Definition:Empty Set/Existence

From ProofWiki
Jump to navigation Jump to search


Some authors have problems with the existence (or not) of the empty set:

If $A, B$ are disjoint, then $A \cap B$ is not really defined, because it has no elements. For this reason we introduce a conventional empty set, denoted $\O$, to be thought of as a 'set with no elements'. Of course this is a set only by courtesy, but it is convenient to allow $\O$ the status of a set.
The best attitude towards the empty set $\O$ is, perhaps, to regard it as an interesting curiosity, a convenient fiction. To say that $x \in \O$ simply means that $x$ does not exist. Note that it is conveniently agreed that $\O$ is a subset of every set, for elements of $\O$ are supposed to possess every property.
Now some students are bothered with the notion of an "empty set". "How", they say, "can you have a set with nothing in it?" ... The empty set is only a convention, and mathematics could very well get along without it. But it is a very convenient convention, for it saves us a good deal of awkwardness in stating theorems and proving them.

Such a philosophical position is considered by many mathematicians to be a timid attitude harking back to the mediaeval distrust of zero.

In any case, its convenience cannot be doubted:

One may regard [the vacuous set] as a zero element that is adjoined to the collection of "real" subsets.
One practical advantage in admitting $\O$ as a set is that we may wish to talk about a set without knowing a priori whether it has any members.
The courtesy of regarding this as a set has several advantages ... In allowing $\O$ the status of a set, we gain the advantage of being able to talk about a set without knowing at the outset whether or not it has any elements.

Other sources allow the definition of the empty set, but because of the way natural numbers are defined, determine that it is neither finite nor infinite.