David Schrittesser: How complex must a set be in order to be non-measurable?

Pozor, seminar bo ob nestandardnem času v petek 6. 11. 2015 ob 11h.

How complex must a set be in order to be non-measurable?

David Schrittesser, University of Copenhagen 

Abstract: We all know that you can use the axiom of choice to construct a great many exotic sets: e.g. a set which is not Lebesgue measurable, or basis for the reals considered as a vector space over the rationals (a Hamel basis).

One can reasonably ask at what level of complexity such sets appear. Since choice was involved, it would seem that we can say absolutely nothing about the "effectiveness" of their construction.

Surprisingly, you can construct a model of set theory where such sets have suprisingly constructions. For example, you can have a co-analytic Hamel basis. In other models, all definable sets are measurable, and there can be no definable Hamel basis (although the axiom of choice holds!).

This talk will be about set theory, independence and forcing; Nonetheless, I it will be very introductory and I shall not assume any familiarity whatsoever with these notions.