# Page 202

**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.