On strong forms of reflection in set theory

Radek Honzik

pp. 125-134

in: Carolin Antos, Radek Honzik, Claudio Ternullo, Sy D. Friedman (eds), The hyperuniverse project and maximality, Berlin, Springer, 2018

Abstract

In this paper we review the most common forms of reflection and introduce a new form which we call sharp-generated reflection. We argue that sharp-generated reflection is the strongest form of reflection which can be regarded as a natural generalization of the Lévy reflection theorem. As an application we formulate the principle sharp-maximality with the corresponding hypothesis IMH^# . IMH^# is an analogue of the IMH (Inner Model Hypothesis, introduced in Friedman (Bull Symb Log 12(4):591–600, 2006)) which is compatible with the existence of large cardinals.