Many-sorted abstract types
M. Downward · 2004
Unfortunately, allowing subsets to be members of sets in the same way as individual objects leads to the problem now described. If subsets are allowed to be members of sets, might it then be argued that the set of all woodpeckers is a member of itself? The question of self-membered sets led Russell to examine the set R of sets s that do not contain themselves, i.e. the set of sets described by the comprehension R = Is I s e: s} A particular subset k belongs to set R if and only if it is not a member of itself: but the set R itself qualifies and we might substitute it for k in the above mutual equivalence to give and we are immediately led into a contradictory statement, now known as the Russell paradox.