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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 mathematicians. Note that "theorem" is distinct from "theory".
When stated formally, a theorem has two parts:
- A description of a formal language and a list of assumptions (axioms) in that language.
- A statement in the formal language which is to be proved.
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.
In general, a statement with a trivially simple derivation is not called a theorem. Other statements may be called by the following terms.
- A Lemma is 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 already proven. A proposition B is a corollary of a proposition or theorem A if B can be deduced quickly and easily from A.
- A Proposition is a statement not associated with any particular theorem. This term sometimes connotes a statement with a simple proof.
- 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.
The following types of statements are not theorems and are typically offered without proof.
- An Axiom is a statement that is considered "self-proved", that is, a statement that is not dependent on the conditions of any other statement.
- A Postulate is a statement that is accepted without proof in certain context. Every axiom is a postulate, but theorems can be used as postulates in certain situations.
These statements form the foundation from which theorems are proved. and every theorem must be proved on the basis of axioms, postulates, and other theorems.
A statement which is believed to be true but has not been proven is sometimes known as a Conjecture or Hypothesis. 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.
A key property of theorems is that they possess proofs, not that they are “true.” A statement which is considered obvious and is presented without proof is called an axiom instead. Gödel's incompleteness theorem establishes very general conditions under which a formal system will contain a true statement for which there exists no proof 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) and a process of inference that 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. Proof theory is a field of mathematics which studies formal axiom systems and the proofs that can be performed within them.
See also
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