Global reflection principles

We consider a structural reflection principle that seeks to go beyond the traditional reflection principles of Mahlo, Levy, Bernays, et al. The latter are all firmly intra-constructible that is they produce justifications for only small large cardinals consistent with Gödel’s constructible universe L. Our Global Reflection Principles by contrast ensure that the universe of set theory has unboundedly many measurable Woodin cardinals. This is a hypothesis which (with variants) is used by current set theorists to establish many absoluteness (and consistency) results concerning the universe V. It is argued that the Principles are justified by appealing to a Cantorian Absolute perspective, together with a mereological view of its classes as parts: thus we may distinguish the mathematico-set theoretic part of the realm, the sets, from the classes. As there are no ’super-parts’ a hierarchy over and above the universe V does not threaten.