Theorem

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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. This meaning of the word "theorem" is distinct from the meanings of the word "theory". In the context of mathematics, a theory is often taken to be a collection of (proven) theorems, rather than a plausible conjecture.

Logically speaking, many theorems are of the form: if A, then B. Such a theorem does not state that B is always true, only that B must be true if A is true. In this case A is called the hypothesis of the theorem (note that "hypothesis" here is something very different from a conjecture) and B the conclusion. The theorem "If n is an even natural number then n/2 is a natural number" is a typical example in which the hypothesis is that n is an even natural number and the conclusion is that n/2 is also a natural number.

It is common in mathematics to choose a number of hypotheses that are assumed to be true within a given theory, and then declare that the theory consists of all theorems provable using those hypotheses as assumptions. In this case the hypotheses that form the foundational basis are called the axioms (or postulates) of the theory. The field of mathematics known as proof theory studies formal axiom systems and the proofs that can be performed within them.

A key property of theorems is that they possess proofs, not merely that they are true. In order to produce a theorem it is necessary to demonstrate the existence of a proof of the theorem's statement from the axioms. Although the proof is necessary to produce a theorem, it is not usually considered part of the theorem. Although more than one proof may be known for a single theorem, only one proof is required to establish a theorem.

In order to be proven, a theorem must be expressible as a precise, formal statement. Theorems are often stated informally, however, 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. Thus most theorems and proofs in mathematics are expressed in the natural language of the mathematician rather than in a completely symbolic form.

[edit] Terminology

Theorems are often further specified by use of one of several terms. The actual label "theorem" is often reserved for the most important results, while other results are called by different terminology.

• A Proposition is a statement not associated with any particular theorem. This term sometimes connotes a statement with a simple proof.
• A Lemma is a "pre-theorem", 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 other theorem or definition. That is, proposition B is a corollary of a proposition A if B can be deduced quickly and easily from A.
• 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.

There are other, less commonly used terms attached to proven statements. These terms are usually used by convention, so that certain results are referred to by historic or customary names. Such names include:

• Rule, used for certain theorems such as Bayes' rule and Cramer's rule that establish useful formulas.
• Law. Examples include the law of large numbers, the law of cosines, and Kolmogorov's zero-one law. The word law can also refer to an axiom, a rule of inference, or, in probability theory, a probability distribution.
• Principle. Examples include Harnack's principle, the least upper bound principle, and the pigeonhole principle.

A few well-known theorems have even more idiosyncratic names. The name Division algorithm is used for a theorem expressing the outcome of division in the natural numbers and more general rings. The name Banach–Tarski paradox is used for a theorem in measure theory that is paradoxical in the sense that it contradicts common intuitions about volume in three-dimensional space.

A statement which is believed to be true but has not been proven is known as a Conjecture (sometimes a conjecture is also called a Hypothesis, but, of course, with a different meaning from the one discussed above). 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, as with Goldbach's conjecture. Other famous conjectures include the Collatz conjecture and the Riemann hypothesis.

[edit] See also

• Metatheorem
• List of theorems
• Mathematics for a list of famous theorems and conjectures.
• Gödel's incompleteness theorem, that establishes very general conditions under which a formal system will contain a true statement for which there exists no proof within the system.
• Inference