实数完备性的基本定理为什么是 6 个？ 第1页

灵剑 在评论里已经说了实数的8个关于完备性的命题。这里补充第9个，你可能想不到，那就是

介值定理（Intermediate Value Theorem, IVT）

关于它在实数完备性中（completeness of the real number）的逻辑地位，wiki是这样说的：

The intermediate value theorem

The intermediate value theorem states that every continuous function that attains both negative and positive values has a root. This is a consequence of the least upper bound property, but it can also be used to prove the least upper bound property if treated as an axiom. (The definition of continuity does not depend on any form of completeness, so this is not circular.)

简单概括一下，介值定理与那一大堆定理都是等价的。

（细心的读者会发现这段描述中IVT其实是被当做零点存在定理了，但事实上零点存在定理和介值定理是等价的。）