# The best is not there

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My favorite interpretation of Eduardo Scarpetta’s play “Miseria e nobiltà” (“Poverty and Nobility”) is the one by Totò, the Prince of laughter. The video above shows an extract from the comedy film taken from the play at the end of which Totò pronounces the words “la migliore non c’è” (“the best is not there”), speaking about chairs in his house. I find this line hilarious for its spectacular absurdity.

It is evident that, among the chairs of a house, there must be the best one, although this might not be unique. In fact, if we let $S$ be the non-empty finite set of chairs in a house, we can show by induction that it contains its supremum, $\sup S$, namely there is a best chair.

• If $S$ contains a single chair $x$, then $\sup S = x \in S$.
• Suppose that, if $S$ contains $n$ chairs, then $\sup S = \max S \in S$. Now, assume $S$ contains $n+1$ chairs and let $x$ be an element of $S$. By the induction hypothesis, as $S^\prime = S \setminus {x}$ contains $n$ chairs, it contains its supremum, let’s call it $x^\prime \in S^\prime$. Then:
$$\sup S = \sup \left( S^\prime \cup \{x\} \right) = \sup \{x^\prime, x\} = \max \{x^\prime, x\} \in S$$

as $x^\prime \in S^\prime \subseteq S$ and $x \in S$.

Would the line “la migliore non c’è” still sound spectacularly absurd if Totò lived in the Grand Hilbert Hotel assuming there is a chair in each room?