We encounter _Russell's Paradox_ if we try to define a [set of sets](Sets%20of%20sets.md) whose elements are sets and only sets that are _not_ members of themselves. Consider a set $S,$ such that $S=\{X:X\notin X\}.$ In this case $S$ both belongs to itself and also doesn't. This is because $X$ may still be any set, by the definition, however, if $X$ is only sets that don't contain themselves, then $S$ cannot be in $S.$ Thus, we must conclude that $S=\{X:X\notin X\}$ is not a valid set. ^ea388b # Resolution of Russell's Paradox Russell's paradox implies that the _set of all sets_ does not exists since it comes about when we try to define a paticular kind of [set of sets.](Sets%20of%20sets.md) ^f069c5 %%This is roughly how it's stated in pg. 4 of Jech.%% %%You still need to integrate problem 1.1.2 from David Barrington's text.%% #MathematicalFoundations/SetTheory