Math 2.0

statement axiom

The successor function is injective

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.

Explanations

L1 human

Two different numbers never have the same next number. If you know the number after nn, then you know nn.

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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
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  1. Natural numbers object
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  4. The successor function is injective statement

Sources

  1. Web https://en.wikipedia.org/wiki/Peano_axioms, Historical second-order formulation

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