theorem

A proposition proved by logical deduction from one or more initial premises. Although geometrical theorems are the most widely known, theorems exist in all branches of mathematics. A simple theorem which is proved and then used towards the proof of another theorem is known as a lemma. If the theorem is ‘statement p implies statement q’, the converse is ‘statement q implies statement p’. The converse of a theorem is not always true. For example, if two triangles are congruent, they are equal in area, but two triangles that are equal in area are not necessarily congruent.

Proving theorems is a central activity of mathematicians. Note that "theorem" is distinct from "theory".

When stated formally, a theorem has two parts:

A description of a formal language and a list of assumptions (axioms) in that language.

In order to produce a theorem it is necessary to demonstrate the existence of a proof of the statement from the axioms. The proof is necessary to produce a theorem but is not considered part of the theorem. Thus a single theorem may have more than one proof, although only one is required to establish a theorem. A theorem is often stated informally when the intended audience is believed to be able to produce the formal version from the informal one. It is common for an informal but rigorous argument to be given showing that a formal proof of the statement from the axioms could be constructed, without an actual formal proof being given.

In general, a statement with a trivially simple derivation is not called a theorem.

A Lemma is a statement that forms part of the proof of a larger theorem. The distinction between theorems and lemmas is rather arbitrary, since one mathematician's major result is another's minor claim. Gauss' lemma and Zorn's lemma, for example, are interesting enough per se that some authors present the nominal lemma without going on to use it in the proof of any theorem. A proposition B is a corollary of a proposition or theorem A if B can be deduced quickly and easily from A. A Proposition is a statement not associated with any particular theorem.

The following types of statements are not theorems and are typically offered without proof.

An Axiom is a statement that is considered "self-proved", that is, a statement that is not dependent on the conditions of any other statement. Every axiom is a postulate, but theorems can be used as postulates in certain situations.

These statements form the foundation from which theorems are proved. Every theorem must be proved on the basis of axioms, postulates, and other theorems (or lemmata, corollaries, or propositions).

A statement which is believed to be true but has not been proven is sometimes known as a Conjecture or Hypothesis.

A key property of theorems is that they possess proofs, not that they are “true.” A statement which is considered obvious and is presented without proof is called an axiom instead. Gödel's incompleteness theorem establishes very general conditions under which a formal system will contain a true statement for which there exists no proof within the system.

As noted above, a theorem must exist in the context of some formal system. This will consist of a basic set of axioms (see axiomatic system) and a process of inference that allows one to derive new theorems from axioms and other theorems that have been derived earlier. In mathematical logic, any provable statement is called a theorem. Informally speaking, most such theorems are not of any particular interest; Proof theory is a field of mathematics which studies formal axiom systems and the proofs that can be performed within them.