In set theory, a discipline within mathematics, the aleph numbers are a
sequence of numbers used to represent the cardinality of infinite sets. 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 α, as described below.
The concept goes back 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 ω or ω0, 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 the set of all square numbers, the set
of all cubic numbers, the set of all fourth powers, ...
the set of all perfect powers, the set of all prime powers,
the set of all even numbers, the set of all odd numbers,
the set of all prime numbers, the set of all composite numbers,
the set of all integers, the set of all rational numbers,
the set of all constructible numbers, the set of all algebraic numbers,
the set of all computable numbers, the set of all definable numbers,
the set of all binary strings of finite length, and
the set of all finite subsets of any given countably infinite set.
These infinite ordinals: ω, ω+1, ω·2, ω2, ωω and ε0 are among the countably
infinite sets. For example, the sequence of all positive odd integers followed by
all positive even integers {1, 3, 5, 7, 9, ..., 2, 4, 6, 8, 10,
...} 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 ω1 or Ω. This ω1 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 AC we can show one of the
most useful properties of the set ω1: any countable subset of ω1 has an upper
bound in ω1. 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. 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. 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 ω1.
In popular books is sometimes incorrectly defined to be , but this is
wrong if the continuum hypothesis fails. 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-ω
Conventionally the smallest infinite ordinal is denoted ω, and the cardinal
number is the least upper bound of among alephs.
Aleph-ω 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-α for general α To define for arbitrary ordinal number
, we must define the successor cardinal operation, which assigns to any cardinal
number ρ the next larger well-ordered cardinal ρ+.
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 function is a bijection
between the ordinals and 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(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. does not have the same cardinality of S in general, but all its
elements do.) See also
Regular cardinal References
Notes Sierpiński, Wacław, Cardinal and ordinal
numbers., Polska Akademia Nauk Monografie Matematyczne 34, Warsaw:
Państwowe Wydawnictwo Naukowe, MR 0095787
External links Hazewinkel, Michiel, ed., "Aleph-zero",
Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
Weisstein, Eric W., "Aleph-0", MathWorld.