Instructor: Laszlo Babai
Scribe: Ivona Bezakova
Question: How should we define ?
Trouble: two conventions conflict: and .
We shall argue that this conflict can be resolved
and that is the reasonable choice.
Argument 1: empty products.
The convention is consistent with the conventions
and
.
(Why are these the only reasonable interpretations of
empty sums and products?) The ``empty product '' rule
is used in conventions like and .
Argument 2: combinatorial interpretation of powers.
Let and be two finite sets. The number of functions
is clearly . Therefore, (there is only one function
,
namely the empty function).
Argument 3: the limit of .
Let us consider . This limit does not exist. In fact, subsequences can converge to any number between 0 and 1.
Nonetheless, the limit is ``almost well defined.''