Logic Seminar

This work is motivated by the Lecomte-Zeleny result giving a level by level version of the Kechris-Solecki-Todorcevic G0-dichotomy, proved for the first three levels. We present a new approach, based on forcing, allowing to prove a general level by level weak form of this dichotomy. It also gives a new proof of the (strong) dichotomy at the level three. If time permits, another new proof of the latter result, in terms of representation of Borel sets in the sense of Debs-Saint Raymond, will be described. Our forcing method also allows to recover other results from the field, like the Matrai theorem relating descriptive complexity and Baire category, and a complexity result about filters and ideals due to Debs and Saint Raymond. This is joint work with Greenberg, Turetsky and Zeleny.