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Consider a piece of information. I know it and so do you. Moreover, I know that you know it, and you know that I know it. Further, I know that you know that I know it, you know that I know that you know it, and so on. Each stage entails a further degree of knowledge because it is quite possible in principle, for example, for Tom to know that Jane knows something without Jane knowing that Tom knows that she knows it. But there is surely an abundance of cases where two people (say A and B) in close contact are mutually aware of a simple inescapable fact F (for example, that they both live in a certain house) and, if you asked either "Is it true that ... A knows that B knows that B knows that A knows that ... knows F?", to any degree of recursion, then the answer would be "yes, obviously" (assuming that the questionee had not fallen asleep in the meantime). What bothers me here is that this implies an infinite amount of knowledge. How can that be possible?

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3 Answers

up vote 5 down vote accepted

The most immediate reply to your worry is that, for every relevant application of the notion of common knowledge, what counts is the potential beliefs of agents.

There is an infinite hierarchy of potential beliefs of the kind you consider, but only a finite (indeed, small) number of them will ever need to be entertained by actual agents in actual situations in social life.

For much more on this.

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Link down...... –  Pacerier Feb 26 '13 at 19:33
    
It works here... –  Schiphol Mar 3 '13 at 21:15
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Just use induction and self-reference:

You and I know the same thing. Let us call this being 0-aware of the fact. We also know that we have the same conception of this fact, which makes us 1-aware (we each know that the other knows). If this proof is true, then if you are K-aware of my knowledge, I am (K+1)-aware of your knowledge (and vice versa). Thus, by induction we are N-aware for arbitrary N, after having understood this proof. I understand the proof, and so do you.

Infinite regress is best avoided by understanding of the pattern and generalizing. Otherwise, as finite beings with brains of finite capacity, we eventually get lost in the mechanics of recursing.

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I'm not sure that this does "imply an infinite amount of knowledge". Infinite recursion does not necessarily result in an infinite total. 1+1/2+1/4+1/8+1/16+... to an infinite level of recursion still results in just less than 2, not infinity.

In relation to your question, each "piece" of knowledge seems "smaller" (less meaningful, less relevant?) than the previous one, and sufficiently so that the sum of all those pieces is not significantly "larger" than the first few terms.

In practice, there are many environments where we communicate information and don't require confirmation. There are still relatively many where we require confirmation, but very few that I can think of where the communication isn't considered complete until we've confirmed receipt of the confirmation. If these pieces of information were "equal", there would be many more situations where there was significant recursion of confirmations.

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Information is atomic: at least one bit is entailed in any extra information. Also, the fact that an item of knowledge is less relevant, or even completely useless, does not diminish its information content. –  John Bentin Nov 21 '12 at 17:55
    
"One bit" holds relevance when talking about computer systems, but not the human brain. Knowing that we both know X and are aware of each others' knowledge of X (recursively, ad infinitum) is still one "piece" of knowledge to the human brain, whereas a computer would have to hold a bit at a time, infinitely, which it can't do. So yes, reducing the meaning contained in each level of recursion does reduce the "amount" of knowledge added to the total with each recursion. Of course, using the words amount, size etc are just approximations anyway when talking about the brain. –  Ryno Nov 22 '12 at 10:23
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