Kurt Gödel Research Center

2017 seminar talk: $\text{NS}_{\omega_1}$ saturated and a $\Sigma^{1}_{4}$-definable wellorder on the reals

Talk held by Stefan Hoffelner (KGRC) at the KGRC seminar on 2017-05-11.

Abstract

The investigation of the saturation of the nonstationary ideal $\text{NS}_{\omega_1}$ 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 $\text{NS}_{\omega_1}$ 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 “$\text{NS}_{\omega_1}$ 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 $\text{NS}_{\omega_1}$: In a universe with a measurable cardinal and $\text{NS}_{\omega_1}$ saturated, it is impossible to have a $\Sigma^1_3$-wellorder. This leads naturally to the question whether there is a universe in which $\text{NS}_{\omega_1}$ is saturated and its reals have a $\Sigma^1_{4}$-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 $\Sigma^1_{4}$-definable wellorder on the reals in which $\text{NS}_{\omega_1}$ is saturated.

This is joint work with Sy-David Friedman.