What is an Ordinal Number?

One of the basic concepts of set theory is the generalization of the first and second order numbers that are used daily. The concept of ordinal number is based on the concept of well-ordered set, and well-ordered set is a special case of partial-ordered set and total-ordered set.

The first is 0; the second is a successor of a certain ordinal , = {}, called

Let ( < ) be an ordinal sequence, and specify the order of any two elements < , i >, < , j > in the set A = (Figure 1) as follows: < , i > << , J >; if and only if i < j or i = j and < ; then < A , <;> constitutes one

Partial and perfect order

Order is a very important type of binary relationship (see mapping).

Let R be a binary relationship defined on A that meets the following conditions:

x R x (reflexive) for all x A ;

For all x , y A , we can get x = y from x R y and y R x (anti-symmetry);

For all x , y , z A , x R y and y R z can be used to obtain x R z (transitiveness), which is called R is a partial order defined on A , also called semi-order. Partial order R is usually recorded as or

Ordinal

, b ) Read as before b .

Set A, together with the partial order defined above, is called a partial order set, and is written as < A , >. The normal size relationship on the real number set, the contained relationship between the sets, and the divisibility relationship between the natural numbers are all examples of partial order. Let be the partial order on A. If we define a relation <on A such that x < y if and only if x y and x y , then the relation <; satisfies the condition:

does not hold for any x A

can be obtained from x < y and y < z . At this time <; is called strictly partial order. Conversely, let <; be strictly partial order. If x y is defined if and only if x < y or x = y , must be a partial order.

Therefore, it is enough to discuss only one of partial order and strict partial order. Let < A , > be a partially ordered set. If x 0 A and there are no other x in A such that x x 0, then x 0 is called a minimal element (prime) of A. If x 0 x for all x A , then x 0 is called the smallest element (prime) in A, and the positive integer set is the smallest element under the partial order of divisibility, but if it is limited to integers greater than 1, then There are only minimal elements (each prime number) and no minimum elements.

In this way, the maximum element and the maximum element can be defined. Let x be a subset of the partially ordered set < A , >. If A exists, so that for all x x , there is x , then is called a lower bound of x (about A ). If all lower bounds of x on A have a maximum element 0, then 0 is called the lower exact bound of x (on A ) and recorded as inf x . Similarly, the upper bound and the upper bound can be defined, the latter being written as sup x .

A partial order on A, if you add the condition for all x , y A , there is always x y or y x (at least one holds), it is called total order on A , also called line order . < A , > is called a fully ordered set. Obviously, in the total order set x < y , x = y , x > y , the three must be one and only one. A real number set and any of its subsets are examples of a fully ordered set under normal relations.

For a fully ordered set < A , > If any non-empty subset of condition A is added, it has a minimum element, which is called is a good order on A , and < A , > is called a good order set. Finite sets arranged in any order, natural number sets in natural order, all natural number sets {1, 3, 5, ..., 2, 4, 6, ... An example of a well-ordered set. But the whole integer, the interval [0,1], is not a well-ordered set. Let < A , 1>, < B , 2>; be two partial ordered sets, if there exists a bijective from A to B such that for all x , y A , x 1 y if and only if ( x ) 2 ( y ), the two partial ordered sets are called order isomorphisms, and they are denoted as AB. For example, the odd and even sets are ordered isomorphic, but none of the three well-ordered sets listed above are ordered isomorphic. ^[1]