my forty years on his shoulders - Department of Mathematics

23function symbols, together with the relevant axioms. Precisely this approach wasadopted by (Quine, 1940, 1951, Chapter 7).However, in so doing, one has expanded the language of T. Accordingly,two choices are apparent.The first choice is to make sure that as one adds new sorts and newrelevant relations and function symbols and new axioms to T, associated withsyntax, one also somehow has already appropriately treated, directly, the newsyntactic objects and axioms beyond T that arise when one is performing thisaddition to T.The second choice is to be content with adding the new sorts and newrelevant relations and function symbols and new axioms to T, associated with thesyntax of T only - and not try to deal in this manner with the extended syntaxthat arises from this very process. This is the choice made in (Quine, 1940, 1951,Chapter 7).We lean towards the opinion that the first choice is impossible to realize inan appropriate way. Some level of indirection will remain. Perhaps the level ofindirection can be made rather weak and subtle. Thus we lean towards theopinion that it is impossible to construct extensions of, say, PA that directly andadequately formalize their entire syntax. We have not tried to prove such animpossibility result, but it seems possible to do so.In any case, the second choice, upon reflection, turns out to be whollyadequate for casting what may be called “direct second incompleteness”. Thisformulation asserts that for any suitable theory T, if T’ is the (or any) extension ofT through the addition of appropriate sorts, relations, functions, and axioms,