Axioms, Theorems, and Proofs
What counts as a fact, and how do we certify it?
Definitions in ProofTalk:
- Axiom (Def 2.1) — a statement we assume to be true without proof.
Usually very basic and self-evident.
- Theorem (Def 2.2) — a statement that has been proven to be true.
- Proof (Def 2.3) — a sequence of inferences from axioms and previously
proven theorems that establishes a new theorem.
- Lemma — a small theorem, usually proven to support a bigger theorem.
- Corollary — a theorem that follows immediately from another theorem.
A proof game has two players:
- The proponent (P) — claims the statement and tries to prove it.
- The opponent / skeptic (O) — challenges the claim, tries to find holes.
In a valid proof, the proponent always has a move; the opponent cannot refute
it. If the opponent successfully attacks, the claim is not proven.
- 0.0sAxioms, Lemmas, Theorems
- 1.2sThree cyan axiom nodes appear at the bottom: P1, P2, P3
- 2.6sTwo amber lemmas sit above: L1 and L2
- 4.8sA green theorem box crowns the pyramid
- 6.5sArrows connect axioms to lemmas to theorem
- 8.4sAxiom-to-lemma arrows glow cyan
- 9.7sLemma-to-theorem arrows glow amber
- 11.0sAxioms are assumed; lemmas + theorems are proved