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= Aleph number =
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Introduction
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In mathematics, and in particular 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 are named after the
symbol used to denote them, the Hebrew letter aleph (\aleph) (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 \aleph_0 (read
'aleph-naught' or 'aleph-zero'; the term 'aleph-null' is also
sometimes used), the next larger cardinality is aleph-one \aleph_1,
then \aleph_2 and so on. Continuing in this manner, it is possible to
define a cardinal number \aleph_\alpha for every ordinal number
\alpha, as described below.
The concept and notation are due to Georg Cantor, Miller quotes :
"His new numbers deserved something unique. ... Not wishing to invent
a new symbol himself, he chose the aleph, the first letter of the
Hebrew alphabet...the aleph could be taken to represent new
beginnings..."
who defined the notion of cardinality and realized that infinite sets
can have different cardinalities.
The aleph numbers differ from the infinity (\infty) 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 (applied to a function or sequence that "diverges to
infinity" or "increases without bound"), or an extreme point of the
extended real number line.
{{anchor|Aleph-null}}Aleph-naught
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\aleph_0 (aleph-naught, 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 \omega or \omega_{0}
(where \omega is the lowercase Greek letter omega), has cardinality
\aleph_0. A set has cardinality \aleph_0 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
* 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 (in the geometric sense),
* 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: \omega, \omega+1, \omega\cdot2, \omega^{2},
\omega^{\omega} and \varepsilon_{0} are among the countably infinite
sets. For example, the sequence (with ordinality �·2) 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 (with cardinality \aleph_0) of positive
integers.
If the axiom of countable choice (a weaker version of the axiom of
choice) holds, then \aleph_0 is smaller than any other infinite
cardinal.
Aleph-one
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\aleph_1 is the cardinality of the set of all countable ordinal
numbers, called \omega_{1} or sometimes \Omega. This \omega_{1} is
itself an ordinal number larger than all countable ones, so it is an
uncountable set. Therefore, \aleph_1 is distinct from \aleph_0. The
definition of \aleph_1 implies (in ZF, Zermelo-Fraenkel set theory
'without' the axiom of choice) that no cardinal number is between
\aleph_0 and \aleph_1. If the axiom of choice is used, it can be
further proved that the class of cardinal numbers is totally ordered,
and thus \aleph_1 is the second-smallest infinite cardinal number.
Using the axiom of choice we can show one of the most useful
properties of the set \omega_{1}: any countable subset of \omega_{1}
has an upper bound in \omega_{1}. (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 \aleph_0: every finite set of
natural numbers has a maximum which is also a natural number, and
finite unions of finite sets are finite.
\omega_{1} 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 \sigma-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 \omega_{1}.
Every uncountable coanalytic subset of a Polish space X has
cardinality \aleph_1 or 2^{\aleph_0}.
Continuum hypothesis
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The cardinality of the set of real numbers (cardinality of the
continuum) is 2^{\aleph_0}. 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
{{block indent|2^{\aleph_0}=\aleph_1 }}
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
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Aleph-omega is
:\aleph_\omega = \sup \{ \aleph_n : n \in \omega \} = \sup \{ \aleph_n
: n \in \left\{\,0,1,2,\dots\,\right\}\, \}
where the smallest infinite ordinal is denoted �. That is, the
cardinal number \aleph_\omega is the least upper bound of
{{block indent|\left\{\,\aleph_n :
n\in\left\{\,0,1,2,\dots\,\right\}\,\right\}.}}
\aleph_\omega 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 2^{\aleph_0} = \aleph_n,
and moreover it is possible to assume 2^{\aleph_0} is as large as we
like. We are only forced to avoid setting it to certain special
cardinals with cofinality \aleph_0, meaning there is an unbounded
function from \aleph_0 to it (see Easton's theorem).
Aleph-<math>\alpha</math> for general <math>\alpha</math>
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To define \aleph_\alpha for arbitrary ordinal number \alpha, we must
define the successor cardinal operation, which assigns to any cardinal
number \rho the next larger well-ordered cardinal \rho^{+} (if the
axiom of choice holds, this is the next larger cardinal).
We can then define the aleph numbers as follows:
{{block indent|\aleph_{0} = \omega}}
{{block indent|\aleph_{\alpha+1} = \aleph_{\alpha}^+}}
and for λ, an infinite limit ordinal,
{{block indent|\aleph_{\lambda} = \bigcup_{\beta < \lambda}
\aleph_\beta.}}
The α-th infinite initial ordinal is written \omega_\alpha. Its
cardinality is written \aleph_\alpha.
In ZFC, the aleph function \aleph is a bijection from the ordinals to
the infinite cardinals.
Fixed points of omega
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For any ordinal α we have
In many cases \omega_{\alpha} 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
{{block indent|\omega,\ \omega_\omega,\ \omega_{\omega_\omega},\
\ldots.}}
Any weakly inaccessible cardinal is also a fixed point of the aleph
function. This can be shown in ZFC as follows. Suppose \kappa =
\aleph_\lambda is a weakly inaccessible cardinal. If \lambda were a
successor ordinal, then \aleph_\lambda would be a successor cardinal
and hence not weakly inaccessible. If \lambda were a limit ordinal
less than \kappa , then its cofinality (and thus the cofinality of
\aleph_\lambda) would be less than \kappa and so \kappa would not be
regular and thus not weakly inaccessible. Thus \lambda \geq \kappa
and consequently \lambda = \kappa which makes it a fixed point.
Role of axiom of choice
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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.)
See also
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* Regular cardinal
* Ordinal number
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Original Article:
http://en.wikipedia.org/wiki/Aleph_number
