— notes — 2 min read
An axiom or postulate is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments.
A proof is a sequence of logical deductions from axioms and previously proved statements that concludes with the proposition in question.
There are several common terms for a proposition that has been proved. The different terms hint at the role of the proposition within a larger body of work:
i. Important true propositions are called theorems. ii. A lemma is a preliminary proposition useful for proving later propositions. iii. A corollary is a proposition that follows in just a few logical steps from a theorem.
Euclid’s axiom-and-proof approach, now called the axiomatic method.