定义，公理，定理，推论，命题和引理的区别

一、概念

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.

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).

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").

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).

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.

Identity ((恒)等式) - a mathematical expression giving the equality of two (often variable) quantities (trigonometric identities, Euler's identity)

定义 - 对数学术语的精准明确的描述。它通过描述全部属性，且只有绝对成立的属性,来阐释词语。

公理/假定 - 是被假定不证自明的陈述。这些是证明所有定理的基础(欧几里得的五个公设，泽梅洛-弗伦克尔公理，皮亚诺公理)

定理 - 用严格的数学推理证明的数学命题。在数学论文中，定理一词常被用来表示最重要的结果。

引理 - 是辅助证明定理的次级结论。它是证明定理的一块垫脚石。引理偶尔可以独自发挥作用。(佐恩引理、Urysohn引理、Burnside引理、Sperner引理)

推论 - 是依赖给定的定理引出的(通常简短的)结论(我们经常说“这是定理a的一个推论”)。

悖论 - 在一组定义与公理下同时可证实也可证伪的陈述。悖论经常被用来显示一个有缺陷的理论的不一致性(罗素悖论)。这个术语也用作描述给定规则集下导出的反直觉结果(巴纳赫-塔尔斯基悖论，阿拉巴马悖论，加布里埃尔的号角)。

命题 - 经过证明的通常较为有趣的结论，但一般没有定理重要。

猜想 - 一个未经证实但被认为是正确的命题(Collatz猜想、哥德巴赫猜想、孪生素数猜想)。

断言 - 经过证明的断言。它经常被用作非正式引理。

恒等式 - 两个(通常是可变的)量相等的数学表达式(三角恒等式，欧拉恒等式)

二、补充

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。

三、属性

• 定义、公理不可证明，定理必定经过证明

• 推论是根据公理或定理而推导出来的真命题.

参考资料：

定义、公理、定理、推论、命题和引理的区别 - 百度文库 (baidu.com)https://wenku.baidu.com/view/4cd882af770bf78a64295454.html[离散数学]定理,引理,推论与诸如此类概念的异同 - AstatineAi - 博客园 (cnblogs.com)https://www.cnblogs.com/AstatineAi/p/difference-between-theorem-lemma-and-corollar.html