One of the most useful properties of the whole numbers
is that every nonempty subset has a least element;
this allows us to begin a process of "counting"
by successively choosing least elements:
0, 1, 2, 3, 4, ...
Any (totally) ordered set which has this property is
said to be wellordered. Using a wellordering,
we can define a notion of "counting" for
sets of arbitrary size, not just ones with
finitely many objects!
Let % denote the empty set.
Consider the following sequence of sets:
%, {%},
{%, {%}},
{%, {%}, {%,{%}}}, ...
You can verify that these have 0, 1, 2, 3,... elements in them, respectively, and that each member of the sequence is the SET of all the sets that came before it.
Formally an ordinal number is any set which
is (i) transitive (every member is a subset) and
(ii) strictly wellordered by the membership relation.
For example, consider {%,{%},{%,{%}}}.
The member {%,{%}} is in fact a subset consisting of the
two elements % and {%}. The set is also wellordered
because % is a member of {%}, and % and {%} are both
members of {%,{%}}.
The sets defined above are ordinals.
One can show that every ordinal S
has a successor which is S union {S}.
Moreover, every element of an ordinal is an ordinal,
and the union of any set of ordinals is an ordinal.
If we call the set % as "0", the
next set as "1", etc., then consider the union all the
sets {0,1,2,...}.
This is another ordinal called "omega" and it is the
first nonfinite ordinal. It has a successor:
omega union with {omega}, often called "omega + 1".
More ordinals can be obtained by continuing this
succession, and taking the union of all these
ordinals yields an ordinal we call "omega times 2".
Continuing this succession yields an ordering
something like:
0, 1, 2, ..., omega, omega+1, omega+2, ...,
(omega)(2), (omega)(2)+1,...
Somewhere beyond this there is the
first uncountable ordinal.
And there are many more ordinals than these!
Two wellordered sets have the same order type if there
is a 11 correspondence between them that preserves order.
A surprising fact is that any wellordered set has
the same order type as one of the ordinals!
Moreover, a famous theorem known as the
WellOrdering Theorem says that every set
can be wellordered, so ordinals give us a way of
"counting" any set, even if it is not finite!
Presentation Suggestions:
Motivate this subject by having students think about
what "counting" means and how one might systematically
count a set of objects which is uncountable.
The Math Behind the Fact:
Ordinal numbers even have an interesting arithmetic:
we can add two ordinals by concatenating their
order types, and considering the ordinal that represents
the new order type. This addition is not commutative!
For instance, 1 + omega = omega, but
this is not the same as omega + 1.
Multiplication of two ordinals A and B
can be defined as the ordinal representing
the order type of B many copies of A,
concatenated. Thus ordinal multiplication is not
necessarily commutative, either, because (2)(omega) is
(omega) which is not the same order time as (omega)(2).
You can learn more about ordinal numbers in a
course on set theory.
Ordinal numbers form the basis of
transfinite induction which is a generalization
of the principle of induction.
The WellOrdering Theorem (on which the
principle of transfinite induction is based) is
equivalent to the Axiom of Choice.
How to Cite this Page:
Su, Francis E., et al. "Ordinal Numbers."
Math Fun Facts.
<http://www.math.hmc.edu/funfacts>.

