Intersection of Empty Set

Theorem
Consider the set of sets $\mathbb S$ such that $\mathbb S$ is the empty set $\varnothing$.

Then the intersection of $\mathbb S$ is $\mathbb U$:


 * $\displaystyle \mathbb S = \varnothing \implies \bigcap \mathbb S = \mathbb U$

where $\mathbb U$ is the universe.

A paradoxical result.

Proof
Let $\mathbb S = \varnothing$.

Then from the definition:
 * $\displaystyle \bigcap \mathbb S = \left\{{x: \forall X \in \mathbb S: x \in X}\right\}$

Consider any $x \in \mathbb U$.

Then as $\mathbb S = \varnothing$, it follows that:
 * $\forall X \in \mathbb S: x \in X$

from the definition of vacuous truth.

It follows directly that:
 * $\displaystyle \bigcap \mathbb S = \left\{{x: x \in \mathbb U}\right\}$

That is:
 * $\displaystyle \bigcap \mathbb S = \mathbb U$

Comment
Although it appears counter-intuitive, the reasoning is sound.

This result is therefore classed as a veridical paradox.

However, in, the authors declare that:


 * "... an empty intersection does not make sense." (p. 457)

This is slightly at odds with the earlier :


 * "There is no profound problem here; it is merely a nuisance to be forced always to be making qualifications and exceptions just because some set somewhere along some construction might turn out to be empty. There is nothing to be done about this; it is just a fact of life."

In, the author recognizes the result, but does not adopt it:


 * "If one has a given large set $X$ that is specified at the outset of the discussion to be one's "universe of discourse," and one considers only subsets of $X$ throughout, it is reasonable to let $\displaystyle \bigcap_{A \mathop \in \mathcal A} A = X$ when $\mathcal A$ is empty. Not all mathematicians follow this convention, however. To avoid difficulty, we shall not define the intersection when $\mathcal A$ is empty."