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Stanford Encyclopedia of Philosophy
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Proof of Proposition 2.4

Proposition 2.4.
If omega in K*N(E) and E subset F, then omega in K*N(F).

If E subset F, then as we observed earlier, Ki(E) subset Ki(F), so

K 1N ( E ) = intersection
i in N
K i( E ) = intersection
i in N
Ki(F) = K1N(F)

If we now set E prime = KnN(E) and F prime = KnN(F), then by the argument just given we have

Kn+1N(E) = K1N(Eprime) subset K1N(Fprime) = Kn+1N(F)

so we have mth level mutual knowledge for every n greater than or equal to 1.

Hence if omega in infinity
n = 1
KnN(E) then omega in infinity
n = 1

Copyright © 2001 by
Peter Vanderschraaf

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First published: August 27, 2001
Content last modified: August 27, 2001