1. Definition 定义
对某个数学概念\术语（符号、表达）的解释。
An explanation of the mathematical meaning of a word.
2. Lemma 引理
用于证明其他命题的正确陈述。（辅助证明theorem的某个重要的中间结论）
A true statement used in proving other true statements (that is, a less important theorem that is helpful in the proof of other results).
3. Theorem 定理
被证明是正确的陈述。（一般用于为文章中的重要结论做准备）
A statement that has been proven to be true.
4. Proposition 命题
一个对后文的证明不是特别重要却仍然有参考和辅助作用的正确陈述。
A less important but nonetheless interesting true statement.
5. Corollary 推论
从定理或命题中简单推导出来的正确陈述。
A true statment that is a simple deduction from a theorem or proposition.
6. Proof 证明
对命题的解释。
The explanation of why a statement is true.
7. Conjecture 猜想
被认为是正确的陈述。
A statement believed to be true, but for which we have no proof. (a statement that is being proposed to be a true statement).
8. Axiom 公理
某数学研究情境下最为基本的假设（一定是正确的，和 Postulate 相当）。
A basic assumption about a mathematical situation. (a statement we assume to be true).
References:
[1] https://www.google.com.hk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=&cad=rja&uact=8&ved=2ahUKEwjI0JWY3sftAhWuGaYKHVRXAwgQFjABegQICRAC&url=https%3A%2F%2Fusers.math.msu.edu%2Fusers%2Fduncan42%2FAxiomNotes.pdf&usg=AOvVaw2EQncZYPgm4iyNnS-Z-lrJ
[2] https://divisbyzero.com/2008/09/22/what-is-the-difference-between-a-theorem-a-lemma-and-a-corollary/