Full Idea
All the main set theories deny that there is a set of which everything is a member. No interpretation has a domain with everything in it. So the universal quantifier never gets to mean everything all at once; 'all' does not mean all.
Gist of Idea
The universal quantifier can't really mean 'all', because there is no universal set
Source
William D. Hart (The Evolution of Logic [2010], 4)
Book Reference
Hart,W.D.: 'The Evolution of Logic' [CUP 2010], p.111
A Reaction
Could you have an 'uncompleted' universal set, in the spirit of uncompleted infinities? In ordinary English we can talk about 'absolutely everything' - we just can't define a set of everything. Must we 'define' our domain?