>>10889203>Why is considered indeterminate rather than undefined?Because there are several plausible definitions for what it may be that do not yield the same result.
If you think of this in terms of calculus, you could say that it stems from the fact that the limit does not exist.
In fact, taking the limit along appropriate paths, you can make it anything you want: For any , you have
Now there are arguments for why it should be 1 (as Bourbaki, for example, suggests).
For example, , which seems like the most natural limit-based definition of (but, as I explained above, a limit-based definition not really satisfying).
Another reason, more algebraic, is that any empty product (so product with zero terms) "should" be 1 by convention, for the same reason that a sum of no terms should be zero.
It allows one to defined products over any set and many formulas only extend if we set it like this, for example if is a finite collection of real numbers and we have a partition , we would like to say .
For this to work even when one of the sets is empty, we need to set . With this convention, we might reasonably say
