The investigation of the saturation of the nonstationary ideal \(\text_\) has a long tradition in set theory. In the early 1970's K. Kunen showed that, given a huge cardinal, there is a universe in which \(\text_\) is \(\aleph_2\)-saturated. The assumption of a huge cardinal has been improved in the following decades, using very different techniques, by many set theorists until S. Shelah around 1985 realized that already a Woodin cardinal is sufficient for the consistency of the statement “\(\text_\) is saturated”.

Due to work of H. Woodin on the one hand and G. Hjorth on the other, there is a surprising and deep connection between definable wellorders of the reals and the saturation of \(\text_\): In a universe with a measurable cardinal and \(\text_\) saturated, it is impossible to have a \(\Sigma_3\)-wellorder. This leads naturally to the question whether there is a universe in which \(\text_\) is saturated and its reals have a \(\Sigma_4\)-wellorder. In my talk I will outline a proof that this is indeed the case; assuming the existence of \(M_1\) there is a model with a \(\Sigma_4\)-definable wellorder on the reals in which \(\text_\) is saturated.

This is joint work with Sy-David Friedman.