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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's 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.

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