my forty years on his shoulders - Department of Mathematics

9ordinary mathematics. A particularly important type of structure is a structurewhose domain includes absolutely everything.Indeed, it can be argued that the original Fregean conception of logicdemands that quantifiers range over absolutely everything. From this viewpoint,quantification over mathematical domains is a special case, as “being in a givenmathematical domain” is treated as (the extensions of) a unary predicate oneverything.These general philosophical considerations were sufficient for an appliedphilosopher like me to begin reworking logic using structures whose domainconsists of absolutely everything.The topic of logic in the universal domain has been taken up in thephilosophy community, and in particular, by T. Williamson in (Rayo, Williamson2003), and (Williamson 2000, 2003, 2006).We have not yet published on this topic, but unpublished reports on ourresults are available on the web. Specifically, in (Friedman 1999), and in(Friedman 2002a 65-99). We plan to publish a monograph on this topic in the nottoo distant future.3. THE FIRST INCOMPLETENESS THEOREM.The Gödel first incompleteness theorem is first proved in (Gödel 1931). Itis proved there in detail for a specific variant of what is now known as the simpletheory of types (going back to Bertrand Russell), with natural numbers at thelowest type. This is a rather strong system, nearly as strong as Zermelo settheory.