为何闭区间上连续函数最大值的定义属于非直谓定义？

Hey there! Let's unpack this clearly—no jargon overload, promise.

First, let's get straight what an impredicative definition actually means:

• It’s a definition where you describe an object by referencing a collection (set, class, etc.) that the object itself is part of.
• Put simpler: you’re defining something by pointing to a group that includes the very thing you’re trying to define.

Now let’s apply this to the maximum value of a continuous function on [a,b]. The standard definition goes like this:

The maximum value M of f on [a,b] is the largest number such that:

1. There exists some c ∈ [a,b] where f(c) = M;
2. For every x ∈ [a,b], f(x) ≤ M.

Here’s why this is impredicative, broken down step by step:

1. To define M, you first have to consider the entire set of values f takes on [a,b]—let’s call this set S = { f(x) | x ∈ [a,b] }.
2. M is an element of S (since M = f(c) for some c in [a,b], so it’s part of the function’s output set).
3. You’re defining M as the largest element of S—so you’re using the full collection S (which includes M itself) to pick out and define M.

Think of a more everyday analogy: if I say "the tallest person in this room," that’s an impredicative definition too. I’m defining that person by referencing the group of people in the room, which includes the tallest person I’m describing.

Contrast this with a predicative definition, like "the sum of 2 and 3." The number 5 isn’t part of the collection {2,3} I used to define it—so no circularity or self-reference there.

For continuous functions on closed intervals, we know by the Extreme Value Theorem that this maximum M exists, but the way we define M still relies on referencing the set that contains it. That’s the key reason it counts as an impredicative definition.

内容的提问来源于stack exchange，提问作者iMath