# 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.