Alphabeta Math
LemmaStatement: Literature-sourcedProof: AI-adaptedprecheck passverified 2026-08-02 (claude-opus-5)
How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

No natural number equals its own successor

Statement

For every n∈N, n≠σ(n).

Facts & Assumptions

Given: N with distinguished element 0 and successor σ.

[L1]

σ(n)≠0 for all n (P1) and σ is injective (P2) (The von Neumann naturals form a Peano system).

[L2]

The induction principle (The principle of mathematical induction).

Proof

technique · induction
1.1

Base case n=0: σ(0)≠0 by P1, so 0≠σ(0).

baseL1
1.2

Inductive hypothesis: n≠σ(n).

ih
2.1

If σ(n)=σ(σ(n)) then injectivity of σ gives n=σ(n), contradicting the hypothesis; hence σ(n)≠σ(σ(n)).

step 1.2L1
3.1

By induction [L2], n≠σ(n) for all n∈N.

step 1.1step 2.1discharge-induction∎

Depends on

Used by

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources