In a course, the teacher stands up and says that there will be a surprise test before the end of the year. But this is impossible! For look. If the test was given on the last day of class, would anybody be surprised? Why of course not, they know it has to come, and they know there is no other day possible. So it cannot come on the last day. But then on the second to last day, what then? Why they know it cannot be on the last day, and so does the teacher, so it must be today! Where is the surprise? And so it goes back to the first day. It is impossible for the teacher to give a surprise test to this logical group of people.

Yet teachers announce surprise tests, and surprise their students.

I leave you to reflect on the difference between knowledge and meta-knowledge. As I do so, I mention in passing that there is a world of difference between saying, I prove this! and saying, I prove that I can prove this! In fact this distinction is critical. Were it not so, then mathematics would run afoul of Gödel's theorem.

Catch me another day and I may explain how this ties in with the existence of a finite number - trivially proven to be finite - but about which you cannot prove that any number explicitly written out in base 10 can ever be proven to be larger than this one in any consistent axiom system comprehensible by humans. But you wouldn't want that. It might tempt you into studying mathematics, which is a profession that will make you far less money than computers can...

(BTW one such number is busy beaver of 50 million. It is a finite number which we can find no explicit upper bounds for.)

In reply to There ain't no such thing as surprise tests then by tilly
in thread (OT) Meditate on this brain-teaser by japhy

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