Yes I agree, I only tried to show my fascination over how one inevitable or primal notion (that of something universal) is tamed by another (that of consistency) in the context of set theory and domains of discourse.

«No sentence is everywhere true» and «every sentence is both true and false» would both seem to make themselves untrue, leaving one with something universal and some consistency. These very informal thoughts, and the seeming tension between them in set theory is why I found this interesting in the first place, so the in-depth answers from you both are much appreciated!

Thank you for the clear and helpful answers (here and elsewhere on the site), in contrast to my own questions ;)
As a layman, it is interesting to hear that no domain of discourse is truly universal.
So when we say, explicitly or otherwise, that «not everything is within this domain of discourse»,
that which is inside that domain and that which is not, when taken together still do not form a universal domain of discourse but only a larger one.

What I mean is just that the sentence «there is no set of which every set is a member» clearly says something about every set. But how does this not define some kind of collection or set?

Apologies for what may be a naive question, but wouldn’t the statement or conclusion «there is no set of all sets» be all-inclusive in one way or another if it really is true?