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:
- 1951: Nathan Jacobson: Lectures in Abstract Algebra: I. Basic Concepts: Introduction $\S 1$: Operations on Sets:
- 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.
- 1951: Nathan Jacobson: Lectures in Abstract Algebra: I. Basic Concepts ... (previous) ... (next): Introduction $\S 1$: Operations on Sets
- 1965: J.A. Green: Sets and Groups ... (previous) ... (next): $\S 1.3$. Intersection
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $1$: Algebraic Structures: $\S 1$: The Language of Set Theory
- 1968: Ian D. Macdonald: The Theory of Groups ... (previous) ... (next): Appendix: Elementary set and number theory
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 1$: Fundamental Concepts