THEOREM, LEMMA

WHAT IS THE DIFFERENCE BETWEEN A THEOREM, A LEMMA, AND A COROLLARY?

PROF. DAVE RICHESON

Definition — a precise and unambiguous description of the meaning of a mathematical term. It characterizes the meaning of a word by giving all the properties and only those properties that must be true. This is an assignment of language and syntax to some property of a set, function, or other object. A definition is not something you prove, it is something someone assigns. Often you will want to prove that something satisfies a definition. Example: We call a mapping f:X→Yf:X→Yinjective if whenever f(x)=f(y)f(x)=f(y) then x=yx=y.

Theorem — a mathematical statement that is proved using rigorous mathematical reasoning. In a mathematical paper, the term theorem is often reserved for the most important results. This is a property of major importance that one can derive which usually has far-sweeping consequences for the area of math one is studying. Theorems don't necessarily need the support of propositions or lemmas, but they often do require other smaller results to support their evidence. Example: Every manifold has a simply connected covering space.

Lemma — a minor result whose sole purpose is to help in proving a theorem. It is a stepping stone on the path to proving a theorem. Very occasionally lemmas can take on a life of their own (Zorn’s lemma, Urysohn’s lemma, Burnside’s lemma,Sperner’s lemma). This is a property that one can derive or prove which is usually technical in nature and is not of primary importance to the overall body of knowledge one is trying to develop. Usually lemmas are there as precursors to larger results that one wants to obtain, or introduce a new technique or tool that one can use over and over again. Example: In a Hausdorff space, compact subsets can be separated by disjoint open subsets.

Corollary — a result in which the (usually short) proof relies heavily on a given theorem (we often say that “this is a corollary of Theorem A”).

Proposition — a proved and often interesting result, but generally less important than a theorem. This is a property that one can derive easily or directly from a given definition of an object. Example: the identity element in a group is unique.

Conjecture — a statement that is unproved, but is believed to be true (Collatz conjecture, Goldbach conjecture, twin prime conjecture).

Claim — an assertion that is then proved. It is often used like an informal lemma.

Axiom/Postulate — a statement that is assumed to be true without proof. These are the basic building blocks from which all theorems are proved (Euclid’s five postulates,Zermelo-Fraenkel axioms, Peano axioms).  I would appreciate community input on this, but I haven't seen this word used in any of the texts/papers I read. I would assume that this is synonymous with proposition.

Identity — a mathematical expression giving the equality of two (often variable) quantities (trigonometric identities, Euler’s identity).

Paradox — a statement that can be shown, using a given set of axioms and definitions, to be both true and false. Paradoxes are often used to show the inconsistencies in a flawed theory (Russell’s paradox). The term paradox is often used informally to describe a surprising or counterintuitive result that follows from a given set of rules (Banach-Tarski paradox, Alabama paradox, Gabriel’s horn).

Hypothesis - An educated prediction that one makes based on their experience. I have rarely actually seen this word in use in mathematical texts or literature, although it is quite common in the physical sciences to make hypotheses. I could be wrong on this one. What makes something a hypothesis is that it is not proven, but it is conjectured. Once it is proven or disproven, it ceases to be a hypothesis and either becomes a fact (backed by a theorem usually) or there is some interesting counterexample to demonstrate how it is wrong.