Monday, December 11, 2006

Tarski and Proper Classes

A "set" is a class which is a member of a class. A proper class is a class which is not a member of anything. In one established theory about classes, viz. the theory expounded by Monk, proper classes are asserted to exist. According to other theories, e.g., Zermelo-Frankel set theory, all classes are sets. Here I will argue for the following claim:

If Tarski's semantical account in Logic, Semantics, and Metamathematics for what he called 'the language of the calculus of classes' (called C) is correct, then Monk's theory is false.

In particular, Tarski's definition of truth for C is inconsistent with Monk's assertion that there are proper classes.

If Tarski's definition of truth for C is correct then the truth conditions of each sentence of C are detrimental. Still there remains the question: which sentences of C are true? Different theories about classes constitute different answers. A theory about classes in C can be construed as a particular subset of C's sentences, viz., the theory's theses. If the theory is an axiom theory this set consists of the theory's axioms and all logical consequences of its axioms.

C is a first order language whose non-logical constant is a sign for class inclusion. Any theory about all classes is formulable in C as long as the theory's primitive non-logical constants are definable solely in terms of the sign for class inclusion. Monk's is such a theory, for its primitives are all definable in terms of class inclusion and it is a theory about all classes. My argument for this is as follows. Suppose Monk's really is a theory about some but not all classes. Then a language L is appropriate to Monk's theory only if L's variables range over some but not all classes. But then Monk's assertion of proper classes is verified by the fact that among the values of variables of L are classes that are members of none of the values of those variables. But this point is trivial, for no one doubts that there are classes among which some are not members of any others. Since it is not this triviality that is intended by the Monk assertion of proper classes, Monk's theory is about all classes.

It is true that according to the standard characterization of an interpretation of a first-order language an interpretation consists of a non-empty class, called the domain of discourse, plus a function that assigns the usual things to the non-logical constants of the language. When one goes to the definition of truth under that interpretation, it is plain that the domain of discourse is the class of those entities which are the values of the variables under that interpretation. But a Tarski-style definition of truth for a first-order language does not require a domain of discourse for either language or metalanguage. All that is required is a specification of the objects over which the language's variables range.

No comments: