[borisp]

Right, the proof of the statement "everyone dies no matter what" is by induction, but in any real application, the islanders/villagers perform a finite chain of reasoning.

A good illustration is to imagine that the inhabitants had received this announcement in the mail, and that no one ever talks about the mail. In that situation, everyone goes along happily as if nothing ever happened (except in the specific case where one person has blue eyes), because no one knows what everyone else has gotten in the mail.

Specifically, if there are N people with blue eyes on the island, the phrase "Everyone knows that everyone knows that everyone knows that ... (N times) ...someone has blue eyes" changes from false to true. In other words, the knowledge that "someone has blue eyes" changes from private to public.

I think this is neat because it illustrates that just because something is on everyone's mind does not mean that they do anything about it, or that they can draw conclusions from it. The public announcement is something tangible and it has consequences, even in the event that it announces nothing new. I see this as analogous to when a poker book is written, in that everyone can have a common strategy, but once it is printed in a book, the strategy inevitably evolves, although it seems as if nothing new has been introduced.