Set theories with a true universal class
In ZF (Zermelo-Frankel set theory)there is no universal set. In the weak Bernays theory we can get a class of all sets but not a universal class (since classes are not members). Naive set theory can be formulated as the axiom schema
(F)(Ef)(x)(M(x,f)<->Fx)where M(x,f) is the membership relationship signifying that x is a member of f, and F is a wff free in one variable in which the primitive relationship is M. Naive set theory is inconsistent. Weak Bernays theory takes the primitiive relationship as
M1(x,y) :=: M(x,y) & Zx & Zywhere Zx is ‘x is Zermellian’, i.e. the cardinality of x is a ZF cardinal. Non-zermellian sets are commonly called classes. Strong Bernays theory takes the primitive relationship as
M2(x,y) :=: M(x,y) & ZxI.e. one can quantify over membership in classes, but classes cannot be members. There is a third possibility; one can take the primitive relationship as
M3(x,y) :=: M(x,y) & ZyI.e. classes can be members but you cannot quantify over membership in classes. The resulting theory is a consistent weak extension over ZF with a true universal class — the universal class contains all classes, including itself.
The resulting theory is quite different from Quine’s NF. In Nf the comprehension schema does not restrict the membership relationship. Instead the suite of formulas in the schema is restricted by requiring the formulas to be stratified.
This page was last updated October 1, 2005.