Intersection of Empty Set

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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.

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:

$\displaystyle \bigcap \mathbb S = \mathbb U$

$\blacksquare$

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

This result is therefore classed as a veridical paradox.