# 2013 seminar talk: Large Cardinals and Lightface Definable Wellorders without GCH

Talk held by Peter Holy (Bristol University, UK) at the KGRC seminar on 2013-01-10.

### Abstract

We investigate the possible coexistence of wellorders of $H_{\kappa^+}$ which are lightface definable over $H_{\kappa^+}$ for inaccessible cardinals $\kappa$ with a failure of the GCH at $\kappa$. Assuming SCH it is possible to obtain a lightface definable wellorder of $H_{\kappa^+}$ for every inaccessible $\kappa$ while preserving all cofinalities, the continuum function and all supercompact cardinals. This is joint work with Sy Friedman and Philipp Lücke, who jointly obtained a similar result for boldface (instead of lightface) definable wellorders.