Math 2.0

Mathematical logic and foundations Theorem

Well-ordering principle

Builds on Order on the natural numbersMathematical induction

Statement

Every nonempty set of natural numbers has a least element. That is, if A⊆ℕA \subseteq \mathbb{N} and A≠∅A \neq \emptyset, then there is an aa in AA with a≤xa \le x for each xx in AA.

Remarks

The least element is unique, because the order on ℕ\mathbb{N} is antisymmetric. The principle follows from strong induction, and conversely it implies the principle of mathematical induction. The set of the integers does not have this property: the set of negative integers has no least element.

Reading path

To understand this page from the ground up, read these 8 pages in order. The foundations come first.

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Definition by recursion, concept, no evidence levelDefinition by recursi…conceptMathematical induction, concept, no evidence levelMathematical inductionconceptNatural numbers, object, no evidence levelNatural numbersobjectPeano axioms, concept, no evidence levelPeano axiomsconceptSuccessor function, definition, no evidence levelSuccessor functiondefinitionAddition of natural numbers, definition, no evidence levelAddition of natural n…definitionOrder on the natural numbers, definition, no evidence levelOrder on the natural …definitionWell-ordering principle, statement, no evidence levelWell-ordering princip…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Definition by recursion concept
  2. Mathematical induction concept
  3. Natural numbers object
  4. Peano axioms concept
  5. Successor function definition
  6. Addition of natural numbers definition
  7. Order on the natural numbers definition
  8. Well-ordering principle statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Well-ordering_principle, lead section
  2. Web https://en.wikipedia.org/wiki/Mathematical_induction, Relationship to the well-ordering principle
About this page
Tag01BX15HSRR
URIm2://01BX15HSRR
Kindstatement, theorem
WritingW1 stated
MSC03F30
StatusThis copy of the site has no build data. Run m2 status 01BX15HSRR.
UsesOrder on the natural numbers, Mathematical induction
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