Math 2.0

Mathematical logic and foundations Axiom

The successor function is injective

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 all natural numbers mm and nn, if S(m)=S(n)S(m) = S(n), then m=nm = n.

Remarks

This is one of the Peano axioms. In a construction of ℕ\mathbb{N}, for example in set theory, it is a theorem. An equivalent form is: if m≠nm \neq n, then m+1≠n+1m + 1 \neq n + 1.

Reading path

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

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Natural numbers, object, no evidence levelNatural numbersobjectPeano axioms, concept, no evidence levelPeano axiomsconceptSuccessor function, definition, no evidence levelSuccessor functiondefinitionThe successor function is injective, statement, no evidence levelThe successor functio…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. The successor function is injective statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Peano_axioms, Historical second-order formulation
About this page
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Kindstatement, axiom
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UsesSuccessor function, Peano axioms

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