The Well-Ordering Principle and the Pop Quiz Paradox

Last night in class, we went over a proof of the Well-Ordering Principle, and tonight, it occurs to me how much that proof reminds me of what was possibly the first inductive "proof" I ever encountered: the pop-quiz paradox.

First, the well-ordering principle. This just states that, if you have a non-empty set of natural numbers, it has a least element. The natural numbers, N, are these ones:

{1, 2, 3, ...}

i.e., the counting numbers.

So if you take any subset of those, it will have a smallest element. Duh, right?

The well-ordering principle is actually equivalent to the principle of mathematical induction, which enables a certain type of proof, which is the type I am going to give of the well-ordering principle, as follows:

Let S be a non-empty subset of the natural numbers.
Assume, for a contradiction, that S has no least element.
Then 1 can't be in S, because otherwise, since 1 is the smallest natural number, it would be the least element of S.
Suppose, for some number k, that all of the numbers {1, 2, 3, ... k} are not in the set.
Then k+1 can't be in the set, or it would be the least element.
Thus, by the principle of mathematical induction, S is empty, a contradiction.

That is what is called a "proof by induction."

Now, the pop quiz paradox:

Let's say a teacher says she is going to give a pop quiz next week. It is impossible for the teacher to give a pop quiz in a specific week without the students' knowing what day it will be on. If she waits until Friday, the students will know it must be on Friday since it hasn't happened yet, so Friday is out. But since Friday is out, Thursday is also out - if she hasn't given it by Thursday, the students will know it must be on Thursday. But if Thursday and Friday are both out, then so is Wednesday, ...

It is worth noting that, while the well-ordering principle is true, it is, empirically speaking, not impossible to give a pop quiz, even with advance notice of the week in which it is to occur.