Set theories with a true universal classIn 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 & Zy
where 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) & Zx
I.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) & Zy
I.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. |