# Common knowledge as a fixed point

I read on a wikipedia page that from the modal logic formalization CK can be formulated as a fixed point. If it also holds for the set theory formalization? If it does, where I can find about it?

Edited: Common knowledge is used to be thought about like a limit of "I know that you know that I know that you know that I know..."

A nice example is given here.

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## 1 Answer

Yes, there is also a set-theoretic formulation of common knowledge as a fixed point. Common knowledge of an event $E$ is the greatest fixed point of the function $f_E(X)=K^1(E\cap X)$, where $K^1(Y)$ denotes first-order mutual knowledge of $Y$, i.e., that everyone knows $Y$. The existence of a greatest fixed point is guaranteed by the Knaster–Tarski theorem. You can read about this account of common knowledge in the Stanford Encyclopedia of Philosophy.

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Thank you, bounty worked ) –  Ilya May 26 '11 at 10:16