Two axioms, one idea
Induction and well-ordering look unrelated — one is about climbing up, the other about a floor at the bottom. But each can be proved from the other: if you couldn't induct, there'd be a smallest failure; if there were a floorless descending set, induction would break. Mathematics is full of these equivalences, and recording them as explicit graph edges is exactly what Ouroboros is for.
Two doors into the same room. Whichever you walk through, you end up with the same power over the natural numbers.
From well-ordering, prove induction: suppose P(0) holds and P(n) implies P(n+1), yet P fails somewhere. The set of failures has a least element m. m isn't 0, so P(m-1) holds... but then P(m) holds. Contradiction.
Knowing two principles are equivalent means you can use whichever is more convenient — and trust nothing is lost. Equivalence edges like this are how the knowledge graph exposes hidden structure.
Level 1 The precise statement
Over the natural numbers the Axiom of Induction and the Well-Ordering Principle are logically equivalent: each can be derived from the other.
Level 2 The proof
Level 3 What it stands on (2 direct)
- Axiom of Induction (axiom)
- Well-Ordering Principle (axiom)
- Induction and well-ordering are inter-derivable (proposition)
Level 4 The verified record
This page is generated from a machine-checked node. The kernel confirms its dependencies resolve, nothing is circular, and it grounds in axioms (foundation: peano). The content hash below makes tampering evident.
Nothing yet — this is a frontier result.
Nothing depends on it yet, so its failure would be contained.