This is an interesting generalisation of the Blue-Eyed Logicians puzzle. If you are not familiar with the original problem, you may wish to attempt that first!
The problem. Let be an infinite cardinal. A population of logicians, of potentially assorted eye colours, living on an
eye-land island. Everyone can see everyone else’s eyes, but not their own.
In this society, it is an awful taboo to have blue eyes. Anyone who can deduce that they have blue eyes leaves the island on a midnight ferry, never to return – but because it’s such a taboo, there can be no discussion of eye colour.
Unbeknownst to them, every single one of the logicians has blue eyes. Oh dear….
But because there can be no discussion, no logician can ever deduce this, and they all live in perfect harmony.
One day, an oracle proclaims that ‘There is someone on the island with blue eyes’. Call this day 0.
Question: What happens next? We assume that the logicians are immortal, and allow times to run over – that is, all ordinal-valued times. So after days , there’s a day . This is followed by days , and then , etc.
I (believe that I) have a solution, which will be posted here in due course.