Intersection of Empty Set

From ProofWiki
Jump to: navigation, search

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:

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

$\blacksquare$


Paradoxical Implications

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

This result is therefore classed as a veridical paradox.


However, in 2008: Paul Halmos and Steven Givant: Introduction to Boolean Algebras, the authors declare that:

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

This is slightly at odds with the earlier 1960: Paul R. Halmos: Naive Set Theory:

"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 2000: James R. Munkres: Topology (2nd ed.), 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."


1967: George McCarty: Topology: An Introduction with Application to Topological Groups accepts this result, but cautiously:

"A natural mnemonic for these extreme cases is that $\bigcap \mathcal S$ "grows larger" as $\mathcal S$ "grows smaller", and $\bigcup \mathcal S$ grows smaller as $\mathcal S$ grows smaller. No other convention is possible, but the case $\mathcal S = \varnothing$ will often be treated redundantly by itself in definitions and proofs, as a reminder of the null case."


1999: András Hajnal and Peter Hamburger: Set Theory dismiss it casually:

"As usual, we adopt the convention that in case $A = \varnothing$ the expression $\bigcap A$ is defined only in case we work with the subsets of an underlying set $X$. In this case we put $\bigcap A = X$."


Sources