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Location: Chapter 22, additional questions after Problem 17 and before "Kestrels and infinity"

Open Questions

Is there a way to formally tackle the problem of (non-) egocentricity of the Sage bird and the Mockingbird? What can be said, and how, about the questions of whether

\Theta \Theta \stackrel{?}{=} \Theta

and

MM \stackrel{?}{=} M\;.

Discussion

For the Sage bird, there does not seem to be a way of deriving any evident contradiction (such as the ones met in this chapter, KI=I, KI=K, K=I, KK=K) by assuming \Theta\Theta=\Theta: all one seems to get is an infinitely-nested tower of statements of the form:

\Theta=\Theta\Theta=\Theta(\Theta\Theta)=\Theta(\Theta(\Theta\Theta))=\Theta(\Theta(\Theta(\Theta\Theta)))=...

which does not seem to lead anywhere.

A for the Mockingbird, again all one can apparently do is to start by assuming M=MM and, by applying the definition of M, deriving ... that MM=MM, not a great progress indeed.

For \Theta and M, is it possible to prove that they are not egocentric? Is it possible to prove that they are egocentric? Are there further assumption that can be made, such as a lower bound on the number of different birds or some inequalities (e.g. M \neq K), which would help? Is it possible to prove that the above (non-) egocentricity cannot be proven? To me these are, as it stands, open questions still.