Abstract

By way of beginning a consideration of the prospects for a consistent and complete nominalistic foundation of arithmetic, I argue that self-reference is formally possible in nominalistic languages. Establishing the formal possibility of self-reference is a crucial step in the proof of Godel's incompleteness theorems. My argument rests on a proof that self-reference is formally possible, with respect to the atomic universal formulas, in what I characterize as a strictly nominalistic language. ;I construct the proof as follows. First, I construct a calculus which is to serve as the calculus of concreta once it is appropriately interpreted. The construction takes place in the syntactical portion of a meta-language which treats the characters of the calculus as concrete individuals. ;Next, I construct a model-theoretic interpretation of the calculus in which the universe of discourse of the model is restricted to concrete individuals. The semantic portion of the meta-language employed in its construction does employ a set-theory with individuals and a nominalistic theory of qualities which are understood to be abstract individuals. It is within the semantic portion of the meta-language that a universe of concreta is defined. The model-theoretic interpretation of the calculus of concreta is achieved by assigning the free variables of the formulas of the calculus to certain objects in the universe of concreta and providing a definition of satisfaction for those formulas. ;Finally, on the basis of the model-theoretic interpretation of the calculus of concreta, I prove that for any atomic universal formula O which contains at most one free occurrence of a variable and which is satisfiable in the model, there is a formula Y which also contains at most one free occurrence of that variable and which is satisfiable in the model just in case it satisfies O. Y is a self-referring formula, and it says of itself that it O's. I conclude that self-reference is formally possible in the calculus of concreta. ;I also argue that by showing self-reference to be formally possible in a given language, one has taken an initial step toward an incompleteness proof for that language. By having shown self-reference to be formally possible in strictly nominalistic languages, I believe I have made some progress towards establishing Godel's incompleteness results for a nominalistic foundation of arithmetic