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.

Or"di*nal (?), a. [L. ordinalis, fr. ordo, ordinis, order. See Order.]

1.

Indicating order or succession; as, the ordinal numbers, first, second, third, etc.

2.

Of or pertaining to an order.

Or"di*nal, n.

1.

A word or number denoting order or succession.

2. Ch. of Eng.

The book of forms for making, ordaining, and consecrating bishops, priests, and deacons.

A book containing the rubrics of the Mass.

[Written also ordinale.]

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