Editing Theorem (section)

==Terminology==
A number of different terms for mathematical statements exist; these terms indicate the role statements play in a particular subject. The distinction between different terms is sometimes rather arbitrary, and the usage of some terms has evolved over time.
* An ''[[axiom]]'' or ''postulate'' is a fundamental assumption regarding the object of study, that is accepted without proof. A related concept is that of a ''[[definition]]'', which gives the meaning of a word or a phrase in terms of known concepts. Classical geometry discerns between axioms, which are general statements; and postulates, which are statements about geometrical objects.{{sfn|Wentworth|Smith|1913|loc=[https://archive.org/details/planegeometry00gwen/page/n25/ Articles 46-7]}} Historically, axioms were regarded as "[[Self-evidence|self-evident]]"; today they are merely ''assumed'' to be true.
* A ''[[conjecture]]'' is an unproved statement that is believed to be true. Conjectures are usually made in public, and named after their maker (for example, [[Goldbach's conjecture]] and [[Collatz conjecture]]). The term ''hypothesis'' is also used in this sense (for example,  [[Riemann hypothesis]]), which should not be confused with "hypothesis" as the premise of a proof. Other terms are also used on occasion, for example ''problem'' when people are not sure whether the statement should be believed to be true. [[Fermat's Last Theorem]] was historically called a theorem, although, for centuries, it was only a conjecture.
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* A ''theorem'' is a statement that has been proven to be true based on axioms and other theorems.
* A ''[[proposition]]'' is a theorem of lesser importance, or one that is considered so elementary or immediately obvious, that it may be stated without proof. This should not be confused with "proposition" as used in [[propositional logic]]. In classical geometry the term "proposition" was used differently: in [[Euclid]]'s [[Euclid's Elements|''Elements'']] ({{circa|300&nbsp;BCE}}), all theorems and geometric constructions were called "propositions" regardless of their importance.
* A ''[[lemma (mathematics)|lemma]]'' is an "accessory proposition" - a proposition with little applicability outside its use in a particular proof. Over time a lemma may gain in importance and be considered a ''theorem'', though the term "lemma" is usually kept as part of its name (e.g. [[Gauss's lemma (polynomial)|Gauss's lemma]], [[Zorn's lemma]], and [[Fundamental lemma (Langlands program)|the fundamental lemma]]).
* A ''[[corollary]]'' is a proposition that follows immediately from another theorem or axiom, with little or no required proof.{{sfn|Wentworth|Smith|1913|loc=[https://archive.org/details/planegeometry00gwen/page/n25 Article 51]}} A corollary may also be a restatement of a theorem in a simpler form, or for a [[special case]]: for example, the theorem "all internal angles in a [[rectangle]] are [[right angle]]s" has a corollary that "all internal angles in a ''[[square]]'' are [[right angle]]s" - a square being a special case of a rectangle.
* A ''[[generalization]]'' of a theorem is a theorem with a similar statement but a broader scope, from which the original theorem can be deduced as a [[special case]] (a ''corollary'').{{efn|Often, when the less general or "corollary"-like theorem is proven first, it is because the proof of the more general form requires the simpler, corollary-like form, for use as a what is functionally a lemma, or "helper" theorem.}}

Other terms may also be used for historical or customary reasons, for example:

* An ''[[Identity (mathematics)|identity]]'' is a theorem stating an equality between two expressions, that holds for any value within its [[Domain of a function|domain]] (e.g. [[Bézout's identity]] and [[Vandermonde's identity]]).
* A ''rule'' is a theorem that establishes a useful formula (e.g. [[Bayes' rule]] and [[Cramer's rule]]).
* A ''[[Law (mathematics)|law]]'' or ''[[principle]]'' is a theorem with wide applicability (e.g. the [[law of large numbers]], [[law of cosines]], [[Kolmogorov's zero–one law]], [[Harnack's principle]], the [[least-upper-bound principle]], and the [[pigeonhole principle]]).{{efn|The word ''law'' can also refer to an axiom, a [[rule of inference]], or, in [[probability theory]], a [[probability distribution]].}}

A few well-known theorems have even more idiosyncratic names, for example, the [[Euclidean division|division algorithm]], [[Euler's formula]], and the [[Banach–Tarski paradox]].