Differences between Theorem, Lemma and Corollary, etc.

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.

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.

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).

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.

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-Frankel axioms, Peano axioms).

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).

定义（Definition）、定理（Theorem）、命题（Proposition）和引理（Lemma）相互关系与区别

定义和公理是任何理论的基础，定义解决了概念的范畴，公理使得理论能够被人的理性所接受。
        定理和命题就是在定义和公理的基础上通过理性的加工使得理论的再延伸，我认为它们的区别主要在于，定理的理论高度比命题高些，定理主要是描述各定义（范畴）间的逻辑关系，命题一般描述的是某种对应关系（非范畴性的）。而推论就是某一定理的附属品，是该定理的简单应用。
       引理就是在证明某一定理（或命题）时所用到（或计算得到）的其它定理（或结果）。
        在实际论文过程中应用较多的是定义、引理和命题。一般的情况是先（计算）推导出一些重要的引理，然后依据一些引理及其他得到一些命题。引理更多是确定的性解或重要结果，命题更多的是一些描述性的结论。

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Differences between Theorem, Lemma and Corollary, etc.

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