Theorem
Calculus & Advanced MathA 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.
Example
The Pythagorean Theorem says $a^2 + b^2 = c^2$ for right triangles. It is not just "usually true," it is always true and has been proven hundreds of different ways.
Key Insight
Theorems are the permanent buildings of mathematics. Once proven, they never need to be re-checked for exceptions.
Definition
A theorem is a statement derived by logical deduction from axioms and previously proven theorems (lemmas). A proof is the argument demonstrating this derivation. Minor theorems are called lemmas; direct consequences are corollaries.
Example
Theorem: there are infinitely many prime numbers. Proof (Euclid): assume finitely many primes $p_1, \ldots, p_n$. Consider $N = p_1 \times \ldots \times p_n + 1$. $N$ is not divisible by any $p_i$, so it has a prime factor not on the list. Contradiction.
Key Insight
A theorem is only as strong as its proof. The history of mathematics includes statements believed true for centuries that turned out to be false (and vice versa).
Definition
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. By Godel's Incompleteness Theorem, sufficiently powerful systems contain true statements that are not theorems.
Example
Godel constructed a sentence $G$ in arithmetic that asserts 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
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.