Yo. Wsup. I watched a video about three years ago where this guy suppossedly provedthe nonexistence of the universal set. I can't find it now but what he said (rather quickly) was that from Cantor, every set is a subset. Therefore, there is no universal set. 1) Is this valid? 2) RW Implications? Is the Universe then, really a universe? BTW I'm only about to complete engineering math so dont be too complex. Thanks, Mamooie
There are no set of all sets within ZFC (the commonly used and acknowledged axioms for ordinary mathematics). The reason for this is that the existence of a universal set leads to contradiction. It would by the axiom of separation (an axiom of ZFC that essentially says that you can form new set from a former one by specifying the properties of the elements you pick) lead to the well-known Russell's paradox. Alternatively, as you mentioned, the universal set must contain itself (or else it does not contain all sets), and that violates the axiom of regularity, but this is not nearly as enlightening. These are technical difficulties due to our choice of axioms, we simply cannot speak of the set of all sets in ZFC. We do however frequently refer to the class of all sets (and classes of other things). Classes are objects which naturally does not have all the properties sets have, but in return you can define a class merely by specifying the properties of its elements. Proper classes are classes of sets that do not form sets themselves, and of course the universal class is such class. Classes are not formalized in ZFC. Note that this has nothing to do with the physical universe, sets (and classes) are purely mathematical constructions.