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 theorem has two parts, stated in a formal language – a set of assumptions, and a conclusion that can be derived from the given assumptions according to the inference rules of the formal system comprising the formal language. The proof, though necessary to the statement's classification as a theorem, is not considered part of the theorem.
In general, a statement must not have a trivially simple derivation to be called a theorem. Less important statements are called:
- lemma: 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.
- corollary: a proposition that follows with little or no proof from one already proven. A proposition B is a corollary of a proposition or theorem A if B can be deduced quickly and easily from A.
- proposition: a result not associated with any particular theorem.
- claim: a very easily proven, but necessary or interesting result which may be part of the proof of another statement. Despite the name, claims are proven.
- remark: similar to claim. Usually presented without proof, which is assumed to be obvious.
A mathematical statement which is believed to be true but has not been proven is known as a conjecture. Gödel's incompleteness theorem establishes very general conditions under which a formal system will contain a true statement for which there exists no derivation 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), as well as a process of inference, which 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; 'theorem' used in this sense is a technical term indicating that a derivation exists and has none of the subjective connotations of importance as when the term is used in general mathematics.
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