What is the limit of "I know that you know, and I know you know that I know", and so on?

In philosophy and formal logic, this is called "common knowledge". See http://en.wikipedia.org/wiki/Common_knowledge_(logic). Basically, something is "common knowledge" if A) everyone knows it, and B) everyone knows that it is common knowledge. Note that there is essentially no distinction between "X is common knowledge" and "I/you know that X is common knowledge", so there is no point in carrying out the iteration of "I know that you know that..." any further.

This is called 'common knowledge'. There is a well-known logic puzzle that illustrates the concept: https://xkcd.com/blue_eyes.html A group of people with assorted eye colors live on an island. They are all perfect logicians -- if a conclusion can be logically deduced, they will do it instantly. No one knows the color of their own eyes. Every night at midnight, a ferry stops at the island. Any islanders who have figured out the color of their own eyes then leave the island, and the rest stay. Everyone can see everyone else at all times and keeps a count of the number of people they see with each eye color (excluding themselves), but they cannot otherwise communicate. Everyone on the island knows all the rules in this paragraph. On this island there are 100 blue-eyed people, 100 brown-eyed people, and the Guru (she happens to have green eyes). So any given blue-eyed person can see 100 people with brown eyes and 99 people with blue eyes (and one with green), but that does not tell him his own eye color; as far as he knows the totals could be 101 brown and 99 blue. Or 100 brown, 99 blue, and he could have red eyes. The Guru is allowed to speak once (let's say at noon), on one day in all their endless years on the island. Standing before the islanders, she says the following: "I can see someone who has blue eyes." Who leaves the island, and on what night? Followup: 1) If the Guru had said nothing, would anyone have left the island? 2) If your answer to #1 is "yes", what additional information was provided by the Guru? They could *all* see someone with blue eyes.

People seem to be able to think about 4 levels. Dennett, in his book, The Intentional Stance, talks about a hierarchy of intentional systems. First order intentionality is that I can think about stuff. I want chocolate. Second order intentionality is that you can know what I think about stuff. You know that I want  chocolate. This is called a theory of mind. Third order intentionality is that Alice thinks that you know that I like chocolate. Fourth order intentionality is that Bob believes that Alice thinks that you know that I like chocolate. Some non-human animals seem to be able to handle second order intentionality. Adults tend to get stuck beyond 4th order. We cannot handle thinking about Carol who needs Bob  to Believe that Alice thinks that you know that I like chocolate. http://en.wikipedia.org/wiki/Intentional_stance

Let's be logical. You've missed a few steps, and confused the question in your explanation by asking a different question. Your question: I know that you know, and you know that I know – misses a lot of intermediate steps. The explanatory question I know and you know, and I know you know.  You know that I know that you know. I know that you know that I know that you know – is a bit more detailed, but still misses many details of the question. The detailed, properly sequenced list of statements is as follows: 1.   I know.  or You know. can happen independently from (either can come first) a.      I know where you are. b.     You know where I am. c.      The dog knows where the cat (or the rabbit, or the female dog) is. Note: Even this first layer bring with it the possibility for error.  I might ‘know’ where the dog is. But I might be wrong.  The dog might ‘know’ where the cat is – but the cat may have escaped without his knowledge.   2.      I know that you know.  The cat, the rabbit and the female dog know that the dog knows where they are.  Animals are predator and prey, so they can understand this far. Again we have the possibility for error.  You might think that I know, but you might be wrong. The cat might ‘fear’ (or know in his heart) that the dog knows where he is hiding. But he might be wrong.   3.      I know that I know – or the same ‘you know that you know’ (you missed this step). To get to this step requires knowledge of self. Dogs, cats, rabbits are not aware of themselves, as far as we know, and thus cannot attain this level. This level, or the expression of this level, probably requires language.  I don’t know how an animal could demonstrate that it “knows that it knows”, without language. I don’t know what age humans normally attain this level of knowing, but it requires a fairly sophisticated level of intelligence and independence.  This layer is also fraught with error. There are many situations where we might ‘know that we know’, but we have two opportunities to be incorrect, because our knowledge might be wrong, or my assumption that ‘I know’ might be wrong. We need to examples to explain these possible errors. a.      I might know that 12x13 = 144.  That is incorrect knowledge. b.     I might ‘know’ that I know what 12x13 is, but when someone asks me – I find that I really don’t know but I can quickly calculate, so I don’t need to know.    4.      You know that I know that you know. Note: this requires 3. That is to say, it requires you to know that you know. Then it jumps to the next level, knowing that I also know that you know. But it has not yet jumped to the level of “I know that you know that you know”.  I just know that you know.   However for you to know that I know that you know – requires additional knowledge.  There must be some way, some sign, so that ‘you know’ – that I know that you know. I have to tell you, or indicate in some way.  Of course as in all cases, there is possibility for error – so you can’t  just assume that because you ‘think’ I know that you know, that you are correct.  For you to know requires some communications.    We’ll leave errors aside at this point, but be aware that the possibility for error rise as we rise up the ladder.   5.      You know that I know that I know (you missed this possibility). These steps have two more requirements.  a.      First of all, it requires you to attain the level of independence, such that you realize that other people can know that they know things.  We know that dogs don’t ‘know that they know where the cat is’, but as we mature, we recognize that sometimes, people can know that they know something.  And we might also recognize, if we think about it, that sometimes people don’t ‘know they know something’, even though they do possess the knowledge. Dogs, cats and small children presumably, cannot rise to this level of knowing.    b.     It also requires some information flow between me and you. You might have seen me demonstrate not just the knowledge, but also that I know that I know. Maybe I told you that I know how to get to the store.  Until this knowledge passes from me to you – you don’t know that I know that I know - even if you know that I know.   The next steps are simply extensions of statement 4, and all subsequent steps can be represented as extensions of statement 4. Here’s the ladder. 6.                                                            I know that you know that I know that you know.                                         You know that I know that you know that I know that you know.                        I know that you know that I know that you know that I know that you know. You know that I know that you know that I know that you know that I know that you know. As we step up each layer, we can’t just ‘assume’ that what we know is correct. In each step up the ladder, the person who is represented in the additional component MUST receive some indication from the other person, before they can ‘know’.  As each layer goes higher, we don't KNOW, what the other person knows, until we receive evidence. We can't KNOW what the other person knows until they tell us what they know. We can assume, but that would be silly, and likely incorrect, introducing more errors.  So, when does it end? It probably ends at suppertime, or when the Star Trek rerun starts (I suspect even Spock would not be interested in carrying this on for long), or when one participant or the other gets tired of the game, because both must be active participants to 'know that I know that..." and confirm their knowledge of the higher level to the other, correctly. There is another end to this sequence.  Either party might step out of the ladder, and ask the philosophical question: When I know that I know…. Are there two different ‘me’s? Is there a me that knows something – and another me that ‘knows I know’? Is that other me a higher level of me?  As I climb up the ladder, am I creating new instances of ‘me’ for every step up. And to continue, careful, ladders are treacherous… When I recognize that there might be a different ‘me’ at each step in the ladder, I might convince myself that each of those ‘me’s is one person: me.  But wait a minute. What about the person who convinced myself that all of those ‘me’s are one person? Is the me that watches me watching me – a different me? And what if one of those other ‘me’s starts an argument, or doubts that ‘I know’ the next layer, or begins to question the lower layers, the foundation.   To tell the truth, it’s difficult to tell which ‘me’ was the ‘doubting Thomas’ – where does doubt begin? Next thing you know, you’ll be realizing that the top level ‘me’ is not the ‘me’ that you see when you look in the mirror. Or not the ‘me’ that quickly makes assumptions about what the other person knows.  Your top level me (or your "I'm tired of this game me") might be someone who insists on stepping outside of the ladder, checking the foundation, challenging the assumptions. Maybe this is the first step to enlightenment – you have to step away from the ladder before you can take that first step.   ps. I have no idea how the blue eyes answers relate to this question.  Maybe it's an off-by-one-hundred error?