If one holds "too strongly" to axioms like powerset and separation, or proceeds naively with comprehension, one gets an inconsistent presentation of a universal set. However, those seem like easy problems to fix (e.g. say that the universal set V has no subsets (because these can't be fully separated out of it in the first place), only elements, and then there is no powerset above V that somehow has to be an element of V also, etc.).
But consider variations on the theme in terms of fuzzy sets (with nonbinary degrees of elementhood relations) or multisets (with nontrivial iterations of the same element inside a given set):
- Start by inflating the Continuum to the "size" of a proper class. Then use surreal numbers over the concept of elementhood degrees and say that there is some U such that 0 ∈0/U U, 1 ∈1/U U, 2 ∈2/U U, etc. So now one has a universal fuzzy set, i.e. a fuzzy set with all elements (except for zero, I guess; so perhaps it would be better to call this set "couniversal"). However, imagine then some W such that 0 ∈W/0 W, 1 ∈W/1 W, 2 ∈W/2 W, and so on. Again, a (co)universal set (zero again poses a "degeneracy" issue on this scheme; but perhaps then you might start with the index of variation at 1). And there are many such alternations besides.
- Let there be some universal multiset M such that it has the element 0 with a multiplicity of 0, 1 with a multiplicity of 1, and so on. But one might also define another MM as having all its elements with absolute multiplicity (i.e. MM has absolutely infinitely many copies of all of its elements to its name).
- Go to the zone of fuzzy multisets and define some UM as satisfying various permutations of the parameters from (1) and (2), then some WMM satisfying other combined parameters, and so on and on.
(3) seems especially "troubling," in that one might have a universal fuzzy multiset with, say, two copies of some element, one whose elementhood degree is different from that of the other copy, and so on and on. Perhaps such schemes easily lead to contradictions, but even apart from that danger, would such entities result in indeterminate or even indeterminable "forms" of unrestricted quantification? I guess my OP question is: if it possible to vary the elementhood relation so wildly, would this fact undermine the meaningfulness of talk of unrestricted quantification?
