定义、公理、定理、推论、命题和引理的区别

I prepared the following handout for my Discrete Mathematics class (here’s a pdf version). Definition（定义）— 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（定理）— 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（引理）— 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（推论）— 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（命题）— a proved and often interesting result, but generally less important than a theorem.
Conjecture（猜想）— a statement that is unproved, but is believed to be true (Collatz conjecture, Goldbach conjecture, twin prime conjecture).
Claim（断言）— an assertion that is then proved. It is often used like an informal lemma. Axiom/Postulate（公理/假定）— 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).
Identity（恒等式）— a mathematical expression giving the equality of two (often variable) quantities (trigonometric identities, Euler’s identity).
Paradox（悖论）— 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).
首先、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。

其次、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。

最后、引理就是在证明某一定理时所必须用到的其它定理。

而在一般情况下，就像前面所提到的定理的证明是依赖于定义和公理的。

定义就是规定意义，相当于取名字，定理就是根据定义和公理推导演绎出来的命题。

公理就是人们通过实际生活观察到的一些人们共同赞同的但又无法证明的；
根本差别在于:定义不可证明,而定理一定是经过了证明的!
数学就是在定义和公理(经验的总结,不需证明,如过两点可画一条直线)基础上,演绎出的一整套定理组成的逻辑体系.(演绎的过程就是证明定理)
定义:对概念的内涵或语词的意义所做的简要而准确的描述
定理:通过理论证明能用来作为原则或规律的命题或公式。