Math 2.0

Mathematical logic and foundations Axiom

Zero is not a successor

Builds on Successor functionPeano axioms

Statement

Let S ⁣:ℕ→ℕS \colon \mathbb{N} \to \mathbb{N} be the successor function of the natural numbers. For each natural number nn, S(n)≠0S(n) \neq 0.

Remarks

This is one of the Peano axioms. In a construction of ℕ\mathbb{N}, it is a theorem. With the injectivity of the successor function, it shows that the numbers 0,S(0),S(S(0)),…0, S(0), S(S(0)), \dots are all different.

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

Show the path as a graph
Natural numbers, object, no evidence levelNatural numbersobjectPeano axioms, concept, no evidence levelPeano axiomsconceptSuccessor function, definition, no evidence levelSuccessor functiondefinitionZero is not a successor, statement, no evidence levelZero is not a success…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. Natural numbers object
  2. Peano axioms concept
  3. Successor function definition
  4. Zero is not a successor statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Peano_axioms, Historical second-order formulation
About this page
Tag0D752BG155
URIm2://0D752BG155
Kindstatement, axiom
WritingW0 stub
MSC03F30
StatusThis copy of the site has no build data. Run m2 status 0D752BG155.
UsesSuccessor function, Peano axioms

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