Math 2.0

Mathematical logic and foundations Theorem

Strong induction

Builds on Mathematical inductionOrder on the natural numbers

Statement

Let P(n)P(n) be a property of natural numbers nn. Suppose that for each natural number nn, if P(k)P(k) holds for every natural number k<nk < n, then P(n)P(n) holds. Then P(n)P(n) holds for every natural number nn.

Remarks

The hypothesis for n=0n = 0 asks for a proof of P(0)P(0) with no assumption, because no natural number is less than 00. The proof applies ordinary induction to the property "P(k)P(k) holds for every k<nk < n". A standard use is the proof that each integer n≥2n \ge 2 is a product of primes.

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 …definitionStrong induction, statement, no evidence levelStrong inductionstatement
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. Strong induction statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Mathematical_induction, Complete (strong) induction
About this page
Tag0KMVFSHFGZ
URIm2://0KMVFSHFGZ
Aliasescomplete-induction, course-of-values-induction
Kindstatement, theorem
WritingW0 stub
MSC03F30
StatusThis copy of the site has no build data. Run m2 status 0KMVFSHFGZ.
UsesMathematical induction, Order on the natural numbers

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