At the suggestion of the Program Coordinator I am speaking to you today on recent progress in and the current state of descriptive set theory, with an emphasis on the problems that remain open. Rather than attempting to give a cursory survey of the whole field, I have thought it would be more useful if I concentrate on certain parts of the theory that I regard as especially fundamental and in which the time seems ripe for further progress.
Descriptive set theory historically has had as its purpose the study of the properties of the continuum. It was founded in the years 1898-1906 by the three eminent French analysts Borel, Lebesgue, and Baire, and germinated in the context of the great debates associated with the discovery of the paradoxes and with the introduction of the axiom of choice. It was the period of the celebrated cinq lettres between these three and Hadamard (who also made contributions to the theory and to whom may be traced the origin of its famous uniformization problem).

Download File Size:17.88 MB

---