theorem

A theorem is a statement that has been proven to be true based on axioms and previously established theorems. In mathematics and formal logic, a theorem is not accepted on intuition or authority but is derived through a proof, a logical argument that starts from accepted assumptions and proceeds step by step to the conclusion. The status of a theorem relies on the consistency and sufficiency of the underlying axioms and definitions.

The term Theorem comes from the Greek theorein, meaning to look at or examine. The concept has its roots in ancient Greek mathematics, most notably in Euclid’s Elements, where propositions were presented with proofs. Over time, the use of the term has broadened to many areas of mathematics and, informally, to statements with strong explanatory power in other disciplines.

A proof is a sequence of logical deductions that verifies the truth of a theorem. Theorem proofs can be direct, indirect, or by methods such as contrapositive, contradiction, or mathematical induction. Theorems are often accompanied by corollaries, lemmas, and conjectures, which help structure mathematical arguments.

Examples include the Pythagorean theorem, which relates the sides of a right triangle; the Fundamental Theorem of Calculus, which links differentiation and integration; and Fermat’s Last Theorem, stating that a certain equation has no nontrivial whole-number solutions. Other well-known theorems include the infinitude of primes and the Fundamental Theorem of Arithmetic.

In formal systems, a theorem must be derivable from axioms. Gödel’s incompleteness theorems show that, in sufficiently powerful systems, there are true statements that cannot be proven within the system. Theorems thus reflect both the power and the limits of formal reasoning, and they are central to the organization and communication of mathematical knowledge.