What does it mean for two distinct people to know a fact?

Let there be two people A and B. They both encounter a strange object O. What do they know about each others knowledge about O?

More strictly, we are speaking of a situation where, say, you and me encounter object O. I know that you know about existence of O, and you know that I know about existence of O; I know that you know that I know about existence of O, and you know that I know that you know about existence of O; and so on...

The issue here is that, traditionally, when we both see O, we simultaneously acknowledge in such a way that if we were to ask either of us any of the question (from above) of the form, "do you know that I know that you know that I know that you know .... about existence of O", response will be, correctly, "Yes". Let's call this a stable condition.

Now consider an alternative situation. Situation: A encounters O. So A knows O. At:

• t=0: A knows 0.
• t=1: B is privately told, "A knows O".
• t=2: A is privately told, "B knows that A knows O"
• t=3: B is privately told, "A knows that B knows that A knows 0" and so on...

Do A & B, at some time t, reach stable condition? Of course, we are setting these values to be true at time t, t+1, t+2, ... but I am unable to figure out: after many alternating knowledge of each others knowledge bases, why is either of them unable to confidently assert that both know everything (or reach a stable condition) -it makes sense for lower levels, but shouldn't higher level degrees simply "collapse" -all I want is each to know that both know that both know O, so is there some redundancy after a level? Or why is this situation logically different from first situation?

• Comments are not for extended discussion; this conversation has been moved to chat. Dec 26, 2020 at 14:32