Theorem
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 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. 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 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. 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.
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 one to derive new theorems from axioms and other theorems that have been derived earlier. In propositional logic, any proven statement is called a theorem. Informally speaking, most such theorems are not of any particular interest; what makes a mathematical result worth the title 'theorem' is not really an easy matter.
See also
- mathematics for a list of famous theorems and conjectures.
- list of theorems
- Gödel's incompleteness theorem