Toward useful type-free theories. I

Abstract

There is a distinction betweensemantical paradoxeson the one hand andlogicalormathematical paradoxeson the other, going back to Ramsey [1925]. Those falling under the first heading have to do with such notions astruth, assertion(orproposition), definition, etc., while those falling under the second have to do withmembership, class, relation, function(and derivative notions such ascardinalandordinal number), etc. There are a number of compelling reasons for maintaining this separation but, as we shall see, there are also many close parallels from the logical point of view.The initial solutions to the paradoxes on each side—namely Russell's theory of types for mathematics and Tarski's hierarchy of language levels for semantics— were early recognized to be excessively restrictive. The first really workable solution to the mathematical paradoxes was provided by Zermelo's theory of sets, subsequently improved by Fraenkel. The informal argument that the paradoxes are blocked in ZF is that its axioms are true in thecumulative hierarchy of setswhere (i) unlike the theory of types, a set may have members of various (ordinal) levels, but (ii) as in the theory of types, the level of a set is greater than that of each of its members. Thus in ZF there is no set of all sets, nor any Russell set {x∣x∉x} (which would be universal since∀x(x∉x) holds in ZF). Nor is there a set of all ordinal numbers (and so the Burali-Forti paradox is blocked).