This is in fact the logically meaningful thing to do. All must be monotone decreasing in set inclusion. Other similar definitions are that min of an empty set

This is in fact the logically meaningful thing to do. All must be monotone decreasing in set inclusion. Other similar definitions are that min of an empty set is infinity and max of an empty set is minus infinity. Min is monotone decreasing over set inclusion hence min of empty set must be the maximum possible value. Max is montone increasing in ser inclusion hence max of empty set must be the smallest possible value.