Everything about Ordinal Number totally explained
In
set theory,
ordinal,
ordinal number, and
transfinite ordinal number refer to a type of
number introduced by
Georg Cantor in
1897 to accommodate
infinite sequences and to classify sets with certain kinds of
order structures on them. Ordinals are an extension of the
natural numbers different from
integers and from
cardinals.
Well-ordering is
total ordering with
transfinite induction, where transfinite induction extends
mathematical induction beyond the finite. Ordinals represent equivalence classes of well orderings with order-isomorphism being the equivalence relationship. Each ordinal is taken to be the set of smaller ordinals. Ordinals may be categorized as: zero, successor ordinals, and limit ordinals (of various
cofinalities). Given a class of ordinals, one can identify the α-th member of that class, for example one can index (count) them. A class is closed and unbounded if its indexing function is continuous and never stops. One can define addition, multiplication, and exponentiation on ordinals. The
Cantor normal form is a standardized way of writing down ordinals. There is a many to one association from ordinals to cardinals. Larger and larger ordinals can be defined, but they become more and more difficult to describe. Any ordinal number can be made into a
topological space by endowing it with the
order topology.
Ordinals extend the natural numbers
A
natural number (which, in this context, includes the number
0 (number)) can be used for two purposes: to describe the
size of a
set, or to describe the
position of an element in a sequence. When restricted to finite sets these two concepts coincide; there's only one way to put a finite set into a linear sequence, up to isomorphism. When dealing with infinite sets one has to distinguish between the notion of size, which leads to
cardinal numbers, and the notion of position, which is generalized by the ordinal numbers described here. This is because, while any set has only one size (its
cardinality), there are many nonisomorphic well-orderings of any infinite set, as explained below.
Whereas the notion of cardinal number is associated to a set with no particular structure on it, the ordinals are intimately linked with the special kind of sets which are called
well-ordered (so intimately linked, in fact, that some mathematicians make no distinction between the two concepts). A well-ordered set is a totally ordered set (given any two elements one defines a smaller and a larger one in a coherent way) in which there's no infinite
decreasing sequence (however, there may be infinite increasing sequences); equivalently, every non-empty subset of the set has a least element. Ordinals may be used to label the elements of any given well-ordered set (the smallest element being labeled 0, the one after that 1, the next one 2, "and so on") and to measure the "length" of the whole set by the least ordinal which isn't a label for an element of the set. This "length" is called the
order type of the set.
Any ordinal is defined by the set of ordinals that precede it: in fact, the most common definition of ordinals
identifies each ordinal
as the set of ordinals that precede it. For example, the ordinal 42 is the order type of the ordinals less than it, for example, the ordinals from 0 (the smallest of all ordinals) to 41 (the immediate predecessor of 42), and it's generally identified as the set, the smallest ordinal not less than 3.
The set of finite ordinals is infinite, the smallest infinite ordinal: ω.
The set of countable ordinals is uncountable, the smallest uncountable ordinal: ω1.Further Information
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