Say I want to know whether 3 is a root of $x^2-8$. Let $S$ be the set of all roots of $x^2-8$, together with all functions from finite sets to finite sets. I think to myself - if I search through this set and find the number 3, then 3 has to be a root of $x^2-8$, because the only numbers in the set are roots - all the other members of the set are functions. I make a list of all the sets in $S$ written as part of the cumulative hierarchy starting with the empty set, and I find that 3 is an element of this set. So 3 must be a root right?