my forty years on his shoulders - Department of Mathematics

3We will not attempt to properly discuss the full impact of Gödel's workand all of the ongoing important research programs that it suggests. This wouldrequire a book length manuscript. Indeed, there are several books discussing theGödel legacy from many points of view, including, for example, (Wang 1987,1996), (Dawson 2005), and the historically comprehensive five volume set (Gödel1986-2003).In sections 2-7 we briefly discuss some research projects that are suggestedby some of his most famous contributions.In sections 8-11 we discuss some highlights of a main recurrent theme inour own research, which amounts to an expansion of the Gödel incompletenessphenomena in new critical directions.The incompleteness phenomena lie at the heart of the Gödel legacy. Somecareful formulations, informed by some post Gödelian developments, arepresented in sections 3,4,5.One particular issue that arises with regard to incompleteness has been adriving force for a considerable portion of <strong>my</strong> work over the last <strong>forty</strong> <strong>years</strong>. Thishas been the ongoing search for necessary uses of set theoretic methods innormal mathematics.By way of background, Gödel’s first incompleteness theorem is anexistence theorem not intended to provide a mathematically intelligible exampleof an unprovable sentence.Gödel’s second incompleteness theorem does provide an entirelyintelligible example of an unprovable sentence - specifically, the cruciallyimportant consistency statement. (Remarkably, Gödel demonstrates by a briefsemiformal argument, that the sentence he constructs for his first incompleteness