Theorem
A theorem is a mathematical statement that has been rigorously proved to be true from axioms and previously established results.
Definition
A theorem is a mathematical fact that has been proven to be definitely, always true; it is not a guess, it has been carefully shown to follow from rules that are already accepted. More precisely, a theorem is a statement derived by logical deduction from axioms and previously proven theorems (lemmas), with a proof as the argument demonstrating this derivation; minor theorems are called lemmas, and direct consequences are corollaries. In formal logic, a theorem is a formula derivable from the axioms of a formal system using the rules of inference; by Godel's Completeness Theorem, a statement is a theorem of first-order logic iff it is true in all models, while Godel's Incompleteness Theorem shows that sufficiently powerful systems contain true statements that are not theorems.
Example
The Pythagorean Theorem, $a^2 + b^2 = c^2$ for right triangles, is not just "usually true," it is always true and has been proven hundreds of different ways. Euclid proved there are infinitely many primes by assuming finitely many, $p_1, \ldots, p_n$, and considering $N = p_1 \times \ldots \times p_n + 1$, which has a prime factor not on the list, a contradiction. Godel constructed a sentence $G$ in arithmetic asserting its own unprovability: $G$ is true in the standard model but not provable from the Peano axioms, a theorem that cannot be a theorem.
Key Insight
Theorems are the permanent buildings of mathematics; once proven, they never need to be re-checked for exceptions, but a theorem is only as strong as its proof, and history includes statements believed true for centuries that turned out to be false. Incompleteness shows the limits of the axiomatic method: no consistent formal system can prove all mathematical truths, making the gap between "true" and "provable" a permanent feature of mathematics.