A theorem is a proposition that has been or is to be proved on the basis of explicit assumptions. Proving theorems is a central activity of mathematicians. Note that "theorem" is distinct from "theory".

A key property of theorems is that they possess proofs, not merely that they are true. Logically speaking, everything that is proved is something in the form: if A, so B. In other words, only implications are proved, its impossible to prove that B is always true, but what may be possible to prove is that B is true if A is true. In this case A is called the Hypothesis of the theorem (note that "hypothesis" here is something very different from Conjecture).

It is usual to choose a number of hypotheses which are assumed to be true within a given theory, and require everything else that is called true in the theory to be an implication of the stated assumptions. In this case the group of hypotheses that form a foundational basis are called axioms (or postulates) of the theory. Such a theory is called an Axiomatic System, and Formalism is the word given to the ideology of using only of axiomatic models. Proof theory is a field of mathematics which studies formal axiom systems and the proofs that can be performed within them.

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.

Terminology

Theorems are often further specified by use of one of several terms, saving the use of the term "theorem" only for important theorems.

• A Proposition is a statement not associated with any particular theorem. This term sometimes connotes a statement with a simple proof.
• A Lemma is a "pre-theorem", 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 Corollary is a proposition that follows with little or no proof from one other theorem or definition. A proposition B is a corollary of a proposition or theorem A if B can be deduced quickly and easily from A.
• A Claim is a necessary or independently interesting result which may be part of the proof of another statement. Despite the name, claims must be proved.

Less commonly used terms for proved statements include:

• Rule, for example Bayes' rule, or Cramer's rule. The terms lemma or theorem are preferred.
• Law, for example the law of large numbers, or Kolmogorov's zero-one law. The terms lemma or theorem are preferred, especially since law can also refer to an axiom, a rule of inference, or, in probability theory, a probability distribution.
• Principle, for example Harnack's principle, or the pigeonhole principle.
• Algorithm as in division algorithm. Note that this usage is highly unusual and distinct from the term algorithm as used in computer science.
• Paradox is a statement which is provably true but which appears to defy simple intuition. For example, the Banach–Tarski paradox is a true statement and the valid consequence of, among other things, the axioms of set theory. The name paradox draws attention to its seemingly unintuitive conclusion. The term paradox, however, can also refer to fallacies and other false (but seemingly true) statements.

A statement which is believed to be true but has not been proven is known as a Conjecture (sometimes conjectures are also called Hypothesis, but, of course, with a different meaning from the one already defined here). To be considered a conjecture, a statement must usually be proposed publicly, at which point the name of the proponent may be attached to the conjecture. Other times, a name is attached even though the person named did not make the conjecture. Famous conjectures include the Collatz conjecture and the Riemann hypothesis.

See also

• Metatheorem
• List of theorems
• Mathematics for a list of famous theorems and conjectures.
• Gödel's incompleteness theorem, that establishes very general conditions under which a formal system will contain a true statement for which there exists no proof within the system.
• Inference