A universal set, or a set that contains all sets, exists.

The statement "a universal set or a set that contains all sets exists" is considered false in the context of set theory. This statement is related to Russell's paradox, which is a logical paradox discovered by philosopher and logician Bertrand Russell. Russell's paradox arises when considering sets that do not contain themselves. If we assume that there is a universal set containing all sets, we run into a logical contradiction when considering whether this universal set contains itself or not. If the universal set contains itself, then it shouldn't, according to the definition of the universal set. If it doesn't contain itself, then it should, again leading to a contradiction. To avoid such logical issues, set theory typically avoids the assumption of a universal set that contains all sets. Instead, set theories use axioms and principles to carefully define the existence and properties of sets without leading to paradoxes like Russell's paradox.