I thought that "common knowledge" means something which everybody knows, and everybody knows that everybody knows it, and everybody knows that everybody knows that everybody knows, ad infinitum. However I can't see isomorphism between that and the definition you have used. Are they the same, or it's only a confusing coincidence in terminology?
Here is a proof of the equivalence between my definition and Aumann's for "common knowledge". I'm assuming some familiarity with set partitions.
Aumann's definition is in terms of the Kolmogorov model of probability. In particular, a proposition is identified with the set of possible worlds in which the proposition is true.
Let P₁ be that partition of the possible worlds such that two worlds share the same block in P₁ if and only if I condition on the same body of knowledge when computing posterior probabilities in the two worlds*. Let P₂ be the...
I've written up a 2-page explanation and proof of Aumann's agreement theorem. Here is a direct link to the pdf via Dropbox.
The proof in Aumann's original paper is already very short and accessible. (Wei Dai gave an exposition closely following Aumann's in this post.) My intention here was to make the proof even more accessible by putting it in elementary Bayesian terms, stripping out the talk of meets and joins in partition posets. (Just to be clear, the proof is just a reformulation of Aumann's and not in any way original.)
I will appreciate any suggestions for improvements.
Update: I've added an abstract and made one of the conditions in the formal description of "common knowledge" explicit in the informal description.
Update: Here is a direct link to the pdf via Dropbox (ht to Vladimir Nesov).
Update: In this comment, I explain why the definition of "common knowledge" in the write-up is the same as Aumann's.
Update 2020-05-23: I fixed the Dropbox link and removed the Scribd link.