In
set theory, an ordinal is a
set that represents the most
basic concept of
order.
Given the following three properties (notation
explained here):
Trans (x) <-> ( (a ∈ b & b ∈ x) -> a ∈ x )
"x is transitive if and only if whenever a is in b
and b is in x, a is in x."
Alt (x) <-> ( (a ∈ x & b ∈ x & a != b) <-> (a ∈
b v b ∈ a) ).
"x is alternate if and only if for each pair of distinct
elements of x, one is also an element of the other."
Fund (x) <-> ( (a ⊆ x & a != 0) -> ((Eb) (b e a &
b ∩ a = 0)) ).
"x is well-founded if and only if each nonempty subset a
of x contains an element containing no elements from a."
This, by the way, is exactly what the Axiom of Foundation asserts for every set.
Ordinals are sets that have all of these properties:
Ord (x) <-> ( Trans (x) & Alt (x) & Fund (x) ).
The empty set 0 is of course an ordinal.
I'm not going to go into the whole structure of ordinal theory here;
however, it is important to mention an intersting consequence of the Trans property which sets the stage for the general structure of ordinals:
Trans (x) <-> ( (a ∈ x) <-> (a ⊆ x) )
"x is transitive if and only if every element of x is also a
subset of x, and vice versa."
This means we can build up ordinals starting from the empty set 0:
0: 0
1: {0}
2: {0, {0}}
3: {0, {0}, {0, {0}}}
4: {0, {0}, {0, {0}}, {0, {0}, {0, {0}}}}
and so on. Notice the sets are labeled with numbers next
to them. I did this because building up ordinals in this way is one way
of defining the natural numbers.
But notice, we can also
convert the goofy sets to:
0 = 0
1 = {0}
2 = {0, 1}
3 = {0, 1, 2}
4 = {0, 1, 2, 3}
5 = {0, 1, 2, 3, 4}
and so on.
This leads further to the primary consequence of the definition of ordinals
and the axiom of infinity: Accepting the class of all natural numbers
to be a set means that this set is itself an ordinal.
Yes, infinity IS a number.
Source of learning: Axiomatic set theory, Paul Bernays, 1968.