3 research outputs found
The first measurable can be the first inaccessible cardinal
In [7] the second and third author showed that if the least inaccessible
cardinal is the least measurable cardinal, then there is an inner model with
. In this paper we improve this to and
show that if is a -supercompact cardinal, then there is a
symmetric extension in which it is the least inaccessible and the least
measurable cardinal.Comment: 12 page
Critical Cardinals
We introduce the notion of a critical cardinal as the critical point of sufficiently strong elementary embedding between transitive sets. Assuming the axiom of choice this is equivalent to measurability, but it is well-known that choice is necessary for the equivalence. Oddly enough, this central notion was never investigated on its own before. We prove a technical criterion for lifting elementary embeddings to symmetric extensions, and we use this to show that it is consistent relative to a supercompact cardinal that there is a critical cardinal whose successor is singular