Ralf Bader (Oxford): “From continuity to completeness”

Abstract: Continuity conditions appear to be innocuous and have not received much scrutiny. Completeness, by contrast, is a substantive and contentious condition that has been questioned on numerous occasions. Surprisingly, however, continuity implies completeness. There are two types of continuity conditions that jointly rule out incompleteness. This paper puts forward a proof that clearly exhibits how the different continuity conditions differ and how they jointly imply completeness. It then distinguishes two different models of incompleteness, namely in terms of ordering gluts and ordering gaps, and shows under what conditions which type of continuity condition can be rejected.