# Definition:Empty Set/Existence

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## Definition

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

- 1965: J.A. Green:
*Sets and Groups*: $\S 1.3$:

*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.*

- 1968: Ian D. Macdonald:
*The Theory of Groups*: Appendix:

*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.*

- 2000: James R. Munkres:
*Topology*(2nd ed.): $1$: Set Theory and Logic: $\S 1$: Fundamental Concepts

*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: Volume $\text { I }$: 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.*

- 1965: Seth Warner:
*Modern Algebra*: $\S 1$:

*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.*

- 1975: T.S. Blyth:
*Set Theory and Abstract Algebra*: $\S 1$:

*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.

## Sources

- 1951: Nathan Jacobson:
*Lectures in Abstract Algebra: Volume $\text { I }$: 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 $\text I$: 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