my forty years on his shoulders - Department of Mathematics

6At the outer limits, normal mathematics is conducted within completeseparable metric spaces. (Of course, we grant that it is sometimes convenient touse fluff - as long as it doesn’t cause any trouble). Functions and sets arenormally Borel measurable within such so called Polish spaces. In fact, the setsand functions normally considered in mathematics are substantially nicer thanBorel measurable, generally being continuous or at least piecewise continuous - ifnot outright countable or even finite. 1We now know that the incompleteness phenomena do penetrate thebarrier into the relatively concrete world of Borel measurability - and even intothe countable and the finite world - with independence results of a mathematicalcharacter.In sections 8-11 we discuss <strong>my</strong> efforts concerning such concreteincompleteness, establishing the necessary use of abstract set theoretic methodsin a number of contexts, some of which go well beyond the ZFC axioms.Yet it must be said that our results to date are very limited in scope, anddemand considerable improvement. We are only at the very beginnings of beingable to assess the full impact of the Gödel incompleteness phenomena.1 Apparently, nonseparable arguments are being used in the proofs of certainnumber theoretic results such as Fermat’s Last Theorem. We have beensuggesting strongly that this is an area where logicians and number theoristsshould collaborate in order to see just how necessary such appeals tononseparable arguments are. We have conjectured that they are not, and thatEFA = IΣ 0 (exp) = exponential function arithmetic suffices. See (Avigad 2003).