Well-ordering principle

Well-ordering principle

• Each non-empty subset of the set of natural numbers has a minimum element, that is, the natural numbers form a well-ordered set under their standard size relationship
• In axiomatic set theory, the set of natural numbers is defined as the smallest inductive set (the set containing 0 and the successor of every element in itself), and it can be shown that all n satisfying {0,...,n} is a well-ordered set The set composed is an inductive set, and thus the set of natural numbers itself. From this, it can be deduced that the set of natural numbers is also a well-ordered set.
• The well-ordering theorem is one of the equivalent forms of the axiom of choice
  • Its content is: for any set S, there is a binary relation R on S such that <S, R> is a well-ordered set
• significance
  • The significance of the principle of good order is mainly that the so-called "least counterexample method" can be used in the proof, which is equivalent to the combination of the method of contradiction and mathematical induction.
• General description: Let "<" be a relation on the set S that satisfies the following properties:
  1. Given x, y, and z in S, if x<y and y<z, then x<z;
  2. Given x and y in S, exactly one of the following three possibilities is true:
     1. x<y
     2. x=y
     3. y<x
  3. If A is any non-empty subset of S, then there is an element x in A such that for all y in A, there is x <= y (that is, x < y or x = y)
  • This relation is called a well-ordered relation of S
• Applications in computer algorithms
  • If we can map the states x of a computation to an element f(x) belonging to a well-ordered set S such that each step of the computation transforms a state x into a state y, with f(y)<f(x ), the algorithm must terminate.

Among the computer algorithms, the most classic (personally think ^_^)

设S对于<是良序，且设P(x)是关于S的元素x的一个命题。
求证：如果在对于所有的y < x,P(y)为真的假定下能证得P(x)为真，则P(x)对S中得所有x为真。
证明：
令A是使得P(x)为假的所有的x的集合。
如果A非空，则它含有最小的元素x0。
因此，P(y)对于所有y < x0为真。
但这意味着P(x0)为真，所以x0不在A中（与假设矛盾）
因此：A必须为空 => 即P(x)总是为真。

The above can also be used for the proof of P(1)

对于所有正整数 y < 1,如果P(y)为真，则P(1)为真。