CounterexampleConstruction: AI-adaptedVerification: 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.

Injectivity of the successor is independent of the other Peano axioms

Statement refuted

Refuted claim: (P2) injectivity of $σ$ follows from (P1) $0$ is not a successor together with (P3) induction. The witness is the three-element system $N = {0, 1, 2}$ with $σ (0) = 1$ , $σ (1) = 2$ , $σ (2) = 2$ .

Facts & Assumptions

Given: $N = {0, 1, 2}$ , base point $0$ , $σ (0) = 1$ , $σ (1) = 2$ , $σ (2) = 2$ .

[L1]

The Peano axioms (Peano system).

Counterexample

technique · direct

1.1

P1 holds: the values of $σ$ are $σ (0) = 1$ , $σ (1) = 2$ , $σ (2) = 2$ , none of which is $0$ , so $σ (x) \neq 0$ for all $x$ .

givenL1

1.2

P3 holds: if $S \subseteq N$ with $0 \in S$ and $S$ closed under $σ$ , then $0 \in S$ forces $1 = σ (0) \in S$ , then $2 = σ (1) \in S$ , so $S = {0, 1, 2} = N$ ; thus induction holds.

givenL1

1.3

P2 fails: $σ (1) = 2 = σ (2)$ but $1 \neq 2$ , so $σ$ is not injective.

givenL1

2.1

So $(N, 0, σ)$ satisfies P1 and P3 but not P2, refuting the claim; note the failure permits the pathology $σ (2) = 2$ (a number that is its own successor) and a finite model in which distinct numerals collapse.

step 1.1step 1.2step 1.3∎

Depends on

Peano system

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

Peano axioms (Wikipedia) (standard reference, not scraped)
Mathematical induction (Wikipedia) (standard reference, not scraped)
Peano axioms (Encyclopedia of Mathematics) (standard reference, not scraped)