A theorem is a statement which can be proven true within some logical framework. Proving theorems is a central activity of mathematics. Note that 'theorem' is distinct from 'theory'.
A theorem generally has a set-up - a number of conditions, which may be listed in the theorem or described beforehand. Then it has a conclusion - a mathematical statement which is true under the given set up. The proof, though necessary to the statement's classification as a theorem is not considered part of the theorem.
In general mathematics a statement must be interesting or important in some way to be called a theorem. Less important statements are called:
- lemma: a statement that forms part of the proof of a larger theorem. Of course, 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 for some authors to stop at the nominal lemma without going on to use that result in any "major" theorem.
- corollary: a statement which follows immediately or very simply from a theorem. A proposition A is a corollary of a proposition or theorem B if A can be deduced quickly and easily from B.
- proposition: a result not associated with any particular theorem.
- claim: a very minor, 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. Probably 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
As noted above, a theorem requires some sort of logical framework, this will consist of a basic set of axioms (see axiomatic system), as well as a process of inference, which allows to derive new theorems from axioms and other theorems that have been derived earlier. In propositional logic, any proven statement is called a theorem.
- mathematics for a list of famous theorems and conjectures.
- Gödel's incompleteness theorem