HOME

TheInfoList



OR:

The
von Neumann Von Neumann may refer to: * John von Neumann (1903–1957), a Hungarian American mathematician * Von Neumann family * Von Neumann (surname), a German surname * Von Neumann (crater), a lunar impact crater See also * Von Neumann algebra * Von Ne ...
cardinal assignment is a
cardinal assignment In set theory, the concept of cardinality is significantly developable without recourse to actually defining cardinal numbers as objects in the theory itself (this is in fact a viewpoint taken by Frege; Frege cardinals are basically equivalence cla ...
that uses
ordinal number In set theory, an ordinal number, or ordinal, is a generalization of ordinal numerals (first, second, th, etc.) aimed to extend enumeration to infinite sets. A finite set can be enumerated by successively labeling each element with the leas ...
s. For a
well-order In mathematics, a well-order (or well-ordering or well-order relation) on a set ''S'' is a total order on ''S'' with the property that every non-empty subset of ''S'' has a least element in this ordering. The set ''S'' together with the well-or ...
able set ''U'', we define its
cardinal number In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. ...
to be the smallest ordinal number
equinumerous In mathematics, two sets or classes ''A'' and ''B'' are equinumerous if there exists a one-to-one correspondence (or bijection) between them, that is, if there exists a function from ''A'' to ''B'' such that for every element ''y'' of ''B'', the ...
to ''U'', using the von Neumann definition of an ordinal number. More precisely: :, U, = \mathrm(U) = \inf \, where ON is the
class Class or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used differently ...
of ordinals. This ordinal is also called the initial ordinal of the cardinal. That such an ordinal exists and is unique is guaranteed by the fact that ''U'' is well-orderable and that the class of ordinals is well-ordered, using the
axiom of replacement In set theory, the axiom schema of replacement is a schema of axioms in Zermelo–Fraenkel set theory (ZF) that asserts that the image of any set under any definable mapping is also a set. It is necessary for the construction of certain infinite ...
. With the full
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
, every set is well-orderable, so every set has a cardinal; we order the cardinals using the inherited ordering from the ordinal numbers. This is readily found to coincide with the ordering via ≤''c''. This is a well-ordering of cardinal numbers.


Initial ordinal of a cardinal

Each ordinal has an associated
cardinal Cardinal or The Cardinal may refer to: Animals * Cardinal (bird) or Cardinalidae, a family of North and South American birds **'' Cardinalis'', genus of cardinal in the family Cardinalidae **'' Cardinalis cardinalis'', or northern cardinal, t ...
, its cardinality, obtained by simply forgetting the order. Any well-ordered set having that ordinal as its
order type In mathematics, especially in set theory, two ordered sets and are said to have the same order type if they are order isomorphic, that is, if there exists a bijection (each element pairs with exactly one in the other set) f\colon X \to Y suc ...
has the same cardinality. The smallest ordinal having a given cardinal as its cardinality is called the initial ordinal of that cardinal. Every finite ordinal (
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
) is initial, but most infinite ordinals are not initial. The
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
is equivalent to the statement that every set can be well-ordered, i.e. that every cardinal has an initial ordinal. In this case, it is traditional to identify the cardinal number with its initial ordinal, and we say that the initial ordinal ''is'' a cardinal. The \alpha-th infinite initial ordinal is written \omega_\alpha. Its cardinality is written \aleph_ (the \alpha-th
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 a ...
). For example, the cardinality of \omega_=\omega is \aleph_, which is also the cardinality of \omega^, \omega^, and \epsilon_ (all are
countable In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers ...
ordinals). So we identify \omega_ with \aleph_, except that the notation \aleph_ is used for writing cardinals, and \omega_ for writing ordinals. This is important because arithmetic on cardinals is different from arithmetic on ordinals, for example \aleph_^ = \aleph_ whereas \omega_^ > \omega_. Also, \omega_ is the smallest
uncountable In mathematics, an uncountable set (or uncountably infinite set) is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal num ...
ordinal (to see that it exists, consider the set of
equivalence class In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
es of well-orderings of the natural numbers; each such well-ordering defines a countable ordinal, and \omega_ is the order type of that set), \omega_ is the smallest ordinal whose cardinality is greater than \aleph_, and so on, and \omega_ is the limit of \omega_ for natural numbers n (any limit of cardinals is a cardinal, so this limit is indeed the first cardinal after all the \omega_). Infinite initial ordinals are limit ordinals. Using ordinal arithmetic, \alpha<\omega_ implies \alpha+\omega_=\omega_, and 1 ≤ ''α'' < ω''β'' implies ''α'' · ω''β'' = ω''β'', and 2 ≤ ''α'' < ω''β'' implies ''α''ω''β'' = ω''β''. Using the Veblen hierarchy, ''β'' ≠ 0 and ''α'' < ω''β'' imply \varphi_(\omega_) = \omega_ \, and Γω''β'' = ω''β''. Indeed, one can go far beyond this. So as an ordinal, an infinite initial ordinal is an extremely strong kind of limit.


See also

*
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 a ...


References

* Y.N. Moschovakis ''Notes on Set Theory'' (1994 Springer) p. 198 {{Mathematical logic Cardinal numbers Ordinal numbers