Well-founded crowd

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In mathematics a well-founded set (also well-founded set , well-founded order , terminating order , Noetherian order ) is a semi-ordered set that does not contain any infinite, genuinely descending chains. Equivalently, a semi-ordered set is called well-founded if every non-empty subset contains at least one minimal element .

All well-ordered sets are well-founded because in a well-ordered set every non-empty subset must have a smallest element and the smallest element of a set is always minimal. Unlike well-ordered sets, well-founded sets do not need to be totally ordered . All totally ordered well-founded sets are well ordered.

Noetherian induction

Profound amounts allow the application of noetherian induction, a version of transfinite induction : Is a property of elements of a under an order relation depth amount , and the following statements are true:

  1. is true for all minimal elements of .
  2. If an element of and is true for all , then is also true.

Then true for all elements of .


The whole numbers , the rational numbers, and the real numbers in their natural order each contain infinite descending chains and are therefore not well founded.

The power set of a set with the subset relation as order is well founded if and only if the set is finite . All finite semi-ordered sets are well-founded because finite sets can only have finite chains.

The following sets are well-founded but not totally ordered:

  • the natural numbers with the order
If a divisor of is
  • the set of all pairs of natural numbers with order
if and
If a substring of is
If a part of expression is
  • loads of sets with the order
, if is an element of (really element, not subset!)

Length of descending chains

If and is a well-founded set , then the descending chains at the beginning are all finite, but their length does not have to be restricted. Consider e.g. B. the amount

(where ) with the order

if or

It is z. B. and . is well-founded, but there are descending chains of arbitrary (finite) length at the beginning.

See also