Algebra Seminar talk

2017-10-13 (12:00!)
Fabiana Castiblanco (Universität Münster)
The ordinal $u_2$ and a $\bf{\Delta}^1_3$-thin equivalence relation

Under the existence of sharps for reals the second uniform indiscernible can be defined as $$u_2=\sup\{\omega_1^{+L[x]}: x\in{}^\omega\omega\}$$ Suppose that $\mathbb{P}$ is Sacks, Mathias, Silver, Miller or Laver forcing. In this talk we will see that if every real has a sharp then $u_2^V=u_2^{V^\mathbb{P}}$ and, in fact, $\mathbb{P}$ does not add any new equivalence class to the $\bf{\Delta}^1_3$-thin equivalence relation defined by $xEy\iff \omega_1^{+L[x]}=\omega_1^{+L[y]}$.