The terms you need to know
Axiom (postulate) - an underlying assumption, whose truth does not
Rule of inference - a mean used to draw conclusions from other
Proof of a statement A - a sequence of statements, where each
statement is either an axiom or follows by one of the rules of inference
from other (earlier) statements in this sequence, and where the last
Theorem - a statement with a proof.
Lemma - a theorem used in the proof of other theorems.
Corollary - a theorem that (almost) immediately follows from
Conjecture - a likely-to-be-true statement yet with no proof found.
Goldbach’s (18th century) Conjecture:
“Every even integer greater than 4 is the sum of two primes”.