Why does the empty set matter? I'm trying to understand why the empty set is a useful tool for mathematicians. Are there any nontrivial theorems that could only be achieved by the existence of the empty set, or is the recognition of the empty set just a conventional standard that mathematicians have adopted?

The same reason it's a useful tool in everyday life.
There are lots of questions that ask for sets as answers, like...


*

*Which crew members are still aboard the airplane?

*What dishes are in the sink and need to be put away?

*What problems does this piece of software have that must be fixed before we can release it?

*What songs have you practiced on the piano lately?


And sometimes the answer to one of these questions is, "There aren't any." That's the empty set.
