Common Knowledge > Proof of Lemma 2.16 (Stanford Encyclopedia of Philosophy/Fall2006 Edition)

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Proof of Lemma 2.16

Lemma 2.16.
calligraphic-M

(ω) is common knowledge for the agents of N at ω.

Proof.
Since calligraphic-M is a coarsening of calligraphic-H _i for each i ∈ N, K_i((ω)). Hence, K¹_N((ω) ), and since by definition K_i((ω)) = { ω | _i(ω) ⊆ (ω)} = (ω),

K¹_N((ω)) = ∩
^{i ∈ N} K_i((ω)) = (ω)

Applying the recursive definition of mutual knowledge, for any m ≥ 1,

K^m_N((ω)) = ∩
^{i ∈ N} K_i(K^m−1_N((ω)) = ∩
^{i ∈ N} K_i((ω)) = (ω)

so, since ω ∈ calligraphic-M (ω), by definition we have ω ∈ K*_N((ω)). QED

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