Part 1 seems to have little to do with how I remember the theorem. Here is the abstract of Aumann's paper.
Two people, 1 and 2, are said to have common knowledge of an event $E$ if both know it, 1 knows that 2 knows it, 2 knows that 1 knows is, 1 knows that 2 knows that 1 knows it, and so on. THEOREM. If two people have the same priors, and their posteriors for an event $A$ are common knowledge, then these posteriors are equal.
Your 2 implies the following claim:
Agent 1 knows that agent 2 knows E if and only if agent 2 knows that agent 1 knows E.
This claim is obviously false.
Here is another corollary of your definition:
Suppose P1(w)={w,v}, P2(w)={w,u}. Then at w, I know E={w,v,u}, but at v, I do not know E! So I can distinguish between w and v by checking my knowledge, even though I cannot distinguish between w and v!
Part 3 is correct. Indeed, common knowledge of honesty is a requirement.
Your 2 implies the following claim:
Agent 1 knows that agent 2 knows E if and only if agent 2 knows that agent 1 knows E.
How does it imply that? (It well might, within the context of the agreement theorem. My recollection is that you assume from the start that A1 and A2 have common knowledge of E.)
This claim is obviously false.
Why? If knowledge means "justified true belief", then for agent 1 to know that agent 2 know E, agent 1 must also know E, and vice-versa. This doesn't prove the claim that you say I am making, but goes most of the ...
Recent brainstorming sessions at SIAI (with participants including Anna, Carl, Jasen, Divia, Will, Amy Willey, and Andrew Critch) have started to produce lists of rationality skills that we could potentially try to teach (at Rationality Boot Camp, at Less Wrong meetups, or similar venues). We've also been trying to break those skills down to the 5-second level (step 2) and come up with ideas for exercises that might teach them (step 3) although we haven't actually composed those exercises yet (step 4, where the actual work takes place).
The bulk of this post will mainly go into the comments, which I'll try to keep to the following format: A top-level comment is a major or minor skill to teach; upvote this comment if you think this skill should get priority in teaching. Sub-level comments describe 5-second subskills that go into this skill, and then third-level comments are ideas for exercises which could potentially train that 5-second skill. If anyone actually went to the work of composing a specific exercise people could run through, that would go to the fourth-level of commenting, I guess. For some major practicable arts with a known standard learning format like "Improv" or "Acting", I'll put the exercise at the top and guesses at which skills it might teach below. (And any plain old replies can go at any level.)
I probably won't be able to get to all of what we brainstormed today, so here's a PNG of the Freemind map that I generated during our session.