cardinal number ρ the next larger well-ordered cardinal ρ+ (if the axiom of choice holds, this is the (unique) next larger cardinal).
We can then define the aleph numbers as follows:
ℵ0 = ω
ℵα+1 = (ℵα)+
ℵλ = ⋃ for λ an infinite limit ordinal,
The α-th infinite initial ordinal is written ωα. Its cardinality is written ℵα.
Informally, the aleph function ℵ: On → Cd is a bijection from the ordinals to the infinite cardinals.Formally, in ZFC, ℵ is not a function, but a function-like class, as it is not a set (due to the Burali-Forti paradox).
For any ordinal α we have
α ≤ ωα.
In many cases ωα is strictly greater than α. For example, it is true for any successor ordinal: α + 1 < ωα+1 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
ω, ωω, ωωω, ...,
which is sometimes denoted ωω....
Any weakly inaccessible cardinal is also a fixed point of the aleph function.[8] This can be shown in ZFC as follows. Suppose κ = ℵλ is a weakly inaccessible cardinal. If λ were a successor ordinal, then ℵλ would be a successor cardinal and hence not weakly inaccessible. If λ were a limit ordinal less than κ then its cofinality (and thus the cofinality of ℵλ) would be less than κ and so κ would not be regular and thus not weakly inaccessible. Thus λ ≥ κ and consequently λ = κ which makes it a fixed point.
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 (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.)