Proof of Lemma 2.16: A Supplement to Common Knowledge (Stanford Encyclopedia of Philosophy/Winter 2005 Edition)

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Supplement to Common Knowledge
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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

Return to Common Knowledge

Peter Vanderschraaf
peterv@cyrus.andrew.cmu.edu

Giacomo Sillari
Carnegie Mellon University
gsillari@andrew.cmu.edu

Stanford Encyclopedia of Philosophy Supplement to Common Knowledge Citation Information

Proof of Lemma 2.16

Supplement to Common Knowledge Stanford Encyclopedia of Philosophy

Stanford Encyclopedia of Philosophy
Supplement to Common Knowledge
Citation Information

Supplement to Common Knowledge
Stanford Encyclopedia of Philosophy