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 2008 book, the authors declare that:


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