Suppose there are two spies, A and B. There is also a secret, s. The following situation unfolds:
(1) A learns s.
(2) B learns that A knows s.
(3) A learns that B knows that A knows s.
(4) B learns that A knows that B knows that A knows s.
(5) A learns that B knows that A knows that B knows that A knows s.
(6) This situation goes on forever, with both spies finding more and more information about each other but accomplishing absolutely nothing practical.
I know similar questions have been asked before (and I guess now you know that I know), but these are the specific things I'm wondering about this situation.
- At some point, the cycle of knowing that someone else knows that you know something, etc., becomes unintuitive. What in the world does it mean, in common sense terms, to "know that they know that you know that they know, etc."?
- Is there some point at which adding another layer of "knowledge" ceases to affect that actual knowledge of either party?
- Is it possible to traverse infinite steps of this knowledge cycle in a finite amount of time? Since gaining a piece of knowledge requires conscious thought, and conscious thought requires time, it would seem not.
- There is the case where A and B are the same person. It seems that in this case, this person would have to take a sequence of steps: first knowing s, then knowing they know s, then knowing they know they know s. Again, in this case, each piece of knowledge requires a conscious thought that takes time. This case is interesting because it takes higher order thinking to know that you know something. For example, a cat may know what a window looks like, but has it ever pondered the fact that it knows what a window looks like? Is there some point in the self-knowledge of knowledge that humans are rationally incapable of reaching?