CorollaryStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passverified 2026-08-02 (claude-opus-5)

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Literature-sourced — the exact statement appears in a cited source; only wording and notation differ.
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Addition is a well-defined total operation

Statement

For each $m \in \mathbb{N}$ there is a unique function $s_m : \mathbb{N} \to \mathbb{N}$ with $s_m(0) = m$ and $s_m(\sigma(n)) = \sigma(s_m(n))$ . Writing $m + n := s_m(n)$ , addition $+ : \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ is a well-defined total binary operation satisfying $m + 0 = m$ and $m + \sigma(n) = \sigma(m + n)$ .

Facts & Assumptions

Given: the natural numbers $\mathbb{N}$ with successor $\sigma$ .

[L1]

The recursion theorem: for a set $A$ , an element $a \in A$ , and a function $f : A \to A$ there is a unique $g : \mathbb{N} \to A$ with $g(0) = a$ and $g(\sigma(n)) = f(g(n))$ (The recursion theorem).

[L2]

$(\mathbb{N}, 0, \sigma)$ is a Peano system (The von Neumann naturals form a Peano system).

Proof

technique · direct

1.1

Fix $m \in \mathbb{N}$ ; since $(\mathbb{N}, 0, \sigma)$ is a Peano system [L2], apply the recursion theorem with $A = \mathbb{N}$ , $a = m$ , and $f = \sigma$ : there is a unique function $s_m : \mathbb{N} \to \mathbb{N}$ with $s_m(0) = m$ and $s_m(\sigma(n)) = \sigma(s_m(n))$ .

L1L2

2.1

Define $m + n := s_m(n)$ ; since $s_m$ is a total function on $\mathbb{N}$ for every $m \in \mathbb{N}$ , the assignment $(m,n) \mapsto m + n$ is a well-defined function $\mathbb{N} \times \mathbb{N} \to \mathbb{N}$ .

step 1.1

3.1

The recursion equations become $m + 0 = s_m(0) = m$ and $m + \sigma(n) = s_m(\sigma(n)) = \sigma(s_m(n)) = \sigma(m + n)$ , the defining clauses of addition.

step 1.1step 2.1∎

Depends on

Used by

Addition of natural numbers Definition

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 17 results over 9 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

T. Tao, Analysis I, 3rd ed., §2.1-2.3 (Peano axioms, recursion, arithmetic) (standard reference, not scraped)
Peano axioms (Wikipedia) (standard reference, not scraped)
B. Kaya, MATH 320 Set Theory (METU lecture notes) (standard reference, not scraped)
W. Aitken, MATH 378 Ch. 1: The Peano Axioms (CSU San Marcos) (standard reference, not scraped)