The set with nothing in it has a Scandinavian symbol and odd powers
The familiar slashed-circle sign for the empty set was chosen in 1939 by André Weil of the Bourbaki group, borrowing the letter Ø from the Danish and Norwegian alphabets. That borrowing later caused its own trouble: Scandinavian writers can mistake the symbol for an ordinary letter, so Unicode offers a reversed version for them.
Mathematicians say the empty set, never an empty set, because there is only one. Two sets count as equal when they contain exactly the same members, so any two collections with no members at all are automatically identical. Some textbooks call it the null set, but in measure theory that phrase means something else, a set of size zero that need not be empty. Writing the numeral 0 for it, once an occasional habit, is now considered improper.
Its logic produces results that sound like jokes. The empty set is a subset of every set, since it contains nothing that could fail to belong. Any claim about all of its elements is automatically true, because there is no element to disprove it; hence the saying that everything is true of the members of the empty set. It even counts as a derangement of itself, a shuffle where no item stays in place, as there is no item to stay put.
Conventions follow from this. Adding up the members of an empty collection gives 0, the number that leaves any sum unchanged, while multiplying them gives 1 for the same reason. On the number line every real number is simultaneously an upper and a lower bound for the empty set, so on the extended number line its least upper bound is negative infinity and its greatest lower bound positive infinity, the reverse of what intuition suggests. In topology it is both open and closed.
It also serves as the starting brick of arithmetic. In the usual set-based construction, zero simply is the empty set, and in John von Neumann's scheme each following number is formed by joining the previous one with a set containing it.
Source: Empty set
