As of August 2015, is the “set” of all gold medalists in the 2016 Olympics a set? I think it is since the defining property is very clear. However, given any $x$, we do not know if $x$ is in this “set” at the moment.
I want to compare my example with one provided in the answer: the set of rational numbers is clearly a set. Assume our math knowledge is limited so that we don’t known whether $e$ is rational or not. It seems that being unable to determine membership itself can’t be a reason for not being a set.